E-FACTORS FOR GAUSS-MANIN DETERMINANTS

ALEXANDER BEILINSON, SPENCER BLOCH, AND HÉLÈNE ESNAULT

To Yuri Manin, with gratitude.

Abstract. We define ε-factors in the de Rham setting and calculate the determinant of the Gauß-Manin connection for a family of (affine) curves and a vector bundle equipped with a flat connection.

“Ordentliche Leute pflegten ihren Schatten mit sich zu nehmen, wenn sie in die Sonne gingen.” A. v. Chamisso, from ”Peter Schlemihls wundersame Geschichte”

Table of contents

1. Introduction

2. Determinant lines of Fredholm operators 3. The Heisenberg group and its cousins

4. The ε-factors

5. Some formula (with apologies to Charlotte)

1. Introduction

1.1. Let us first recall briefly the format of the classical theory of εfactors, which grew out of Tate’s thesis and the work of Dwork, Langlands, Deligne, and Laumon.

We consider only the case of function fields, so we fix a prime p, a finite field k of characteristic p, and look at classical local k-fields, i.e., topological k-fields isomorphic to k′((t)) where k′ is a finite extension of k. For a local field F we denote by ω(F) the 1-dimensional F -vector space of differentials; set ω(F)^{×} := ω(F) ∖ {0}. Galois modules and local systems have coefficients in ℚ̄_{ℓ} for a prime ℓ ≠ p.

A classical theory of ε-factors is a rule which assigns to every pair (F, V), F is a local k-field, V a Galois module for F, a continuous function ε(F, V) = ε(V) : ω(F)^{×} → ℚ̄_{ℓ}^{×}, ν ↦ ε(V)_{ν}, which satisfies the following properties:

Date: May 10, 2002.

1991 Mathematics Subject Classification. Primary 14C40 19E20 14C99.

2 ALEXANDER BEILINSON, SPENCER BLOCH, AND HÉLÈNE ESNAULT

(i) Multiplicativity with respect to V: For a short exact sequence 0 → V_{1} → V → V_{2} → 0 one has

(1.1.1)  ε(F, V)_{ν} = ε(F, V_{1})_{ν}ε(F, V_{2})_{ν}.

Thus ε(F, V)_{ν} makes sense for virtual Galois modules.

(ii) Induction: Let F′/F be a finite separable extension, V′ a Galois module for F′, ν ∈ ω(F)^{×} ⊂ ω(F′)^{×}. Then

(1.1.2)  ε(F′, V′)_{ν} = ε(F, IndV′)_{ν}

if V′ is a virtual Galois module of rank 0.

(iii) Product formula: Let X be a smooth projective curve over k, D ⊂ X a divisor, V a local system on U := X ∖ D, ν an invertible 1-form on U. For x ∈ D let F_{x} be the local field at x, and V_{x}, ν_{x} the restrictions of V, ν to F_{x}. Then^{1}

(1.1.3)  ∏_{x∈D} ε(F_{x}, V_{x})_{ν_{x}} = ∏_{i} det(−F r_{k}, H^{i}(Uk̄, V))^{(−1)^{i+1}}.

It follows from a deep theorem of Laumon [L] 3.2 that a theory of εfactors does exist. Namely, for fixed non-trivial character ψ : k → ℚ̄_{ℓ}^{×} and p^{1/2} ∈ ℚ̄_{ℓ}^{×} the function ε(F, V)_{ν} := q_{F}^{−rkV ·v_{F} (ν)/2}ε_{ψ}(O_{F} , Rj_{∗}V, ν) is a theory of ε-factors in the above sense. Here q_{F} is the number of elements of the residue field k_{F}, v_{F} (ν) the valuation of ν, O_{F} the ring of integers in F, and ε_{ψ} is as in [L] (3.1.5.4).^{2}

Remarks. (i) Assume that our (F, V, ν) is such that V is unramified and v_{F} (ν) = 0. Then^{3} (here (−1) is the Tate twist)

(1.1.4)  ε(F, V)_{ν} = det(−F r_{k_{F}} , V (−1)).

(ii) To determine ε one invokes the Gabber-Katz theorem [GK] according to which every Galois module for k((t)) can be extended to a local system on 𝔾_{m} = Spec k[t, t^{−1}] having tame ramification at t = ∞. Therefore, by the product formula, we know ε(V) for arbitrary V if the global Frobenius determinants and ε(V) for tame V ’s happen to be known.

(iii) The r.h.s. in (1.1.3) is the (super) trace of the Frobenius acting on the determinant super line of the complex RΓ(Uk̄, V)[1].

1Here F r_{k} is the geometric Frobenius for k, k̄ is an algebraic closure of k.

2In notation of [D2] one has ε(F, V)_{ν} = ϵ(V, ψ_{ν}, μ_{ν}, 1) det(−F r_{k_{F}} , V^{I}(−1)) where ψ_{ν} : F → ℚ̄_{ℓ} is an additive character f ↦ ψTrRes(fν), μ_{ν} a Haar measure on F self-dual with respect to ψ_{ν}, and I ⊂ Gal(F̄/F) the inertia subgroup.

3To see this compare (1.1.3) for U and U with one point removed.

(iv) For a theory ε of ε-factors and a fixed a ∈ k^{×} the function ν ↦ ε(V)_{aν} is again a theory of ε-factors. So the set 𝔼(k) of theories of ε-factors carries a k^{×}-action.

1.2. Unfortunately, the above story, being global-to-local, does not tell much about the nature of ε-factors. Therefore one is tempted to look for a direct, purely local, construction of ε-factors. If available, such construction would make it possible to use the product formula to compute the global Frobenius determinant.

A related dream is to have a geometric theory of ε-factors. One considers local fields over an arbitrary (not necessary finite) base field k, and we look for a rule E that assigns to every (F, V) as above an invertible (super) local system E(V) on the k-(ind-)scheme ω(F)^{×}. The compatibilities 1.1(i)–(iii) now mean canonical isomorphisms of invertible super local systems. The classical ε is recovered as the “trace of Frobenius” function of E.^{4}

1.3. The aim of this article is to present such geometric theory in the de Rham setting. So our k is now a field of characteristic 0, and instead of étale local systems we consider local systems in the de Rham sense, that is vector bundles equipped with flat connections. Recall that there is a well-known, yet mysterious, analogy between the phenomena of wild ramification (étale setting) and irregularity (de Rham setting). The role of local Galois modules is played by de Rham local systems V = (V, ∇) on a formal punctured disc^{5} Spec F, where F is a k-algebra isomorphic to k′((t)), k′ is a finite extension of k.

So for every such V we define a super line bundle (its degree is the irregularity of ∇) with flat connection E(V) on ω(F)^{×} together with data of canonical isomorphisms parallel to 1.1 (i)–(iii). For example, the product formula looks as follows. Consider X, U, D, V as in 1.1(iv), so V now is a local system on U. Let E(F_{D}, V) be the exterior tensor product of E(F_{x}, V_{x}) for x ∈ D. Then one has a canonical isomorphism of local systems on ω(U)^{×}

(1.3.1)  E(F_{D}, V)|_{ω(U)^{×}} →^{∼} ⊗_{i}(detH_{dR}^{i}(U, V))_{ω(U)^{×}}^{⊗(−1)^{i+1}}.

4To be more precise, E(V) should rather be a twisted local system, an object of an appropriate “ε-gerbe”. A choice of trivialization of the ε-gerbe identifies then E(V) with a plain local system. One can further speculate that in the gerbe setting compatibility (1.1.2) holds for arbitrary V′, and the reason behind the rank 0 condition in (1.1.2) is the absence of a nice trivialization of the ε-gerbe. We are grateful to M. Kapranov for this suggestion.

5Notice that the Stokes structure, which is necessary to determine the analytic local structure of a connection, does not play any role here.

Here the l.h.s. is the restriction of E(F_{D}, V) to ω(U)^{×} ↪ ∏ ω(F_{x})^{×}, the r.h.s. is a constant local system.

Remark. Our ε-factors differ from de Rham counterparts of ε-factors from [L]. For example:

(i) For our de Rham ε-factors compatibility (1.1.2) holds for arbitrary V′ (regardless of its rank).^{6}

(ii) If V is a trivial local system of rank 1 on Spec F then the local system E(V) has a non-trivial ±1 monodromy on the connected components of ω(F)^{×} which consist of ν with odd order of zero.^{7}

If our datum varies nicely^{8} with respect to parameters S (this means, in particular, that irregularity does not jump) then E(F_{s}, V_{s}) form a super line bundle E(F/S, V) on ω(F/S)^{×} equipped with an S-relative connection. If the vector bundle V on the space of the family carries an absolute flat connection extending the S-relative one, then our Srelative connection on E(F/S, V) extends canonically to an absolute flat connection, i.e., E(F/S) becomes a local system. The product formula isomorphism (1.3.1) is compatible with the absolute connections (the l.h.s. carries the Gauß-Manin one). So (1.3.1) yields a formula for the Gauß-Manin determinant, once we are able to compute the ε-connections.

1.4. The ε-line E(V)_{ν} is defined as follows. Let τ_{ν} be the vector field ν^{−1}. Then ∇(τ_{ν}) is a Fredholm operator on the infinite-dimensional k-vector space V.^{9} Let V = V^{+} ⊕ V^{−} be any decomposition of the V into a sum of Fourier “positive” and “negative” parts.^{10} Consider the component of ∇(τ_{ν}) acting on V_{−}. Our E(V)_{ν} is the determinant line of this Fredholm operator. The auxiliary choice of V^{±} is irrelevant.^{11}

To see the product formula isomorphism notice that the composition Γ(U, V) → ⊕V_{x} → ⊕V_{x}^{−} is Fredholm. Now (1.3.1) is the corresponding identification of the determinant lines for τ_{ν} on Γ(U, V) and ⊕V_{x}^{−}.

6So in the de Rham setting the hoped-for ε-gerbe from ftn. 4 is canonically trivialized.

7In accordance with the fermionic nature of E.

8Every variation is nice at the generic point; for more details see 4.4.

9Its index vanishes and the determinant line is canonically trivialized, see e.g. 5.9(a)(iv).

10Precisely, for F = k((t)) this means that for certain F -basis {e_{i}} of V our V^{+} contains Σk[[t]]e_{i} as a k-subspace of finite codimension.

11The determinant line of a Fredholm operator F : U → U does not change if U is modified by a finite-dimensional vector space, or the operator is perturbed by an operator with finite-dimensional image.

The ε-connection on E(V) is defined as follows. First, the multiplicativity of determinant lines with respect to product of operators shows that the action of F^{×} on ω(F)^{×} lifts to an action on E(V) of a Heisenberg group controlled by det(V). The connection on det V identifies the corresponding Heisenberg Lie algebra with the standard one (for trivialized V). Finally, every ν ∈ ω(F)^{×} defines on F a non-degenerate scalar product which yields a “self-adjoint” splitting of the standard Heisenberg Lie algebra; this splitting specifies the horizontal directions for the ε-connection at ν. In fact, for all directions but one (namely, along the fibers of the map ν ↦ Res ν) the ε-connection comes from the action of the group of infinitesimal automorphisms of F on our picture. This provides the absolute ε-connection refered to in 1.3.

The above argument for the product formula is essentially Tate’s proof [T2] of the residue formula. Replacing the first order differential operator ∇(τ_{ν}) by a zero order one you get a proof of Weil’s reciprocity (cf. [ACK]); its infinitesimal version is the residue formula.

1.5. The idea that ε-factors have to do with fermionic determinants is due to Laumon: the introduction to [L] opens with a discussion of Witten’s picture of classical Morse theory as a WKB approximation to quantum mechanics of a super particle. This passage was considered by some as nezabudki^{12}^{,}^{13} for Laumon’s construction itself is based on the mere fact that the Fourier transform of a compactly supported distribution f on a line is smooth, and its asymptotics at ∞ are controlled by the singularities of f.

1.6. The formula for the ε-connection looks as follows. We assume that S = Spec K, K is a field. The group of isomorphism classes of line bundles with connection on S equals Ω_{K/k}^{1}/dlog(K^{×}), so our formula should determine an element in this group.

An absolute connection (V, ∇) on K((t)) is said to be admissible if it admits a lattice V_{O} = ⊕K[[t]] ⊂ V with respect to which the connection has the form (for some m) ∇ = d + A(t) = d + g(t)dt/t^{m} + η/t^{m−1}, g ∈ GL_{n}(K[[t]]), η ∈ Mat_{n}(Ω_{K}^{1}) ⊗ K[[t]]. When (V, ∇) is admissible and irregular, we find^{14} that the class of the connection on E_{t^{−m}dt} equals

(1.6.1)  Res_{t}Tr(g^{−1}dg ∧ η/t^{m−1}) − \frac{m}{2}d log det(g(0)).

12Forget-me-nots from a fable “Nezabudki i Zapyatki” of Koz’ma Proutkoff^{13} beloved by I. M. Gelfand.

13A famous writer and philosopher (1803–1863) whose project “On introducing unanimity in Russia” determined the Russian Way for XX–XXI centuries.

14See 5.6 for a more precise statement for general ν.

The corresponding formula for the global Gauß-Manin determinant was found (in case of admissible singularities and a genus 0 curve) in [BE3]; it was the starting point of the present article.

The Levelt-Turritin theorem helps to reduce (in non-effective way) computation of the ε-connection for arbitrary ∇ to the rank 1 situation (which is always admissible).

1.7. A list of natural questions:

(a) How the above picture is related to the rank 1 story of [BE2]?

(b) Can ε-factors be seen microlocally? What is the relation between the de Rham counterpart of Laumon’s Fourier approach [L] and our picture?

(c) What could be a higher dimensional generalization?

(d) Is there a de Rham version of Tate’s Thesis [T1]? More generally, what about the automorphic counterpart of the whole story?

(e) Can one treat families of connections with jumping irregularity?

1.8. The article is organized as follows. Section 2 reviews the basic formalism of determinant lines for families of Fredholm operators in the algebraic setting. The ideas here go back to [KM] and [T2]. Much of the subject is considered from the analytic point of view in the book [PS] (or in the earlier articles [DKJM], [SW]), but we were not able to find a convenient reference for the algebraic situation. Our exposition (which makes no pretense to originality) is based on localization statements 2.3(ii) and 2.12 borrowed from [Dr].^{15} We tried to make signs take care of themselves (cf. [ACK]) using the language of “super extensions”;^{16} the appendix to sect. 2 (which is a variation on theme of SGA 4 XVII 1.4) stores the required general nonsense (for a thorough discussion of super subject we refer the reader to [BDM]). In the next section we consider the Heisenberg (super) extension of the group ind-scheme k((t))^{×}. Its commutator pairing is the parametric version of the tame symbol from [C].^{17} According to [T2] or [ACK], this format yields automatically the Weil reciprocity. The Heisenberg action controls the dependence of our ε-factors on ν and is responsible for the existence of the ε-connection. The key fact here (see 3.10) is that on the k((t))^{×}-torsor of invertible forms k((t))^{×}dt every bundle equivariant with respect to the action of the Heisenberg group twisted by a local system on Spec k((t)) acquires automatically a plain flat connection.^{18} In the fourth section we define ε-factors and study their basic properties. The final section deals with explicit calculations of εconnections. We explain how E_{ν} depends on ν in the general situation. Then we treat the case of regular singularities (cf. [BE1]), the one of irregular admissible connections (cf. [BE3]), and show that the general computation can be reduced, in principle, to the rank 1 case.

15A typical object we meet is a projective R((t))-module of finite rank, and 2.12 describes it as a topological R-module étale locally on Spec R.

16Borrowed from [BD].

17The Heisenberg group approach to the construction of the Contou-Carrère symbol is a group-theoretic version of Tate’s construction of residue [T2], and it is as simple and efficient as the latter, see 3.2–3.4.

Acknowledgements: It is a pleasure to thank Takeshi Saito for discussions on the analogy of the formula as written in [BE3] with the existing formulae for the ε-factors of L-functions. We also thank Gerd Faltings for explaining to us his own version of a variation of the global method of [BE3]. We are most greatful to Vladimir Drinfeld for advice and help in understanding the subject. We want to thank the referee for a number of corrections and illuminating remarks.

The authors were partially supported by, respectively, NSF grants DMS-0100108 and DMS-0103765, and DFG-Schwerpunkt “komplexe Mannigfaltigkeiten”.

2. Determinant lines of Fredholm operators

In this section we establish the basic structure of Fredholm determinants, working insofar as possible in the category of functors on algebras. Of particular importance is the material on Clifford algebras in (2.14)–(2.18): it will be used in section 3 to define a μ_{2}-structure on a certain determinant line over ω(F)^{×} which, in turn, leads to a connection on the epsilon line.

We write “P ∈ C” for “P is an object of a category C.”

2.1. Super lines and super extensions. Let R be a commutative ring. We denote by M_{R}^{s} the tensor category of super R-modules. So an object of M_{R}^{s} is a ℤ/2-graded R-module M = M ¯0 ⊕ M ¯1. The tensor product as well as the associativity constraint, is the usual tensor product of ℤ/2-graded modules, and the commutativity constraint is a ⊗ b = (−1)^{p(a)p(b)}b ⊗ a where p(a) ∈ ℤ/2 is the degree (or parity) of a, i.e., p(a) = i for a ∈ M^{i}. See e.g. ch.1 of [BDM] for details.

A super R-line (or simply super line) is an invertible object of M_{R}^{s}. Super lines form a Picard groupoid^{19} Pic_{R}^{s}. Explicitly, a super line is a pair (L, p) where L is an invertible R-module and p (the parity of our super line) is a locally constant function Spec R → ℤ/2. In notation we usually abbreviate (L, p) to L and write p = p(L); we also write L · L′ := L ⊗ L′, L/L′ := L ⊗ L^{′−1}. The unit object (R, 0) of Pic_{R}^{s} is denoted by 1_{R}.

18Here the Clifford module picture plays the key role.

19See appendix to this section, A1.

The group of isomorphism classes of super lines π_{0}(Pic_{R}^{s}) is P ic(R)× ℤ/2_{R} where ℤ/2_{R} is the group of locally constant functions p : Spec R → ℤ/2. One has π_{1}(Pic_{R}^{s}) := Aut1_{R} = R^{×}.

The above picture is functorial with respect to morphisms of rings: every f : R → R′ yields a base change morphism of tensor categories f^{∗} : M_{R}^{s} → M_{R′}^{s}, M_{R} ↦ M_{R′} := M_{R} ⊗_{R} R′, hence a morphism of Picard

groupoids f^{∗} : Pic_{R}^{s} → Pic_{R′}^{s}.

Variants: (a) Replacing line bundles in the above definition by μ_{2}- torsors on Spec R we get the Picard groupoid μ_{2}-tors_{R}^{s} of super μ_{2}- torsors. So a super μ_{2}-torsor is the same as a super line L equipped with an isomorphism σ : L^{⊗2} →^{∼} 1_{R} (we identify (L, σ) with the μ_{2}- torsor of sections ℓ of L that satisfy σ(ℓ⊗ℓ) = 1, its parity is the parity of L).^{20}

(b) Replacing ℤ/2 in the above definition by ℤ we get the Picard groupoid Pic_{R}^{ℤ} of ℤ-graded super lines. So a ℤ-graded super line is the same as a pair (L, v) where L is a super line and v : Spec R → ℤ is a locally constant function such that p(L) = v mod 2. As above, we usually abbreviate (L, v) to L; we call v = v(L) the degree or index of the ℤ-graded super line.

We will consider Pic_{R}^{s}-extensions (see A2) of various groupoids Γ and refer to them as super O^{×}-extensions or simply super extensions. Every super extension Γ^{♭} yields a homomorphism Γ → π_{0}(Pic_{R}^{s}) (see A2, Remark (ii)), hence we have the parity homomorphism p : Γ → ℤ/2_{R}.

Notice that Pic_{R}^{s} coincides as a monoidal category^{21} with the product of Pic_{R} and the discrete groupoid ℤ/2_{R}. By A4, a Pic_{R}-extension of Γ is the same as a sheaf of central extensions of Γ by O^{×} on Spec R; we refer to it as a plain O^{×}-extension. Thus^{22} a super O^{×}-extension amounts to a plain O^{×}-extension of Γ together with a homomorphism p : Γ → ℤ/2_{R}.

If Γ is a group and we have its super extension Γ^{♭}, then every pair of commuting elements γ, γ′ ∈ Γ yields {γ, γ′}^{♭} ∈ R^{×} (see A5). Looking at the corresponding plain extension we get {γ, γ′}^{plain} ∈ R^{×}. One has

(2.1.1)  {γ, γ′}^{♭} = (−1)^{p(γ)p(γ′)}{γ, γ′}^{plain}.

20The tensor product in this format is (L, σ) ⊗ (L′, σ′) = (L ⊗ L′, σ · σ′) where (σ · σ′)((ℓ_{1} ⊗ ℓ′_{1}) ⊗ (ℓ_{2} ⊗ ℓ′_{2})) := σ(ℓ_{1} ⊗ ℓ_{2})σ′(ℓ′_{1} ⊗ ℓ′_{2}).

21i.e., we forget about the commutativity constraint.

22Use A3 Remark (ii) and A2 Remark (ii).

We leave it to the reader to define the notion of super O^{×}-extension of a group valued functor on the category of commutative R-algebras.

Below we refer to Pic_{R}^{s}-torsors (see A6) as (neutral) super O^{×}-gerbes or simply super gerbes on Spec R.^{23} As always, a trivialization of a super gerbe is its identification with Pic_{R}^{s}. We also have the notion of super pre-gerbe = pre Pic_{R}^{s}-torsor (see Remark in A6).

Replacing in Pic^{s} by Pic^{ℤ} or μ_{2}-tors^{s} we get the notions of ℤ-graded super O^{×}-extension, resp. super μ_{2}-extension. Same for ℤ-graded super O^{×}-gerbes, super μ_{2}-gerbes, etc.

2.2. Determinant lines. We denote by V_{R} the category of projective R-modules; V_{R}^{𝔣} ⊂ V_{R} is the subcategory of modules of finite rank.

For M ∈ V_{R}^{𝔣} let det M = det_{R} M ∈ Pic_{R}^{ℤ} be the top exterior power of M placed in degree rkM. This ℤ-graded super line is functorial with respect to isomorphisms of M’s, so for f : N →^{∼} M we have det f : det N →^{∼} det M or det f : 1_{R} →^{∼} det M/ det N.

The following standard compatibilities hold:

(i) For every finite family {M_{α}} there is a canonical isomorphism

(2.2.1)  det(⊕M_{α}) = ⊗detM_{α}.

Recall that it is this compatibility that forces us to consider det M as a super line.

(ii) For M equipped with a finite filtration with projective subquotients, there is a canonical isomorphism

(2.2.2)  det M = det(grM).

(iii) There is a canonical isomorphism of super lines

(2.2.3)  det(M^{∗}) = (det M)^{−1},

where M^{∗} := Hom(M, R) is the dual module, defined by a pairing e_{M} : det M · det M^{∗} → R such that for a base^{24} m_{1}, .., m_{n} of M and the dual base m_{1}^{∗}, .., m_{n}^{∗} of M^{∗} one has e((m_{1} ∧..∧m_{n})·(m_{n}^{∗} ∧..∧m_{1}^{∗})) = 1.

Notice that (2.2.3) changes the ℤ-grading to the opposite one, and one has e_{M^{∗}} = e_{M}^{m} (see A1 for the notation).

The following two useful facts about projective modules of infinite rank are due, respectively, to Kaplansky and Drinfeld:

23We skip the word neutral since the subject of this article is R-local, so we can assume that all gerbes are neutral.

24We consider our picture locally on Spec R.

2.3. Proposition. (i) If R is a local ring then every projective Rmodule is free.

(ii) Let M be a projective R-module and K ⊂ M a finitely generated submodule. Then Zariski locally on Spec R one can find a finitely generated submodule P ⊂ M which contains K such that M/P is projective.

Proof. (i) See [K].

(ii) See [Dr] 4.2; we reproduce the proof for completeness sake. Take any x ∈ Spec R; let R_{(x)} be the corresponding local ring. Then M_{(x)} := R_{(x)} ⊗ M is a free R_{(x)}-module by (i). So there is an embedding i_{(x)} : R_{(x)}^{n} ↪ M_{(x)} with projective cokernel whose image contains K_{(x)} ⊂ M_{(x)}. Replacing Spec R by an open affine neighbourhood of x we can assume that i_{(x)} comes from i : R^{n} → M whose image contains K, and then that i is an embedding with a projective cokernel (to see this represent M as a direct summand of a free module to reduce the statement first to the case when M is free, then to the case when M is free of finite rank). □

Remark. If R is a Noetherian ring such that Spec R is connected then, according to [Ba], every projective R-module of infinite rank is free. We will not use this fact.

2.4. Asymptotic morphisms and Fredholm morphisms. For projective modules M, N ∈ V_{R} let Hom_{R}^{𝔣}(N, M) ⊂ Hom_{R}(N, M) be the R-submodule of morphisms whose image lies in a finitely generated R-submodule of M. Set

(2.4.1)  Hom_{R}^{∞}(N, M) := Hom_{R}(N, M)/Hom_{R}^{𝔣}(N, M).

Elements of Hom^{∞} are asymptotic morphisms; for f : N → M the corresponding asymptotic morphism is denoted by f^{∞}. The composition of asymptotic morphisms is well-defined.^{25} So we have an R-category V_{R}^{∞} of projective R-modules and asymptotic morphisms together with the obvious functor V_{R} → V_{R}^{∞} which is the identity on objects. Notice that M ∈ V_{R} lies in V_{R}^{𝔣} if and only if id_{M}^{∞} = 0.

A morphism f is said to be Fredholm if f^{∞} is invertible. Denote by V_{R}^{𝔉𝔯} the category of projective modules and Fredholm morphisms. Let V_{R}^{×}, V_{R}^{∞×} be groupoids of invertible morphisms in V_{R}, V_{R}^{∞}. Thus V_{R}^{×} ⊂ V_{R}^{𝔉𝔯} and the morphism of groupoids V_{R}^{×} → V_{R}^{∞×} extends to the functor V_{R}^{𝔉𝔯} → V_{R}^{∞×}.

25The composition of a morphism from Hom^{𝔣} with any morphism lies in Hom^{𝔣}.

Any base change of a Fredholm morphism f is Fredholm. So for every geometric point x of Spec R we have a Fredholm operator f_{x} : N_{x} → M_{x}.^{26} Denote by i(f)_{x} = i(f_{x}) its index, i(f)_{x} := dim Cokerf_{x} − dim Kerf_{x}. The index depends only on the corresponding asymptotic morphism, so we write i(f)_{x} = i(f^{∞})_{x}.

2.5. Lemma. For a morphism f : N → M of projective modules the following conditions are equivalent:

(i) f is Fredholm,

(ii) Cokerf is finitely generated, for every geometric point x of Spec R the corresponding operator f_{x} : N_{x} → M_{x} is Fredholm, and the function i(f) : x ↦ i(f_{x}) on Spec R is locally constant.

(iii) f can be written as composition

(2.5.1)  N →^{i} N ⊕ Q →^{f̃} M ⊕ P →^{p} M

where P, Q ∈ V^{𝔣}, i is the embedding, p the projection, and f̃ is an isomorphism.

Proof. It is clear that (iii) implies (i) and (ii).

(i)⇒(ii): Since f^{∞} admits a right inverse Cokerf is finitely generated. So there exists Q ∈ V^{𝔣} and a morphism α : Q → M such that π := (f, α) : N ⊕ Q → M is surjective. Then P := Kerπ ∈ V, and (i) assures that it has finite rank. Since i(f) = rk(Q) − rk(P), it is a locally constant function.

(i)⇒(iii): Define α, Q, P as above. A splitting of π yields a projector β : N ⊕ Q → P. Then f̃:= (f, α; β) : N ⊕ Q →^{∼} M ⊕ P defines (2.5.1).

(ii)⇒(iii): We define f̃ as above. Our P is projective, so it remains to check that the conditions of (ii) imply that it is finitely generated. We know that every fiber P_{x} is of finite dimension, and this dimension is locally constant with respect to x. Take any x ∈ Spec R; we want to find a Zariski open x ∈ U ⊂ Spec R such that P |_{U} is finitely generated. By 2.3(ii) we can find U such that P |_{U} can be written as P′ ⊕ P′′ where P′ is finitely generated and P′_{x} = P_{x}, i.e., P′′_{y} = 0. Shrinking U if necessary to assure that dim P′_{y} is constant on U, we see that P′′_{y} = 0 for every y ∈ U. By 2.3(i) one has P′′ = 0. □

Remarks. (a) As follows from 2.5(ii), Fredholm morphisms have local nature with respect to the flat topology.

(b) If I ⊂ R is a nilpotent ideal then f is Fredholm if and only if f_{R/I} : N/IN → M/IM is Fredholm (use 2.5(ii)).

26Here the k_{x}-vector space M_{x} := k_{x} ⊗ M is the fiber of M at x.

(c) If R is Noetherian then the last condition in 2.5(ii) is superfluous. Indeed, the proof of (ii)⇒(iii) proceeds as above, but we just notice that P′′_{x} = 0 implies, after shrinking U if necessary, that P′′_{y} = 0 for every generic point of U. So P′′ = 0 by 2.3(i).

2.6. Relative determinant lines. Of course, there is no way to assign a determinant line to a projective module of infinite rank. However if one has two such modules M, N and an asymptotic isomorphism f^{∞} between them then one can use f^{∞} to mutually cancel infinities in det M, det N so that the ratio det M/ det N (the relative determinant line) is well-defined. Let us explain this Dostoevskian ansatz^{27}^{,}^{28} in more details.

Here is a list of data we look for:

(i) A ℤ-graded super extension V_{R}^{∞♭} of the groupoid V_{R}^{∞×}. Thus for every triple (M, N, f^{∞}), where M, N ∈ V_{R}, f^{∞} is an invertible asymptotic morphism N → M, we want to have a ℤ-graded super line det(M, N, f^{∞}) (the relative determinant line), together with data of composition isomorphisms

(2.6.1)  c : det(L, M, g^{∞}) · det(M, N, f^{∞}) →^{∼} det(L, N, g^{∞}f^{∞})

satisfying the obvious associativity property.

(ii) A splitting of the pull-back of V_{R}^{∞♭} to V_{R}^{×}. This means that for every isomorphism f : N →^{∼} M in V we have a canonical trivialization

(2.6.2)  det f : 1_{R} →^{∼} det(M, N, f^{∞})

compatible with composition of f’s (via (2.6.1)).

(iii) Compatibility with sums. For every (M_{1}, N_{1}, f_{1}^{∞}), (M_{2}, N_{2}, f_{2}^{∞}) we have a canonical isomorphism det(M_{1} ⊕ M_{2}, N_{1} ⊕ N_{2}, f_{1}^{∞} ⊕ f_{2}^{∞}) = det(M_{1}, N_{1}, f_{1}^{∞})·det(M_{2}, N_{2}, f_{2}^{∞}) compatible with the associativity and commutativity constraints. It is nicer to write it then as a canonical identification of ℤ-graded super lines

(2.6.3)  det(⊕M_{α}, ⊕N_{α}, ⊕f_{α}^{∞}) = ⊗ det(M_{α}, N_{α}, f_{α}^{∞})

defined for any finite family {(M_{α}, N_{α}, f_{α}^{∞})}. We want it to be compatible with the canonical isomorphisms from (i), (ii).

27“Let a reptile gobble up another of its kind,” a comment of a hero of “The brothers Karamazov” on his father and a younger brother.

28One may prefer to think of two sumo wrestlers, each of infinite girth, locked in a Fredholm grip, with the match determined by a mere finite inbalance of forces.

(iv) Compatibility with finite determinants. For every M, N ∈ V^{𝔣} there is a canonical identification^{29}

(2.6.4)  det(M, N) = det M/ det N

such that (2.6.1)–(2.6.3) become the usual compatibilities from 2.2.

(v) Base change. Our structure should be compatible with the base change in the obvious manner.

Notice that (iv) and (ii) imply that the degree of the super line det(M, N, f^{∞}) is the index i(f^{∞}) (see 2.4).

2.7. Proposition. Such a structure exists and is unique (up to a unique isomorphism).

Proof. Here is an explicit construction. We will not use the uniqueness statement, so its proof is left to the reader.^{30}

Let us construct the determinant line det(M, N, f^{∞}). Assume for a moment that M satisfies the following assumption, which is fulfilled, e.g., if M is a free R-module:

(∗) For every finitely generated R-submodule T ⊂ M there exists a finitely generated R-submodule P ⊂ M with M/P projective such that T ⊂ P.

Let f : N → M be any lifting of f^{∞}. Since Cokerf is finitely generated (see 2.5.(ii)), we can choose P ⊂ M as in (∗) such that P + f(N) = M. Set Q := f^{−1}(P). Then P, Q ∈ V^{𝔣}.

Lemma-definition. The ℤ-graded super lines det P/ det Q for all choices of f, P are canonically identified, so they can be considered as a single super line. This is our det(M, N, f^{∞}).

Proof of Lemma. We fix f for a moment. Let P′ be another submodule as above, Q′ := f^{−1}(P′). Let us define the canonical identification

ϕ = ϕ_{P P′} : det P/ det Q →^{∼} det P′/ det Q′.

If P′ ⊃ P then P′/P ∈ V^{𝔣} and f yields an isomorphism f̄: Q′/Q →^{∼} P′/P. Our ϕ is the image of det f̄ by the canonical identification det(P′/P)/ det(Q′/Q) = (det P′/ det Q′)/(det P/ det Q) (see (2.2.2)).

If P′ is arbitrary then we can choose P′′ as in (∗) such that P′′ ⊃ P + P′, and set ϕ_{P P′} := ϕ_{P′P′′}^{−1}ϕ_{P P′′}. One checks in a moment that this definition does not depend on the choice of P′′ and ϕ so defined satisfy the transitivity property. Thus all super lines for fixed lifting f of f^{∞} are canonically identified.

29We omitted f^{∞} in the l.h.s. of (2.6.4) for it bears no information: in the present situation one has Hom^{∞}(N, M) = Isom^{∞}(N, M) = 0.

30Hint: use 2.9.

Now let f′ be another lifting of f^{∞}. Choose P as in (∗) so that P + f(N) = M and P ⊃ (f − f′)(N). Then P + f′(N) = M and f^{−1}(P) = f^{′−1}(P). So the super lines for f and f′ computed by means of such P are simply equal. If we change P the identifications ϕ for f and f′ are also equal. These identifications of super lines are transitive with respect to f. We are done. □

In order to eliminate condition (∗) notice that super lines are local objects, so it suffices to define det locally on Spec R. Now one can use 2.3(ii) instead of (∗).

Remark. One can also reduce the general situation to the situation of (∗) directly using the following trick: Choose M′ ∈ V such that M ⊕ M′ is free (and so satisfies (∗)); then det(M, N, f^{∞}) = det(M ⊕ M′, N ⊕ N′, f^{∞} ⊕ id_{M′}^{∞}).

The canonical isomorphisms (2.6.1)–(2.6.3) are defined as follows. We use condition (∗) at will; in truth, one should work locally on Spec R and use 2.3(ii).

To define the composition isomorphism c of (2.6.1) we choose some liftings f, g and compute the determinant lines by means of finitely generated S ⊂ L such that L/S is projective and S + gf(N) = L and P := g^{−1}(S), Q := (gf)^{−1}(S) = f^{−1}(P). Our c is the standard isomorphism (det S/ det P)·(det P/ det Q) →^{∼} det S/ det Q. The independence of choices is immediate.

To define det f of (2.6.2) one computes det(M, N, f^{∞}) using f and P = 0.

To define (2.6.3) you compute the r.h.s. using some f_{α} and P_{α} and then compute the l.h.s. using f = ⊕f_{α} and ⊕P_{α}; then use (2.2.1).

2.6(iv), 2.6(v) are inherent in the construction. The mutual compatibilities of the above canonical isomorphisms are obvious. □

2.8. Remark. There is a more general construction (which is not needed in this article) that assigns a determinant line to every pair (M^{·}, d^{∞}) where M^{·} is a finitely ℤ-graded projective module and d^{∞} is a differential of degree 1 on M^{∞} such that (M^{·}, d^{∞}) is a homotopically trivial complex. The construction above corresponds to M^{·} supported in degrees 0, −1.

2.9. Compatibility with filtrations. Let M, N be projective modules equipped with finite (increasing) filtrations M_{·}, N_{·} such that the associated graded modules grM, grN are projective (i.e., the filtrations can be split). Let f^{∞} : N → M be an asymptotic morphism compatible with filtrations such that grf^{∞} : grN → grM is invertible. Then f^{∞} is also invertible.

Lemma. There is a canonical isomorphism

(2.9.1)  det(M, N, f^{∞}) = det(grM, grN, grf^{∞})

compatible with base change and such that (2.9.1) is the identity map provided that our filtered picture came from a graded one. Such data of isomorphisms is unique.

Proof. Here is a construction of (2.9.1) that makes use of general functoriality from 2.6 only.

Consider the Rees R[t]-module M^{𝔯} := ⊕M_{i}t^{i}. This is a projective R[t]-module equivariant with respect to homotheties of t whose fibers at t = 1 and t = 0 are, respectively, M and grM. Similarly, we have N^{𝔯} and an invertible asymptotic R[t]-morphism f^{∞𝔯} : N^{𝔯} → M^{𝔯}. The corresponding super R[t]-line D := det(M^{𝔯}, N^{𝔯}, f^{∞𝔯}) is equivariant with respect to homotheties of t (by base change), i.e., D is a graded R[t]- module. Its fibers at t = 1 and t = 0 are, respectively, det(M, N, f^{∞}) and det(grM, grN, grf^{∞}). The fiber at 0 has degree i = i(grf^{∞}) = i(f^{∞}) with respect to the action of homotheties. Let D^{i} be i^{th} component of D. This is a super R-line, and D = R[t] ⊗_{R} D^{i}. The restriction to t = 1, 0 identifies D^{i} with det(M, N, f^{∞}) and det(grM, grN, grf^{∞}). The composition of these identifications is (2.9.1).

The properties mentioned in the statement of Lemma, as well as the uniqueness, are immediate. □

Remarks. (i) Here is another, direct, construction of (2.9.1). Let us construct det(M, N, f^{∞}) as in the proof of 2.7 by means of some f and P ⊂ M. Choose f compatible with filtrations and P such that grP ⊂grM is such that grM/grP is projective and grP +gr(f)(grN) = grM; then use (2.2.2) to define (2.9.1). The independence of auxiliary choices is immediate, as well as properties mentioned in the statement of Lemma.

(ii) Isomorphisms (2.9.1) are compatible with (2.6.1) and (2.6.2); if M, N are of finite rank then (2.6.4) reduces them to (2.2.2).

2.10. The Tate extension. Below we will consider set-valued functors on the category of commutative R-algebras which commute with finite inverse limits. The “category” of such functors (called “R-spaces”) is denoted by S_{R}; it is closed under finite inverse limits. For an R-algebra B the corresponding representable functor is denoted by Spec B ∈ S_{R}. We have a ring object O = Spec R[t] ∈ S_{R}, O(R′) = R′, so one can consider O-modules, O-algebras in S_{R}, etc. For X ∈ S_{R} and an R-flat commutative R-algebra B, Spec B =: Y, we have an

R-space Hom(Y, X), Hom(Y, X)(R′) := X(R′ ⊗_{R} B). For B = R[h]/h^{2}

we get the tangent bundle Θ_{X} of X. Since R[h]/h^{2} is naturally an Rmodule in the category of commutative R-algebras A equipped with a morphism A → R, the tangent bundle is an O-module over X, so for any x ∈ X(R) the tangent space Θ_{X,x} (:= the fiber of Θ_{X} at x) is an O-module in S_{R}. For a group R-space G its tangent space at 1 ∈ G(R) is denoted by LieG; this is a Lie O-algebra in S_{R}.^{31}

Every projective R-module M defines an O-module in S_{R} which we denote also by M, M(R′) := M_{R′} := M ⊗_{R} R′. For M, N ∈ V_{R} we

have O-modules Hom(N, M), Hom^{∞}(N, M) in S_{R}, Hom(N, M)(R′) := Hom_{R′}(N_{R′}, M_{R′}), Hom(N, M)^{∞}(R′) := Hom_{R′}^{∞}(N_{R′}, M_{R′}). So for M ∈ V_{R} we have the corresponding O-algebras of endomorphisms and the groups of invertible endomorphisms GL(M), GL^{∞}(M) = GL(M^{∞}). There is a canonical homomorphism GL(M) → GL(M^{∞}); denote by GL^{𝔣}(M) its kernel.

By 2.6(v) the ℤ-graded super extensions Aut_{R′}^{∞♭}(M_{R′}) form the Tate super O^{×}-extension GL^{♭}(M^{∞}) of GL(M^{∞}). For g^{∞} ∈ GL(M^{∞})(R′) we denote the corresponding ℤ-graded super line in GL^{♭}(M^{∞})(R′) by det(M^{∞}, g^{∞}) or simply λ_{g^{∞}}.

Here is a list of basic properties of the Tate super extension:

(i) According to Remark (iii) in A3, the Tate extension depends functorially on M considered as an object of V_{R}^{∞×}, i.e., every asymptotic isomorphism f^{∞} : N → M yields a canonical identification of ℤ-graded super extensions

(2.10.1)  Ad_{f^{∞}}^{♭} : GL^{♭}(N^{∞}) →^{∼} GL^{♭}(M^{∞}).

To write it down explicitly, consider V_{R}^{∞♭} as a plain O^{×}-extension of V_{R}^{∞×} (see 2.1) and choose (locally on Spec R) a generator f̃ of the line det(M, N, f^{∞}). Now for g^{∞} ∈ GL(N^{∞}) the map Ad_{f^{∞}}^{♭} : λ_{g^{∞}} → λ_{Ad_{f^{∞}}(g^{∞})} is the adjoint action of f̃ (in our plain O^{×}-extension) multiplied by (−1)^{p(f^{∞})p(g^{∞})} where p is parity (i.e., the index mod 2).

In particular, the adjoint action of GL(M^{∞}) lifts canonically to an action of GL(M^{∞}) on the Tate extension GL^{♭}(M^{∞}).

31See SGA3 t.1 exp.II, pp.26-27. Here is a sketch of an argument: (a) Check that the addition law on LieG comes from the product on G. (b) To define the Lie bracket of α, β ∈ (LieG)(R) consider them as elements of, respectively, G(R[s]/s^{2}), G(R[t]/t^{2}) ⊂ G(R[s, t]/(s^{2}, t^{2}); then the commutator (α, β) := αβα^{−1}β^{−1} belongs to G(R[h]/h^{2}) ⊂ G(R[s, t]/(s^{2}, t^{2})) where h := st; this is [α, β].

(c) The Lie bracket is obviously skew-symmetric. To see Jacobi consider α, β, γ ∈ (LieG)(R) as elements of, respectively, G(R[s]/s^{2}), G(R[t]/t^{2}), G(R[u]/u^{2}) ⊂ G(R[s, t, u]/(s^{2}, t^{2}, u^{2})). Then [α, [β, γ]] corresponds to (α, (β, γ)) ∈ G(R[v]/v^{2})

where v := stu, and Jacobi follows since (α, β) commutes with (α, γ), α, β.

Remark. The groupoid V^{∞×} (as opposed to V^{×}) does not satisfy the flat descent property (see [Dr] for a discussion of this subject). One can sheafify it formally, and the above property assures that automorphism groups of objects of this stack have a canonical super O^{×}-extension.

(ii) The map GL(M) → GL(M^{∞}), g ↦ g^{∞}, lifts canonically to a homomorphism

(2.10.2) GL(M) → GL^{♭}(M^{∞}), g ↦ det(M, g) ∈ det(M^{∞}, g^{∞}).

Namely, for g ∈ GL(M) one has det(M^{∞}, g^{∞}) = det(M, M, g^{∞}), and det(M, g) is det g from (2.6.2).

Exercises. (a) Show that the restriction of (2.10.2) to the subgroup GL^{𝔣}(M) is the usual determinant det : GL^{𝔣}(M) → O^{×}.

(b) Assume that f^{∞}, g^{∞} ∈ Aut^{∞}(M) commute and f^{∞} can be lifted to f ∈ Aut(M). Then f acts by functoriality on the determinant line det(M, M, g^{∞}). Show that this action is multiplication by {f^{∞}, g^{∞}}^{♭}.^{32}

(iii) For a finite collection {M_{α}} the restriction of GL^{♭}(⊕M_{α}^{∞}) to the subgroup ΠGL(M_{α}^{∞}) ⊂ GL(⊕M_{α}^{∞}) identifies canonically with the Baer product of the Tate super extensions GL^{♭}(M_{α}^{∞}) (see 2.6(iii)). In other words, for every α the embedding GL(M_{α}^{∞}) ↪ GL(⊕M_{α}^{∞}) lifts canonically to an embedding of super extensions GL^{♭}(M_{α}^{∞}) ↪ GL^{♭}(⊕M_{α}^{∞}), and for different α’s the images of these embeddings commute.^{33}

Remark. This property is no longer true if we consider our super extensions as plain O^{×}-extensions (see 2.1).

(iv) Assume that an object M^{∞} ∈ V_{R}^{∞} has filtration which admits a splitting; let B ⊂ GL(M^{∞}) be the subgroup preserving this filtration. Then the restriction B^{♭} of GL^{♭}(M^{∞}) to B is canonically isomorphic to the pull-back of GL^{♭}(grM^{∞}) by the map B → GL(grM^{∞}) by 2.9.

Consider the Lie algebras 𝔤𝔩(M), 𝔤𝔩^{𝔣}(M), 𝔤𝔩(M^{∞}), 𝔤𝔩^{♭}(M^{∞}) of our groups GL(M), etc. The first three of them are just the corresponding algebras of endomorphisms considered as Lie algebras, so 𝔤𝔩(M^{∞}) = 𝔤𝔩(M)/𝔤𝔩^{𝔣}(M), and 𝔤𝔩^{♭}(M^{∞}) is a central extension of 𝔤𝔩(M^{∞}) by O. By (ii) the projection 𝔤𝔩(M) → 𝔤𝔩(M^{∞}) lifts canonically to a morphism of Lie algebras 𝔤𝔩(M) → 𝔤𝔩^{♭}(M^{∞}); on 𝔤𝔩^{𝔣}(M) this morphism is the trace map tr : 𝔤𝔩^{𝔣}(M) → O.^{34} Therefore one has:

(v) 𝔤𝔩^{♭}(M^{∞}) is the push-forward of the extension 0 → 𝔤𝔩^{𝔣}(M) → 𝔤𝔩(M) → 𝔤𝔩(M^{∞}) → 0 by the ad-invariant morphism tr.

32See A5 for notation.

33i.e., for γ ∈ GL^{♭}(M_{α}^{∞}), γ′ ∈ GL^{♭}(M_{α′}^{∞}), α ≠ α′, one has {γ, γ′}^{♭} = 1, see A5.

34Indeed, tr is tangent to det from Exercise (a) above.

(vi) Since 𝔤𝔩^{♭}(M^{∞}) is a central extension of 𝔤𝔩(M^{∞}) we can rewrite its bracket as a pairing [ , ]^{♭} : 𝔤𝔩(M^{∞})×𝔤𝔩(M^{∞}) → 𝔤𝔩^{♭}(M^{∞}). It satisfies the cyclic identity: for every a, b, c ∈ 𝔤𝔩(M^{∞}) one has

(2.10.3)  [ab, c]^{♭} + [ca, b]^{♭} + [bc, a]^{♭} = 0.

Here ab is the product of a, b as endomorphisms of M^{∞}.^{35}

2.11. The topological setting: Tate R-modules. An unpleasant feature of the above formalism is the absence of duality: the dual of a projective module of an infinite rank is not there. A way to recover the duality is to consider the setting of Tate modules.

If R is a commutative ring then a topological R-module is an Rmodule equipped with a topology^{36} which has a base formed by Rsubmodules; we always assume that the topology is complete and separated. Topological R-modules form an additive R-category. A morphism R → R′ defines an obvious functor from the category of topological R′-modules to that of R-modules; its left adjoint is the base change

functor F_{R} ↦ F_{R′} := F_{R} ˆ⊗_{R} R′.

We can consider every plain R-module N_{R} as a topological module with discrete topology. If M_{R} is a projective R-module then its dual M_{R}^{∗} = Hom_{R}(M_{R}, R) equipped with the weak topology (formed by annihilators of finite subsets in M_{R}) is a topological R-module. The canonical map M_{R} → (M_{R}^{∗})^{∗} := Hom^{cont}(M_{R}^{∗}, R) is an isomorphism, and the functor M_{R} ↦ M_{R}^{∗} is fully faithful. For every R → R′ one has M_{R′}^{∗} = (M_{R}^{∗})_{R′}.^{37}

Remark. For any N_{R}, M_{R} as above the image of every continuous morphism M_{R}^{∗} → N_{R} is a finitely generated R-module.^{38}

We call a projective R-module considered as discrete topological Rmodule a discrete Tate R-module. The corresponding duals are called compact Tate R-modules. A topological R-module F_{R} is special Tate R-module if it can be represented as the direct sum of a discrete and a compact Tate R-modules. Following Drinfeld [Dr], we define a Tate R-module as a topological R-module which is a direct summand of a special Tate R-module. Notice that a direct summand of a compact Tate module is a compact Tate module; same is true for discrete Tate modules. The base change sends Tate modules to Tate modules. All base changes of a given Tate R-module considered simultaneously (see 2.10) form a Tate O-module in S_{R}.

35Lift a, b, c to ã, b̃, c̃ ∈ 𝔤𝔩(M). By (v), it suffices to show that [ãb̃, c̃] + [c̃ã, b̃] + [b̃c̃, ã] = 0. This is immediate.

36The topologies we consider are always compatible with the additive structure; “base” means “base of neighbourhoods of 0.”

37To see this represent M_{R} as a direct summand of a free R-module.

38To see this notice that we can assume that M_{R} is a free R-module.

Examples. (i) R((t)) equipped with topology with base t^{n}R[[t]], n ≥ 0, is a special Tate R-module.

(ii) Let F be a finitely generated projective R((t))-module. Then F carries a canonical topology whose base is formed by R[[t]]-submodules of F which generate F as an R((t))-module. F is a Tate R-module.

(iii) Let k be a commutative ring and R ⊂ k[x] the subalgebra of polynomials f = f(x) such that f(1) = f(0). Let F_{R} ⊂ k[x]((t)) be the R-submodule of elements g = g(x, t) such that g(1, t) = tg(0, t) ∈ k((t)). It carries a topology with base F ∩ t^{n}k[x]((t)), n ≥ 0. Then F is a Tate R-module which is not special. Moreover, it is not special Zariski locally on Spec R.

The following result is due to Drinfeld [Dr] 8.1, 4.1:

2.12. Proposition. (i) Let M be a projective R-module. Then every asymptotic projector π^{∞} ∈ End_{R}^{∞}(M), (π^{∞})^{2} = π^{∞}, can be lifted to a true projector π ∈ End_{R}M étale locally on Spec R.

(ii) Every Tate R-module is special étale locally on Spec R.

Remark. The proof actually shows that the statements hold Nisnevich locally.

Proof. (i) Choose any lifting of π^{∞} to End_{R}M. Let us consider M as an R[t]-module where t acts on M by this lifting. We will find (after an appropriate localization of R) a polynomial p = p(t) ∈ R[t] such that (a) p(p − 1) kills M, and (b) p(0) = 0, p(1) = 1. Let π be the action of p on M. This is a projector by (a) which lifts π^{∞} because of (b).^{39}

Choose a finitely generated R-submodule L ⊂ M which contains t(t − 1)(M). There is a monic f(t) ∈ R[t] which kills L, so t(t − 1)f(t) kills M. Localizing Spec R with respect to étale^{40} topology we can assume that f(t) = g(t)h(t) where g(1) and h(0) are invertible, and g(t), h(t) generate the unit ideal. Then M is supported on the union of non-intersecting subschemes tg(t) = 0 and (t − 1)h(t) = 0. Our p(t) is any polynomial which vanishes on the first subscheme and equals to 1 on the second one.

39Indeed, (b) means that p(t) − t is divisible by t(t − 1), so the corresponding asymptotic endomorphism of M vanishes (since the one for t(t − 1) does).

40In fact, Nisnevich

(ii) Let F be a Tate R-module. Choose a special Tate R-module G such that F is a direct summand of G. We will find, after an appropriate localization of R, an open submodule L ⊂ G such that (a) L is a compact Tate module, and (b) P := F/F ∩L is a projective R-module. Then F is a special Tate module. Indeed, a section γ : P → F yields F →^{∼} P ⊕ F ∩ L. Now F ∩ L is a direct summand of L,^{41} hence it is a compact Tate module.

Let Π ∈ EndG be a projector such that Π(G) = F, and G = M ⊕N^{∗} be any decomposition of G into a sum of a discrete and compact Tate modules. Let ϕ be the composition M ↪ G →^{Π} G ↠ M. Then ϕ^{∞} is an asymptotic projector (see Remark in 2.11). By (i), after an appropriate localization of Spec R, one can find a projector π ∈ End_{R}M such that ϕ^{∞} = π^{∞}. The images of the composition N^{∗} ↪ G →^{Π} G ↠ M and of π − ϕ are contained in a finitly generated submodule K ⊂ M (see Remark in 2.11). By 2.3(ii) applied to the images of K in Kerπ and in Imπ one can find finitely generated R-submodules L′ ⊂ Kerπ and L′′ ⊂ Imπ such that Kerπ/L′ and Imπ/L′′ are projective R-modules. Set L := N^{∗} ⊕ L′ ⊕ L′′ ⊂ G. Since L′, L′′ are projective L satisfies (a) above; since F/F ∩ L = Imπ/L′′ it satisfies (b). We are done. □

All constructions of this article are local with respect to the flat topology of Spec R, so we will often tacitly assume that the Tate modules we deal with are special.

Remark. According to [Dr]8.3, Tate R-modules, as opposed to special Tate R-modules, satisfy the flat descent property.

2.13. Duality, lattices, and the Tate extension. Let F be a Tate R-module. We say that an R-submodule L ⊂ F is bounded if for every discrete R-module P and a continuous morphism F → P the image of L in N is contained in a finitely generated R-submodule of P. If F is realized as a direct summand in a special Tate module G = M ⊕ N^{∗}, M, N ∈ V_{R}, then L is bounded if and only if the image of L in M = G/N^{∗} is contained in a finitely generated R-submodule.^{42} Notice that bounded open submodules of F form a base of the topology of F.

For a Tate R-module F its dual F^{∗} is Hom_{R}^{cont}(F, R) equipped with a topology whose base is formed by orthogonal complements to bounded R-submodules of F. Then F^{∗} is again Tate module, and the canonical morphism F → (F^{∗})^{∗} is an isomorphism. To see this notice that duality commutes with (finite) direct sums, and for a special Tate module F = M ⊕ N^{∗}, M, N ∈ V_{R}, one has F^{∗} = N ⊕ M^{∗}. So the duality functor is an anti-equivalence of the category of Tate R-modules. It commutes with the base change.

41A projector Π : G ↠ F yields a projector L ↠ F ∩L, ℓ ↦ Π(ℓ)−γ(Π(ℓ)modL).

42This follows from Remark in 2.11.

For a Tate R-module F a c-lattice in F is an open R-submodule L ⊂ F which satisfies either of the following conditions (to check their equivalence is an exercise for the reader):

(i) L is a compact Tate module and the projection F ↠ F/L admits an R-linear section,

(ii) L is a bounded submodule of F such that F/L is a projective R-module.

Of course, a c-lattice exists if and only if F is a special Tate module. A d-lattice in F is an R-submodule M ⊂ F complementary to some

c-lattice.

Remarks. (i) c-lattices need not form a base of the topology of F. However, by 2.12(ii) and 2.3(ii), they form a base Nisnevich locally.

(ii) If L ⊃ L′ are c-lattices then L/L′ ∈ V_{R}^{𝔣}.

(iii) Every d-lattice is a projective R-module. If M ⊃ M′ are dlattices then M/M′ ∈ V_{R}^{𝔣}.

Let L, L′ be c-lattices in a special Tate module F. Every splitting F/L ↪ F yields a morphism F/L → F/L′. The corresponding asymptotic morphism (see 2.4) is an isomorphism that does not depend on the choice of splitting. Therefore the objects (F/L)^{∞} ∈ V_{R}^{∞}, L is any c-lattice in F, are canonically identified. They form a canonically defined object of the category V_{R}^{∞} which we denote by F^{∞}. Our F^{∞} depends on F in a functorial way. This construction is compatible with the base change.

For L, L′ as above set

(2.13.1)  det(L : L′) := det(F/L′, F/L, id_{F^{∞}}).

One has canonical composition isomorphisms of ℤ-graded super lines det(L : L′) · det(L′ : L′′) →^{∼} det(L : L′′), so we have defined a super pregerbe structure on the set of c-lattices. The corresponding ℤ-graded super gerbe is called the gerbe of c-lattices.

For every L ⊃ L′ there is a canonical identification ι : det(L : L′) →^{∼} det(L/L′) such that for L ⊃ L′ ⊃ L′′ the composition becomes the standard isomorphism det(L/L′) · det(L′/L′′) →^{∼} det(L/L′′).

Remark. A super pre-gerbe structure on the set of c-lattices together with identifications ι and compaitibility with base change is unique up to a unique isomorphism (use Remark (i) above).

Consider the semigroup of endomorphisms g of F such that g^{∞} is invertible. Pulling back the Tate super extension by the homomorphism g → g^{∞} to Aut(F^{∞}) one gets a super extension of our semigroup. In particular, we have a super extension Aut^{♭}(F) of Aut(F).

Notice that if L is a c-lattice such that L + g(F) = F then

(2.13.2)  λ_{g} := λ_{g^{∞}} = det(L : g^{−1}(L)).

If g ∈ Aut(F) then any L will do, and, acting by g on the r.h.s., we get a canonical isomorphism λ_{g} = det(g(L) : L).

Remark. The group Aut(F) acts on the set of c-lattices and on the datum of super lines det(L : L′), hence on the gerbe of c-lattices. The above formula means that Aut^{♭}(F) is the super extension defined by this action, see A6.

If F is an arbitrary Tate R-module then the object F^{∞} is welldefined only after certain étale (or Nisnevich) localization of Spec R (see 2.12(ii)). Since super lines have étale local nature this suffices to define the Tate super extension of the above semigroup hence of Aut(F).

This construction is compatible with base change, so we have a group object GL(F) of S_{R}, GL(F)(R′) := Aut(F_{R′}), and its Tate super extension GL^{♭}(F). Let 𝔤𝔩(F) be the Lie algebra of GL(F), so 𝔤𝔩(F)(R′) is the Lie algebra of endomorphisms of the Tate R′-module F_{R′}; the Lie algebra 𝔤𝔩^{♭}(F) of GL^{♭}(F) is a central extension of 𝔤𝔩(F) by O.

Let L, M ⊂ F be, respectively, a c- and a d-lattice in a (special) Tate module F. Denote by GL(F, L), GL(F, M) ⊂ GL(F) the parabolic subgroups of transformations preserving L, M. One has the standard splittings

(2.13.3) s_{L}^{c} : GL(F, L) → GL^{♭}(F), s_{M}^{d} : GL(F, M) → GL^{♭}(F),

defined as s_{L}^{c}(g) := det(F/L, g), s_{M}^{d}(g′) := det(M, g′), see (2.10.2). If

L ⊕ M →^{∼} F then s_{L}^{c}(g) = s_{M}^{d}(g) for g ∈ GL(F, L) ∩ GL(F, M).

The central extension 𝔤𝔩^{♭}(F) can be described as follows. Set 𝔤𝔩_{c}(F) := Ker(𝔤𝔩(F) → 𝔤𝔩(F^{∞})) and let 𝔤𝔩_{d}(F) ⊂ 𝔤𝔩(F) be the submodule of endomorphisms with open kernel. Both 𝔤𝔩_{c}(F), 𝔤𝔩_{d}(F) are ideals in 𝔤𝔩(F),^{43} and their sum equals 𝔤𝔩(F). Set 𝔤𝔩^{𝔣}(F) := 𝔤𝔩_{c}(F) ∩ 𝔤𝔩_{d}(F). There is a canonical Ad-invariant trace functional tr : 𝔤𝔩^{𝔣}(F) → O. Namely, to compute tr(a) for a ∈ 𝔤𝔩^{𝔣}(F) you find (after possible Zariski localization of R) c-lattices L ⊂ L′ such that a(L) = 0 and a(F) ⊂ L′; then tr(a) is the trace of the induced endomorphism of L′/L.

43Moreover, they are stable with respect to the adjoint action of GL(F).

One has 𝔤𝔩(F^{∞}) = 𝔤𝔩(F)/𝔤𝔩_{c}(F) = 𝔤𝔩_{d}(F)/𝔤𝔩^{𝔣}(F). As follows from 2.10(v), the central extension 𝔤𝔩^{♭}(F^{∞}) is the push-out of 0 → 𝔤𝔩^{𝔣}(F) → 𝔤𝔩_{d}(F) → 𝔤𝔩(F^{∞}) → 0 by tr. Our 𝔤𝔩^{♭}(F) is the pull-back of 𝔤𝔩^{♭}(F^{∞}) by the projection 𝔤𝔩(F) → 𝔤𝔩(F^{∞}).

Here is a c-d-symmetric description. As follows from the above construction, 𝔤𝔩^{♭}(F) admits canonical ad-invariant splittings s_{c}, s_{d} over the ideals 𝔤𝔩_{c}(F), 𝔤𝔩_{d}(F) ⊂ 𝔤𝔩(F) such that on 𝔤𝔩^{𝔣}(F) one has s_{d} − s_{c} = tr. This structure determines 𝔤𝔩^{♭}(F) uniquely.

One has a canonical bijection L ↦ L^{⊥} between the sets of c-lattices in F and its dual F^{∗}. There is a canonical isomorphism of super lines

(2.13.4)  det(L : L′) →^{∼} det(L^{⊥} : L′^{⊥})

compatible with composition, i.e., the pre-gerbes of c-lattices in F and F^{∗} are canonically identified. To define (2.13.4) it suffices, by Remark after (2.13.1), to consider the case L ⊃ L′. Here l.h.s. is det(L/L′), r.h.s. is inverse to det((L/L′)^{∗}), and (2.13.4) is (2.2.3) for M = (L/L′)^{∗}. The compatibility with composition is immediate.

The isomorphism GL(F) →^{∼} GL(F^{∗}), g ↦ ^{t}g^{−1}, is compatible with the actions on the pre-gerbes of c-lattices. So, by Remark after (2.13.2), it lifts canonically to super extensions

(2.13.5)  GL^{♭}(F) →^{∼} GL^{♭}(F^{∗}).

Notice that (2.13.5) changes the sign of the ℤ-grading to the opposite. It interchanges the ideals 𝔤𝔩_{c}, 𝔤𝔩_{d} ⊂ 𝔤𝔩 and the splittings s_{c}, s_{d}.

Remark. The above compatibility with duality can be extended to a larger semi-group of Fredholm endomorphisms as follows. We say that a morphism g : N → M of Tate R-modules is Fredholm if both g^{∞} : N^{∞} → M^{∞}, ^{t}g^{∞} : M^{∗∞} → N^{∗∞} are invertible. Equivalently, this means that (locally on Spec R) one can find c-lattices L ⊂ M, P ⊂ N such that g(N) + L = M, P ∩ Kerg = 0 and g^{−1}(L) ⊂ N, g(P) ⊂ M are again c-lattices. Notice that any pair of isomorphisms N^{∞} →^{∼} M^{∞}, M^{∗∞} →^{∼} N^{∗∞} can be lifted to a Fredholm morphism N → M.

A Fredholm g defines a ℤ-graded super line det(M, N, g). To define it choose L, P as above such that L ⊃ g(P); then det(M, N, g) := det(L/g(P))/ det(g^{−1}(L)/P). The arguments of Lemma-Definition in 2.7 show that our super line does not depend on the auxiliary choice of L, P and actually depends only on (g^{∞}, g^{∗∞}). It is compatible with composition, and there is a canonical isomorphism det(M, N, g) = det(N^{∗}, M^{∗},^{t}g)^{−1} coming from (2.2.3). If g is an isomorphism then our super line is canonically trivialized by det g ∈ det(M, N, g).

Let g be a Fredholm endomorphism of F; set det(F, g) := det(F, F, g). There is a canonical isomorphism

(2.13.6)  λ_{g} · λ_{^{t}g} = det(F, g).

Namely, for c-lattices L, P as above one has λ_{g} = det(L : g^{−1}(L)), λ_{^{t}g} = det(P^{⊥} : (^{t}g)^{−1}(P^{⊥})) = det(P : g(P)) (see (2.13.2) and (2.13.4)), and (2.13.6) is the product of these isomorphisms. If g is invertible then the r.h.s. of (2.13.6) is trivialized by det g, and (2.13.6) amounts to (2.13.5)

2.14. Clifford modules. Here is another description of the Tate extension which is inherently self-dual.

Fix a Tate module F. The Tate module F ⊕ F^{∗} carries a standard hyperbolic symmetric bilinear form. Denote by W = W_{F} the same Tate module considered as a super object placed in odd degree. The above form considered as a bilinear form on W is skew-symmetric; we denote it by < , >. So C := W ⊕ R is a Lie super R-algebra with respect to a bracket whose only non-zero component is < , >.

A Clifford module for W is a C-module^{44} Q such that the C-action is continuous with respect to the discrete topology of Q and 1 ∈ R = C^{0} acts as id_{Q}. The category of Clifford modules is denoted by M_{C}^{s}. For morphisms of R’s one has the obvious base change functors. It is clear that Clifford modules are local objects with respect to the flat topology of Spec R.

For a super R-module M and a Clifford module Q the tensor product M ⊗ Q is a Clifford module in the obvious way. A Clifford module P is invertible if the functor M_{R}^{s} → M_{C}^{s}, M ↦ M ⊗ P, is an equivalence of categories. Denote by Pic_{C}^{s} the groupoid of invertible C-modules. For L ∈ Pic_{R}^{s}, P ∈ Pic_{C}^{s} one has L ⊗ P ∈ Pic_{C}^{s}, so Pic_{C}^{s} carries a canonical action of the Picard groupoid Pic_{R}^{s}.

Assume that F is a special Tate R-module. Let L_{W} ⊂ W be a c-lattice, L_{W}^{⊥} ⊂ W its < , >-orthogonal complement; this is again a c-lattice. If L_{W} ⊂ L_{W}^{⊥} then < , > yields a non-degenerate form on W̄ := L_{W}^{⊥} /L_{W} ∈ V^{𝔣}. Let C̄ = W̄ ⊕ R be the corresponding Clifford Lie super algebra. In other words, L_{W}^{⊥} ⊕ R is the normalizer of L_{W} ⊂ C, and C̄ is the subquotient algebra of C. Let Ms ; C̄ be the category of the corresponding Clifford modules.^{45} Now the functor

(2.14.1)  M_{C}^{s} → Ms ; C̄, Q ↦ Q̄ = Q^{L_{W}} ,

44in the tensor category M_{R}^{s} of super R-modules, see 2.1.

45i.e., an object of Ms ; C̄ is a C̄-module such that 1 ∈ R ⊂ C̄ acts as identity.

is an equivalence of categories^{46} (its inverse is the induction functor). In particular, if L_{W}^{⊥} = L_{W} then C̄ = R, hence M_{C}^{s} →^{∼} M_{R}^{s}. This happens when L_{W} = L ⊕ L^{⊥} where L ⊂ F is any c-lattice and L^{⊥} := (F/L)^{∗} ⊂ F^{∗} is its orthogonal complement.

Equivalences (2.14.1) commute with functors M ⊗ ·. The groupoid Pic_{C}^{s} is a Pic_{R}^{s}-torsor, i.e., a super O^{×}-gerbe. Indeed, for every L_{W} as above such that L_{W}^{⊥} = L_{W} (2.14.1) identifies Pic_{C}^{s} with Pic_{R}^{s}.

Let O(W) ⊂ GL(W) be the subgroup of automorphisms preserving < , >. It acts on C in the obvious way, hence O(W) acts on the category of Clifford modules. Explicitly, for g ∈ O(W) and P ∈ M_{C}^{s} the Clifford module gP equals P as a super R-module, and the C-action on it is w, p ↦ g^{−1}(w)p. This action obviously commutes with functors M ⊗ ·. Therefore O(W) acts on Pic_{C}^{s} as on a super O^{×}-gerbe. Let O^{♭}(W) be the super O^{×}-extension opposite to the one defined by this action (see A6), or, equivalently, the super extension defined by the action on the opposite super gerbe. So for g ∈ O(W) its super line in O^{♭}(W) is P/gP, P ∈ Pic_{C}^{s}. Equivalently, O^{♭}(W) consists of pairs (g, g^{♭}) where g ∈ O(W) and g^{♭} is an automorphism of P, considered as a plain R-module, such that for w ∈ W, p ∈ P one has g^{♭}(wp) = g(w)g^{♭}(p).

If F is an arbitrary (not necessary special) Tate module then we define O^{♭}(W) using 2.12(ii).^{47}

Remarks. (i) For L_{W} as above set O(W, L_{W}) := O(W)∩GL(W, L_{W}); there is an obvious projection O(W, L_{W}) → O(W̄). Then (2.14.1) identifies the restriction of O^{♭}(W) to O(W, L_{W}) with the pull-back of the extension O^{♭}(W̄). In particular, if L_{W} = L_{W}^{⊥} then there is a canonical splitting s_{L_{W}}^{c} : O(W, L_{W}) → O^{♭}(W).

(ii) The above discussion for c-lattices has an immediate d-version. Namely, let M_{W} ⊂ W be a d-lattice (see 2.13) such that < , > vanishes on M_{W}. Then M_{W}^{⊥} ⊂ W is a d-lattice and W̄ := M_{W}^{⊥} /M_{W} ∈ V^{𝔣} carries the induced non-degenerate form; let C̄ be the corresponding Clifford Lie super algebra. Thus M_{W}^{⊥} ⊕ R is the normalizer of M_{W} ⊂ C, and C̄ is the subquotient Lie algebra. We have an equivalence of categories

(2.14.2)  M_{C}^{s} →^{∼} Ms ; C̄, Q ↦ Q_{M_{W}} .

It identifies the restriction of O^{♭}(W) to a parabolic subgroup O(W, M_{W}) := O(W) ∩ GL(W, M_{W}) with the pull-back of the extension O^{♭}(W̄) by the projection O(W, M_{W}) → O(W̄). In particular, if M_{W} = M_{W}^{⊥} then there is a canonical splitting s_{M_{W}}^{d} : O(W, M_{W}) → O^{♭}(W).

46If F is a finitely generated projective R-module then (2.14.1) is the usual Morita equivalence. The general case reduces to this one by 2.3(ii) (see Remark (i)

in 2.13) since Clifford modules are Zariski local objects.

47As we did to define GL^{♭}(F).

2.15. Proposition. The embedding GL(F) ↪ O(W) which assigns to g ∈ GL(F) the diagonal matrix g^{o} with components g,^{t}g^{−1} lifts canonically to a morphism of super extensions

(2.15.1)  GL^{♭}(F) ↪ O^{♭}(W).

Proof. By Remark after (2.13.2), GL^{♭}F) is the super extension defined by the action of GL(F) on the gerbe of c-lattices. The super extension induced from O^{♭}(W) is defined by the action of GL(F) on the gerbe opposite to Pic_{C}^{s}. It remains to identify these two gerbes in a way compatible with the GL(F)-actions.

By Remark in A6 we have to assign to every c-lattice L an invertible Clifford module Q_{L} and define identifications det(L : L′) = Q_{L′}/Q_{L} which satisfy the transitivity property. By Remark after (2.13.1) it suffices to establish the latter identifications for L ⊃ L′.

We define Q_{L} as the Clifford module such that (Q_{L})^{L_{W}} = R where L_{W} := L⊕L^{⊥} (see (2.14.1)). Then Q_{L′}/Q_{L} = (Q_{L′})^{L_{W}}, so the promised identification can be rewritten as a canonical isomorphism

(2.15.2)  det(L : L′) = Q^{L_{W}} /Q^{L′_{W}}

valid for any Q ∈ Pic_{C}^{s}. If L ⊃ L′ then the super lines (Q_{L′})^{L_{W}} , Q_{L′}^{L′_{W}} lie in Q_{L′}^{P}, where P := L′ ⊕ L^{⊥} ⊂ W, which is a module for the Clifford algebra C̄ for W̄ = P^{⊥}/P = L/L′ ⊕ L′^{⊥}/L^{⊥}. Now det(L : L′) = det(L/L′) lies in C̄, and its action transforms Q^{L′_{W}} to Q^{L_{W}}. This is (2.15.2). The transitivity property and compatibility with the GL(F)- action are obvious. □

Remarks. (i) We see that GL^{♭}(F) acts canonically on every Clifford module P. The action of 𝔤𝔩^{♭}(F) can be described explicitly as follows. For a^{♭} ∈ 𝔤𝔩^{♭}(F) over a ∈ 𝔤𝔩(F) its action on P is compatible with the action of W ⊂ C: one has a^{♭}(wp) − w(a^{♭}p) = a(w)p for w ∈ W, p ∈ P. This condition alone does not determine the action of a^{♭}; an extra normalization is needed. Since 𝔤𝔩(F) = 𝔤𝔩_{c}(F) + 𝔤𝔩_{d}(F)^{48} it suffices to consider the cases a^{♭} = s_{c}(a), a ∈ 𝔤𝔩_{c}(F), and a^{♭} = s_{d}(a), a ∈ 𝔤𝔩_{d}(F). The action of s_{c}(a) is uniquely determined by property that it kills P^{L} for a sufficiently large c-lattice L ⊂ F; the action of s_{d}(a) is uniquely determined by property that it kills P^{L^{⊥}} for a sufficiently small c-lattice L ⊂ F.

48See the discussion at the end of 2.13.

(ii) Let L, M ⊂ F be a c- and d-lattice. Set L_{W} := L ⊕ L^{⊥}, M_{W} := M ⊕ M^{⊥} ⊂ W; these are c- and d-lattices in W such that L_{W}^{⊥} = L_{W}, M_{W}^{⊥} = M_{W}, and (2.15.1) identifies sections s_{L}^{c}, s_{M}^{d} of (2.13.3) with restrictions of sections s_{L_{W}}^{c}, s_{M_{W}}^{d} from Remarks (i), (ii) of 2.14.

(iii) Recall that GL^{♭}(F) is a ℤ-graded super extension. This ℤgrading can be described in Clifford terms as follows. Consider a ℤgrading on W with F in degree 1 and F^{∗} in degree −1; then GL(F) ⊂ O(W) is the group of orthogonal automorphisms preserving this ℤgrading. One defines ℤ-graded Clifford modules in the obvious way. The picture of 2.14 remains valid in the ℤ-graded setting, so we have a ℤ-graded super O^{×}-gerbe Pic_{C}^{ℤ} equipped with an action of GL(F) and the corresponding ℤ-graded super extension of GL(F). This defines a canonical ℤ-grading on the pull-back of O^{♭}(W) to GL(F) ⊂ O(W). It is easy to see that (2.15.1) is compatible with the ℤ-gradings.

(iv) The Clifford algebras for F and F^{∗} coincide, hence the corresponding groups O(W) and their super extensions O^{♭}(W) are equal. On the subgroups GL(F), GL(F^{∗}) this identification is the standard isomorphism g ↦ ^{t}g^{−1}; its lifting to the Tate super extensions via (2.15.1) is (2.13.5).

2.16. Scalar products and super μ_{2}-torsors. We suggest reading this subsection simultaneously with 3.6.

Let Φ = Φ(F) ∈ S_{R} be the object of all symmetric non-degenerate bilinear forms on F, or, equivalently, that of symmetric isomorphisms ϕ : F →^{∼} F^{∗} (the form corresponding to ϕ is (a, b)_{ϕ} := (ϕa, b)).

For ϕ ∈ Φ let ϕ^{o} ∈ O(W) be the anti-diagonal matrix with components ϕ, ϕ^{−1}. Let λ_{ϕ} be the super line of ϕ^{o} in O^{♭}(W). One has ϕ^{o2} = 1, so the corresponding identification λ_{ϕ}^{⊗2} →^{∼} O_{ϕ} yields a super μ_{2}-torsor μ_{ϕ} (see 2.1). We have defined a canonical super μ_{2}-torsor μ_{Φ} on Φ.

For ϕ, ϕ′ ∈ Φ(F) set g_{ϕ′,ϕ} := ϕ′^{−1}ϕ ∈ GL(F). One has ϕ′^{o} = g_{ϕ′,ϕ}^{o}ϕ^{o} ∈ O(W). By (2.15.1) this provides a canonical isomorphism of super lines

(2.16.1)  λ_{g_{ϕ′,ϕ}} · λ_{ϕ} →^{∼} λ_{ϕ′}.

Notice that g_{ϕ′′,ϕ} = g_{ϕ′′,ϕ′}g_{ϕ′,ϕ} and identifications (2.16.1) are transitive.

One can rephrase this as follows. Let G_{Φ} be the simply transitive groupoid on Φ, so for every ϕ, ϕ′ ∈ Φ there is a single arrow ϕ → ϕ′ in G_{Φ}. We have a homomorphism of groupoids g : G_{Φ} → GL(F) which sends the above arrow to g_{ϕ′,ϕ}. Let G_{Φ}^{♭} be the g-pull-back of the Tate super extension. Now (2.16.1) is a G_{Φ}^{♭}-action on λ_{Φ}, i.e., a λ_{Φ}-splitting of G_{Φ}^{♭} in the terminology of A2.

Therefore (see A2) G_{Φ}^{♭} comes from a μ_{Φ}-split super μ_{2}-extension G_{Φ}^{μ} of G_{Φ}. If 1/2 ∈ R then any μ_{2}-extension is étale, hence it is canonically trivialized over the formal completion Ĝ_{Φ} of G_{Φ}.^{49} The corresponding trivialization of G_{Φ}^{♭} over Ĝ_{Φ} is a canonical formal rigidification of G_{Φ}^{♭}.

Remarks. (i) The above constructions are compatible with direct sums: if F = ⊕F_{α}, ϕ = ⊕ϕ_{α}, then μ_{ϕ} = ⊗μ_{ϕ_{α}}, and duality: μ_{ϕ^{−1}} = μ_{ϕ}.

(ii) Fix any ϕ ∈ Φ(F). Then the morphism g_{ϕ} : Φ(F) → GL(F), ϕ′ ↦ g_{ϕ′,ϕ}, is injective, and its image consists of all g ∈ GL(F) which are self-adjoint with respect to ( , )_{ϕ}. The involution 𝔰_{ϕ} :=Ad_{ϕ^{o}} preserves GL(F), Φ ⊂ O(W), so it induces involutions of GL(F) and Φ which we denote again by 𝔰_{ϕ}. For g ∈ GL(F), ϕ′ ∈ Φ one has 𝔰_{ϕ}(g) = ϕ^{−1}(^{t}g)^{−1}ϕ (which is the adjunction with respect to ( , )) and 𝔰_{ϕ}(ϕ′) = ϕϕ^{′−1}ϕ. Thus g_{ϕ}(𝔰_{ϕ}(ϕ′)) = (g_{ϕ}(ϕ′))^{−1}. The action of 𝔰_{ϕ} on λ_{Φ} is compatible with the trivialization λ_{Φ}^{⊗2} →^{∼} O_{ϕ}, so 𝔰_{ϕ} acts on μ_{Φ}.

(iii) The action of 𝔰_{ϕ} on 𝔤𝔩(F) interchanges the ideals 𝔤𝔩_{c}(F), 𝔤𝔩_{d}(F) ⊂ 𝔤𝔩(F) and the sections s_{c}, s_{d} (see (2.13.5) and Remark (iv) in 2.15). Our g_{ϕ} identifies the tangent space to ϕ ∈ Φ with the submodule 𝔤𝔩(F)^{ϕ} ⊂ 𝔤𝔩(F) of ( , )_{ϕ}-self-adjoint operators. If 1/2 ∈ R then the canonical formal rigidification of G_{ϕ}^{♭} yields a splitting ∇_{ϕ}^{♭} : 𝔤𝔩(F)^{ϕ} → 𝔤𝔩^{♭}(F) which is 𝔰_{ϕ}-invariant; it is uniquely defined by this property.

(iv) Let L, L_{W} , M, M_{W} be as in Remark (ii) of 2.15. Set

(2.16.2) Φ_{L} := {ϕ ∈ Φ : ϕ(L) = L^{⊥}}, Φ_{M} := {ϕ ∈ Φ : ϕ(M) = M^{⊥}}.

Since Φ_{L}^{o} ⊂ O(F, L_{W}), Φ_{M}^{o} ⊂ O(W, M_{W}) the sections s_{L_{W}}^{c}, s_{M_{W}}^{d} from Remarks (i), (ii) of 2.14 trivialize the restrictions of our μ_{2}-torsor μ_{Φ} to Φ_{L_{W}} and Φ_{M_{W}}. We denote these trivializations also by s_{L}^{c}, s_{M}^{d}. Notice that for ϕ, ϕ′ ∈ Φ_{L} the action (2.16.1) identifies s_{L}^{c}(g_{ϕ′,ϕ})s_{L}^{c}(ϕ) with s_{L}^{c}(ϕ′) (see Remark (ii) of 2.15). Same for L replaced by M.

2.17. Let us describe μ_{ϕ} explicitly assuming that R is a field.

(i) Case p(μ_{ϕ}) = 0.^{50} Then one can find a c-lattice L such that the ϕ-orthogonal complement L_{ϕ}^{⊥} equals L, i.e., ϕ ∈ Φ_{L}. As above, it yields the trivialization s_{L}^{c}(ϕ) of μ_{ϕ}. If F ≠ 0 and we live in characteristic ≠ 2 then the Grassmannian of such L’s has 2 components. We leave it to the reader to check that our trivialization changes sign as we switch the component. Thus μ_{ϕ} is the set of components considered as a μ_{2}-torsor.

(ii) Case p(μ_{ϕ}) = 1. Then one can find a c-lattice L such that L_{ϕ}^{⊥} ⊃ L and L̄ := L_{ϕ}^{⊥}/L has dimension 1. We have the induced ϕ̄ : L̄ →^{∼} L̄^{∗}.

49For ϕ, ϕ′ ∈ Φ(R′) the arrow ϕ → ϕ′ lies in Ĝ_{Φ} ⊂ G_{Φ} if and only ϕ equals ϕ′ modulo some nilpotent ideal I ⊂ R′.

50Here p is the parity of our super torsor, see 2.1.

Let us define a canonical identification

(2.17.1)  μ_{ϕ} = {l̄∈ L̄ : (l̄, l̄)ϕ̄ = 1}.

Indeed, ϕ^{o} preserves L_{W} := L ⊕ ϕ(L) and W̄ := L_{W}^{⊥} /L_{W} = L̄ ⊕ L̄^{∗}, so, by Remark (i) in 2.14, we have μ_{ϕ} = μϕ̄. Take any invertible Clifford module P̄ for W̄. An element σ ∈ μϕ̄ is an automorphism of P̄ as a k-vector space such that σ^{2} = 1 and σ(wp) = ϕ̄^{o}(w)σ(p) for w ∈ W̄, p ∈ P. The restriction of such σ to P̄L̄^{∗} ⊂ P̄ is multiplication by l̄∈ L̄ such that (l̄, l̄)ϕ̄ = 1. This provides (2.17.1).

Appendix

A1.^{51} A Picard groupoid P is a symmetric monoidal category^{52} such that every object in P is invertible, as well as every morphism. We denote the operation in P by · and the unit object by 1_{P}. The commutative group of isomorphism classes of objects in P is denoted by π_{0}(P). Set π_{1}(P) := Aut(1_{P}); this is a commutative group, and for every P ∈ P the identification 1 · P = P yields a canonical isomorphism

π_{1}(P) →^{∼} AutP, a ↦ a · id_{P}.

For P ∈ P the corresponding element of π_{0}(P) is denoted by |P |. For a finite family {F_{α}} of objects of P we usually denote their product by ⊗F_{α}.

Remark. For any commutative group A the category A-tors of A-torsors is a Picard groupoid in the obvious way. For every Picard groupoid P its Picard subgroupoid P^{0} of objects isomorphic to 1_{P} is canonically equivalent to A_{P}-tors via P^{0} →^{∼} π_{1}(P)-tors, P ↦ Hom(1_{P}, P).

For two Picard groupoids P, P′ a morphism ϕ : P → P′ is the same as a symmetric monoidal functor. All morphisms form a Picard groupoid Hom(P, P′): namely, the product of two morphisms ϕ, ψ is (ϕ · ψ)(P) = ϕ(P) · ψ(P).

For P ∈ P its inverse P^{−1} is an object of P together with an identification e : P · P^{−1} →^{∼} 1_{P}; the inverse is determined uniquely up to a unique isomorphism. There are two natural identifications P →^{∼} (P^{−1})^{−1} defined by pairings e^{c}, e^{m} : P^{−1} · P →^{∼} 1_{P} where e^{c} is defined using commutativity, and e^{m} is determined by the condition that e · id_{P} = id_{P} · e^{m} : P · P^{−1} · P →^{∼} P (or, equivalently, that

e^{m} · id_{P^{−1}} = id_{P^{−1}} · e : P^{−1} · P · P^{−1} →^{∼} P^{−1}). Notice that the def-

inition of e^{m} does not use commutativity of ·, and e^{m}/e^{c} = α where α = α(P) ∈ π_{1}(P) is the action of the commutativity symmetry on P · P. Unless stated explicitly otherwise, we use identification e^{c}.

51See SGA 4 XVIII 1.4.

52We always assume that P is equivalent to a small category.

Using commutativity, one extends naturally the map P ↦ P^{−1} to an auto-equivalence of the Picard groupoid P.

For P, Q ∈ P we write P/Q := P · Q^{−1}.

A2. For a category Γ, a P-extension Γ^{♭} is a rule that assigns to every arrow γ in Γ an object P_{γ}^{♭} ∈ P and to every pair of composable arrows γ, γ′ a composition isomorphism c_{γ,γ′} : P_{γ}^{♭} · P_{γ′}^{♭} → P_{γγ′}^{♭}. We demand that c be associative in the obvious sense.

Assume we have a P-bundle L on the set of objects of Γ which is a rule which assigns to every object x in Γ an object L_{x} of P. An action of Γ^{♭} on L is a rule that assigns to every arrow γ : x → x′ an isomorphism P_{γ}^{♭} · L_{x} →^{∼} L_{x′} which satisfies an obvious transitivity property.

Notice that for a given L there is a unique (up to a unique isomorphism) P-extension Γ_{L}^{♭} of Γ acting on L. Indeed, the action amounts to a datum of isomorphisms P_{Lγ}^{♭} := ξ_{x′}/ξ_{x} which identifies the composition in Γ^{♭} with the obvious product. We call Γ_{L}^{♭} the L-split P-extension. So for an arbitrary P-extension Γ^{♭} of Γ we also refer to its action on L as an L-splitting of Γ^{♭}.

Remarks. (i) For every object x in Γ there is a canonical identification

e_{x} : P_{id_{x}}^{♭} →^{∼} 1_{P} defined by c_{id_{x},id_{x}} : P_{id_{x}}^{♭} · P_{id_{x}}^{♭} →^{∼} P_{id_{x}}^{♭}. This isomorphism

identifies every c_{id_{x},γ′} and c_{γ,id_{x}} with the canonical isomorphisms 1_{P} · P_{γ′}^{♭} →^{∼} P_{γ′}^{♭}, P_{γ}^{♭} · 1_{P} →^{∼} P_{γ}^{♭}. For every invertible arrow γ : x → x′ in Γ there is a canonical identification P_{γ^{−1}}^{♭} →^{∼} (P_{γ}^{♭})^{−1} defined by the

isomorphism e_{γ} := e_{x′}c_{γ,γ^{−1}} : P_{γ}^{♭} · P_{γ^{−1}}^{♭} →^{∼} 1_{P}. Notice that e_{γ^{−1}} = e_{γ}^{m}.

(ii) A P-extension Γ^{♭} yields a homomorphism^{53} Γ → π_{0}(P), γ ↦ |P_{γ}^{♭}|. If P is discrete, i.e., A_{P} = 1, then P-extensions are the same as homomorphisms Γ → π_{0}(P).

A3. Let Γ^{♭}, Γ^{♭′} be two P-extensions of Γ. A morphism θ : Γ^{♭} → Γ^{♭′} is a system of morphisms θ_{γ} : P_{γ}^{♭} → P_{γ}^{♭′} compatible with composition; it is clear how to compose morphisms of P-extensions. The Baer product of Γ^{♭}, Γ^{♭′} is a P-extension γ ↦ P_{γ}^{♭} · P_{γ}^{♭′} with the obvious composition rule; the Baer product is associative and commutative in the obvious way. Therefore for small Γ its P-extensions form a Picard groupoid which we denote by Ext(Γ, P). Its unit object 1_{E} is the trivial split extension γ ↦ 1_{P}. One has π_{1}(Ext(Γ, P)) = Hom(Γ, π_{1}(P)).

53For a group A a homomorphism a : Γ → A assigns to every arrow γ an element a(γ) ∈ A so that a(γγ′) = a(γ)a(γ′) for every composable γ, γ′.

For a functor ϕ : Γ′ → Γ and a P-extension Γ^{♭} it is clear what is the pull-back of Γ^{♭} by ϕ. Thus for small Γ, Γ′ we have a morphism of Picard groupoids ϕ^{∗} : Ext(Γ, P) → Ext(Γ′, P).

Remarks. (i) The notion of P-extension of Γ depends on the isomorphism class of Γ, and not on its equivalence class: if ϕ : Γ → Γ′ is an equivalence then ϕ^{∗} need not be an equivalence.^{54}

(ii) The notion of P-extension depends only on monoidal structure of P: the commutativity constraint is irrelevant. However, the commutativity constraint is used in the definition of the Baer product.

(iii) Let Γ^{♭} be a P-extension of a groupoid Γ. Then for x ∈ Γ we have a P-extension Aut^{♭}(x) of the group Aut(x). It depends functorially on x: for every γ : x → x′ the identification Ad_{γ} : Aut(x) →^{∼} Aut(x′) lifts canonically to an isomorphism of P-extensions Ad_{γ}^{♭} : Aut^{♭}(x) →^{∼} Aut^{♭}(x′). Namely, for g ∈ Aut(x) the corresponding identification Ad_{γ}^{♭} : P_{g}^{♭} →^{∼} P_{Ad_{γ}(g)}^{♭} is the composition

P_{g}^{♭} →^{∼} P_{g}^{♭} · (P_{γ}^{♭} · P_{γ^{−1}}^{♭}) →^{∼} P_{γ}^{♭} · P_{g}^{♭} · P_{γ^{−1}}^{♭} →^{∼} P_{Ad_{γ}(g)}^{♭}.

Here the first arrow is id_{P_{g}^{♭}} ·(e_{γ})^{−1}, the second one is the commutativity constraint, the third is the composition map.^{55} For composable γ, γ′ one has Ad_{γγ′}^{♭} = Ad_{γ}^{♭}Ad_{γ′}^{♭}.

A4. Let A be a commutative group. A central A-extension of Γ is a category Γ̃ together with a functor π : Γ̃ → Γ and a homomorphism from A to the automorphisms of the identity functor of Γ̃. We demand that π is bijective on objects, for every x, x′ the projection π : HomΓ̃(x, x′) → Hom_{Γ}(x, x′) is surjective, and the action of A a(γ̃) = a_{x′} γ̃ = γ̃a_{x} is simply transitive along the fibers of this projection. If Γ is a group (i.e., a groupoid with single object), then its central A-extensions coincide with central extensions of Γ by A.

Any Γ̃ as above yields an A-tors-extension Γ^{♭} of Γ: namely, P_{γ}^{♭} is the A-torsor π^{−1}(γ), the composition c is the composition of arrows in Γ̃.

This way we see that A-tors-extensions are the same as central Aextensions.

A5. Suppose that Γ is a group; set A := π_{1}(P). As we have seen in Remark (iii) of A3, the adjoint action of Γ on itself lifts canonically to a Γ-action Ad^{♭} on any P-extension Γ^{♭}. If γ, γ′ ∈ Γ commute then Ad_{γ}^{♭} : P_{γ′}^{♭} →^{∼} P_{γ′}^{♭} is multiplication by an element of A which we denote by {γ, γ′}^{♭}. Equivalently, {γ, γ′}^{♭} ∈ A is the composition

54Consider the case when Γ is a trivial groupoid and Γ′ is an equivalent groupoid with 2 objects.

55The compatibility of Ad_{γ}^{♭} is with product (composition) follows from the fact that the composition map P_{γ^{−1}}^{♭} · P_{γ}^{♭} → 1_{P} is equal to e_{γ}^{m}, see Remark (i) in A2.

P_{γ}^{♭} · P_{γ′}^{♭} →^{c_{γ,γ′}} P_{γγ′}^{♭} = P_{γ′γ}^{♭} →^{c_{γ′,γ}^{−1}} P_{γ′}^{♭} · P_{γ}^{♭} →^{∼} P_{γ}^{♭} · P_{γ′}^{♭}

where the last arrow is the commutativity constraint. If Γ is commutative then { , }^{♭} : Γ × Γ → A is a bimultiplicative skew-symmetric pairing. If P = A-tors then { , }^{♭} is the usual commutator pairing for the corresponding central A-extension.

A6. A P-action on a category C is a functor · : P × C → C equipped with an associativity constraint P ·(P′·C) →^{∼} (P ·P′)·C and 1_{P} ·C →^{∼} C which satisfies the obvious compatibilities. We say that C is a P-torsor if C is non-empty and for every C ∈ C the functor P → C, P ↦ P · C, is an equivalence of categories. For a P-torsor C one has a canonical functor C×C → P, C, C′ ↦ C/C′, together with natural identifications (C/C′) · C′ = C; such functor is unique.

Remark. Let C be a P-torsor. We have then a P-extension of the simply transitive groupoid on its set of objects: for C′, C ∈ C the object P_{C,C′}^{♭} ∈ P corresponding to the (unique) arrow C′ → C is C/C′, the composition is obvious. Conversely, suppose we have a set K together with a P-extension P_{K}^{♭} of the simply transitive groupoid on K.^{56} Then there is a P-torsor C together with a map from K to the objects of C and its lifting to a morphism of P-extensions. Such datum is unique in the obvious sense. We call C the P-torsor generated by pre P-torsor (K, P_{K}^{♭}).

Let Γ be a group. Assume that it acts on a category C, i.e., we have a rule that assigns to every γ ∈ Γ a functor γ : C → C, C ↦ γC, together with natural isomorphisms γ(γ′C) = (γγ′)C, 1C = C; this datum should satisfy the usual compatibilities. If C is also equipped with a P-action then we say that the Γ and P-actions commute if we are given a system of natural isomorphisms γ(P · C) = P · (γC) that satisfy the obvious compatibilities.

For example, assume that P acts on C and we have a P-extension Γ^{♭} of Γ. Then γC := P_{γ}^{♭} · C is a Γ-action on C which commutes with the P-action in the obvious way.

If C is a P-torsor then every Γ-action on C commuting with the P-action (we say then that Γ acts on C as on a P-torsor) arises as above. Precisely, there is a unique, up to a unique isomorphism, Γ^{♭} together with an identification of the corresponding action on C with our action which is compatible with the constraints. Namely, one has P_{γ}^{♭} = (γC)/C (for various C ∈ C these objects are canonically identified in the obvious way) and the composition law is P_{γ}^{♭} ·P_{γ′}^{♭} = (γ′(γC)/γC)· (γC/C) = γ′γC/C = P_{γγ′}^{♭}. We call Γ^{♭} the P-extension defined by the action of Γ on C.

56Which is a rule that assigns to k, k′ ∈ K an object P_{k,k′}^{♭} ∈ P and composition maps P_{k,k′}^{♭} · P_{k′,k′′}^{♭} → P_{k,k′′}^{♭} which satisfy the associativity property.

For example, suppose we have a pre P-torsor (K, P_{K}^{♭}) as in Remark above equipped with a Γ-action (so Γ acts on the simple transitive groupoid on K,^{57} and this action is lifted to P_{K}^{♭}). Then Γ acts on the P-torsor C generated by (K, P_{K}^{♭}), which yields a P-extension Γ^{♭}.

3. The Heisenberg group and its cousins

The purpose of this section is to study various Heisenberg central extensions which arise by pullback from the Tate central extension in 2.10. In 3.5, the superline λ_{ω(F)^{×}} on ω(F)^{×} is constructed and the structure group is reduced to μ_{2}. In particular, this line has a canonical connection. For V a projective F -module of rank n, a connection on det_{F} V leads in 3.8 to an identification of formal groupschemes F̂^{×n♭} ≅ F̂_{(V)}^{×♭}. For E any superline bundle on ω(F)^{×} with an action of F̂_{(V)}^{×♭}, this isomorphism enables us to transfer the connection from λ_{ω(F)^{×}} to E.

3.1. The setting. Let R be a commutative ring and F_{R} a commutative topological R-algebra which is a Tate R-module (see 2.11). Let F_{R}^{×} be the multiplicative group of F_{R}. By base change we have a group valued functor F^{×} ∈ S_{R}, F^{×}(R′) := F_{R′}^{×}. The Lie algebra of F^{×} is F: its R′- points is the additive group of F_{R′}.

Similarly, let Aut(F_{R}) be the group of continuous R-automorphisms of the algebra F; we have a group functor Aut(F) ∈ S_{R}, Aut(F)(R′) := Aut(F_{R′}), which acts on F^{×}. The R-points of the Lie algebra of Aut(F) is the Lie algebra Θ(F_{R}) of continuous R-derivations of F_{R}.

Let F_{R}^{τ} be our F_{R} considered as a Tate R-module, so we have a group functor GL(F^{τ}), R′ ↦ GL(F_{R′}^{τ}) (see 2.13). One has the obvious embeddings

(3.1.1)  F^{×}, Aut(F) ↪ GL(F^{τ}).

They identify the action of Aut(F) on F^{×} with the adjoint action of GL(F^{τ}).

The pull-back of the Tate super extension of GL(F^{τ}) from 2.13 to F^{×} is the Heisenberg super extension F^{×♭} of F^{×}. Its Lie algebra is a central extension F^{♭} of F by O. One deduces an explicit description of F^{♭} from 2.10(iv). Notice that the action of Aut(F) on F^{×} lifts canonically to the (adjoint) action on F^{×♭}.

57In all examples we consider Γ actually acts on K as on a mere set.

The Heisenberg super extension defines the commutator pairing^{58}

(3.1.2)  { , }^{♭} : F^{×} × F^{×} → O^{×}

and its Lie algebra version

(3.1.3)  [ , ]^{♭} : F × F → O.

So the adjoint action of f ∈ F^{×} on F^{×♭} and F^{♭} is^{59}

(3.1.4)  Ad_{f}(g^{♭}) = {f, g}^{♭}g^{♭}, Ad_{f}(a^{♭}) = [f, f^{−1}a]^{♭} + a^{♭}.

Consider the Tate dual F_{R}^{∗} := Hom_{R}^{cont}(F_{R}^{τ}, R) as an F_{R}-module. As follows from (2.10.3) the R-linear morphism

(3.1.5)  d : F → F^{∗}, da(b) := [a, b]^{♭},

is a continuous derivation. It yields a canonical F_{R}-linear morphism ω_{F_{R}} → F_{R}^{∗} where ω_{F_{R}} = ω_{F_{R}/R} is the module of continuous differentials relative to R. Let Res^{♭} : ω_{F_{R}} → R be its composition with evaluation at 1 ∈ F_{R}, i.e., Res^{♭}(bda) := [a, b]^{♭}. Thus the adjoint action of F^{×} on F^{♭} is given by a cocycle

(3.1.6)  F^{×} × F → O, f, a ↦ Res^{♭}(ad log f).

Exercises. (i) Let A → R be a morphism of commutative algebras such that R is finite and flat over A. Then F_{R} is also a Tate Amodule, so we have the two pairings { , }_{R}^{♭} : F_{R}^{×} × F_{R}^{×} → R^{×} and { , }_{A}^{♭} : F_{R}^{×} × F_{R}^{×} → A^{×}. Show that { , }_{A}^{♭} = Nm_{R/A}{ , }_{R}^{♭}.^{60}

(ii) Assume that F is a product of finitely many algebras F_{α} as above. Show that for f = (f_{α}), g = (g_{α}) ∈ F^{×} = ΠF_{α}^{×} one has {f, g}_{F}^{♭} = Π{f_{α}, g_{α}}_{F_{α}}^{♭}.

3.2. The case of F_{R} = R((t)). An important example of the above F is the algebra F_{R} = R((t)) of Laurent power series equipped with the usual topology.^{61} Let us discuss it in more detail.

The language of ind-schemes is convenient here. For us, an indscheme is a functor X on the category of commutative rings which can be presented as a filtering inductive limit of functors R ↦ X_{α}(R) where X_{α} are schemes and transition maps X_{α} → X_{β} are closed embeddings. We say that X is ind-affine if all X_{α} are affine schemes, X is formally smooth if the usual Grothendieck property is satisfied; X is reduced if one can choose X_{α} to be reduced schemes. To every X there corresponds the reduced ind-scheme X_{red} := lim_{⟶}(X_{α})_{red}.

58See 2.1, A5.

59The second formula comes from the description 2.10(v) of F^{♭}: if ã is an operator with open kernel which represents a^{♭} then Ad_{f}(a^{♭}) is represented by fãf^{−1}, so Ad_{f}(a^{♭}) − a^{♭} = tr(fãf^{−1} − ã) = tr([f, ãf^{−1}]) = [f, af^{−1}]^{♭} = [f, f^{−1}a]^{♭}.

60Here Nm_{R/A} : A^{×} → R^{×} is the norm map.

61A base of neighbourhood of 0 is t^{n}R[[t]], n ≥ 0.

For a commutative ring R set F_{R} = R((t)). It is well-known that the multiplicative group F_{R}^{×} is a direct product of the following four subgroups: (i) the subgroup ℤ_{R} of elements t^{d}; (ii) R^{×}; (iii) the subgroup of elements 1 + a_{1}t + a_{2}t^{2} + .., a_{i} ∈ R; and (iv) the subgroup of elements 1 + b_{1}t^{−1} + b_{2}t^{−2} + .. where b_{i} are nilpotent elements of R and almost all b_{i} = 0.

The functor F^{×} : R ↦ F^{×}(R) := F_{R}^{×} is a commutative group indscheme which is ind-affine and formally smooth. The subgroup (i) is represented by the discrete group ℤ, (ii) by 𝔾_{m}, (iii) by the group scheme 𝕎 of (big) Witt vectors, and (iv) by the formal completion 𝕎̂ of 𝕎 (via the change of variable t ↦ t^{−1}). Therefore

(3.2.1)  F^{×} = ℤ × 𝔾_{m} × 𝕎 × 𝕎̂.

One has F_{red}^{×} = ℤ × 𝔾_{m} × 𝕎. The projection F^{×} → ℤ has connected kernel; for d ∈ ℤ let F^{×d} ⊂ F^{×} be the preimage of d.

The functor Aut(F) : R ↦ Aut(F)(R) := Aut(F_{R}) is a group indscheme. The map Aut(F) → F^{×1}, ϕ ↦ ϕ(t), is an isomorphism of ind-schemes.

The next proposition shows that (3.1.2) coincides in the present situation with the inverse to a “parametric” version of the tame symbol introduced by Contou-Carrère [C] (see also [D3] 2.9); the usual tame symbol is its restriction to F_{red}^{×}.

3.3. Proposition. (i) { , }^{♭} is Aut(F)-invariant.

(ii) With respect to decomposition (3.2.1) the non-zero components of { , }^{♭} are:

(a) the ℤ × ℤ-component: one has {t^{m}, t^{n}}^{♭} = (−1)^{mn}.

(b) the 𝔾_{m} × ℤ-component (and its transpose): {r, t^{m}}^{♭} = r^{−m}.

(c) the 𝕎 × 𝕎̂-component (and its transpose): for f(t) ∈ 𝕎 and g(t) ∈ 𝕎̂ one has {f(t), g(t^{−1})}^{♭} = r(f ∗ g). Here ∗ is the product of Witt vectors^{62} which sends 𝕎 × 𝕎̂ to 𝕎̂, and r : 𝕎̂ → 𝔾_{m} is the morphism h(t) ↦ h(1)^{−1}.^{63} (iii) The morphism Res^{♭} is the usual residue Res at t = 0, so the

62Recall (see e.g. [Mu] Lecture 26) that 𝕎 is naturally a commutative ring scheme; the ring structure is uniquely characterized by property that every map w_{i} : 𝕎(R) → R, Σw_{i}t^{i} := −t∂_{t} log f, is a ring homomorphism.

canonical morphism ω_{F} → F^{∗} is an isomorphism.

Proof. (i) is clear since Aut(F) acts on F^{×♭}.

We will prove (ii), (iii) simultaneously.

(a) follows from (2.1.1) since {f, f}^{plain} = 1 for any f, and (b) follows from Exercise (b) in 2.10(ii).

For the remainder of the proof, we assume we are working in the category of ℚ-algebras. We can do this since it suffices to check our statements for the universal R; it has no torsion, so we loose no information tensoring by ℚ.

Let us check that ℤ × 𝔾_{m} is orthogonal to 𝕎 × 𝕎̂. Since we are in char 0 and 𝕎 × 𝕎̂ is connected we can replace this group by its Lie algebra which is the subspace F^{0} ⊂ F of formal power series Σa_{i}t^{i} with a_{0} = 0. The action of homotheties ⊂ Aut(F) via t ↦ at preserves F^{0} and has trivial coinvariants there. Since homotheties fix 𝔾_{m} ⊂ ℤ × 𝔾_{m} we see, by (i), that 𝔾_{m} is orthogonal to F^{0}. The action of homotheties on the quotient ℤ = ℤ×𝔾_{m}/𝔾_{m} is trivial, so the same argument shows that ℤ is also orthogonal to F^{0}, q.e.d.

It remains to compute the restriction of { , }^{♭} to 𝕎 × 𝕎̂. Again, it suffices to check the formulas on the level of Lie algebras.

The group pairing f(t), g(t^{−1}) ↦ r(f ∗ g) from (c) looks as follows. For f(t) = exp(α_{1}t + α_{2}t^{2} + ..), g(t) = exp(β_{1}t + β_{2}t^{2} + ..) one has (f ∗ g)(t) = exp(− ∑ iα_{i}β_{i}t^{i}), so r(f ∗ g) = exp(∑ iα_{i}β_{i}). It is clear that the corresponding infinitesimal pairing is a, b ↦ Res(bda).

So it remains to check (iii), i.e., to show that [t^{m}, t^{n}]^{♭} = mδ_{m,−n}, which is an easy calculation (use presentation 2.10(iv) for F^{♭}). □

Remarks. (i) Here is another convenient formula.^{64} Let f ∈ R[[t]] be a formal power series with invertible constant term, and g ∈ R[t, t^{−1}] a Laurent polynomial with one of coefficients invertible and other nilpotent. Then divg is a Cartier divisor supported at t = 0 and, as follows from Exercise (b) in 2.10(ii), one has

(3.3.1)  {f, g}^{♭} = N_{divg}(f)^{−1}.

(ii) According to [C] the pairing { , }^{♭} is non-degenerate, i.e., it identifies the group ind-scheme F^{×} with its Cartier dual. For an analytic version of this statement see [D3], sect. 4.

63Here 𝕎̂(R) is identified with units in 1 + tR[t], i.e., with polynomials of the form 1 + a_{1}t + ... where a_{i} are nilpotent.

64We are grateful to referee for pointing it out to us.

3.4. The standard splittings and the Weil reciprocity. Let R, F_{R} be as in 3.1. Suppose we have R-subalgebras O_{R}, A_{R} ⊂ F_{R} which are, respectively, c- and d-lattices in F (see 2.13). By base change we can consider it as an embedding O, A ↪ F of O-algebras in S_{R} (see 2.10). Then O^{×}, A^{×} ⊂ F^{×} and (2.13.3) gives the standard splittings

(3.4.1)  s_{O}^{c} : O^{×} → F^{×♭}, s_{A}^{d} : A^{×} → F^{×♭}.

In particular, the restriction of { , }^{♭} to O^{×} and A^{×} vanishes.

This situation occurs in geometry as follows. Let X be a smooth projective family of curves over Spec R, D ⊂ X a relative divisor such that U = X ∖ D is affine. Let O_{D} be the algebra of functions on the formal completion of X at D and F_{D} its localization with respect to an equation of D.^{65} Then F_{D} is a Tate R-module, and O_{D} ⊂ F_{D} is a c-lattice. We have the obvious embedding of R-algebras O(U) ↪ F_{D} which identifies O(U) with a d-lattice in F_{D}. So we have the standard splittings

(3.4.2)  s_{O_{D}}^{c} : O_{D}^{×} → F_{D}^{×♭}, s_{O(U)}^{d} : O(U)^{×} → F_{D}^{×♭}.

Let { , }_{D}^{♭} be the symbol map (3.1.2) for F_{D}. If D is a disjoint union of R-points x_{α} then F_{D} = ΠF_{x_{α}} and 3.3^{66} identifies { , }_{D}^{♭} with the product of (the inverse of) local tame symbols at x_{α}. If D is étale over Spec R then one computes { , }_{D}^{♭} using Exercise (i) in 3.1. So the vanishing of { , }_{D}^{♭} on O^{×}(U) ⊂ F_{D}^{×} is the classical Weil reciprocity law. Passing to Lie algebras one gets, by 3.3(iii), the residue formula.

3.5. ω(F)^{×} and the extended Heisenberg. Let R, F_{R} as in 3.1 and assume that locally in flat topology of Spec R our F_{R} is isomorphic to a product of several copies of R((t)). All our constructions are local, so we tacitly assume that we work flat locally on Spec R.

So, localizing R if needed, we get, by 3.3(iii), a canonical isomorphism ω(F) := ω_{F} →^{∼} F^{∗} of free F -modules of rank 1. Denote by ω(F)^{×} the set of generators, i.e., invertible elements, of ω(F).

The story of 2.16 in our special setting looks as follows.

As in 2.14 set W := F ⊕ F^{∗} = F ⊕ ω(F), so we have the orthogonal group O = O(W) and its super extension O^{♭}. One has an embedding F^{×} ↪ O, f ↦ f^{o} := the diagonal matrix with entries f, f^{−1}. By 2.15 the restriction of O^{♭} to F^{×} is the Heisenberg super extension F^{♭}.

65So O_{D} := lim_{⟵} Γ(X, O/O(−nD)), F_{D} := lim_{⟶} lim_{⟵} Γ(X, O(mD)/O(−nD)).

66Together with Exercise (ii) in 3.1.

Consider an embedding^{67} ω(F)^{×} ↪ O(W) which identifies ν with the anti-diagonal matrix ν^{o} with entries ν, ν^{−1}. The disjoint union of F^{×} and ω(F)^{×} is a subgroup F̃^{×} in O(W) which contains F^{×} as a normal subgroup of index 2; the multiplication table is

(3.5.1)  ν^{o}ν′^{o} = (ν′/ν)^{o}, f^{o}ν^{o} = (f^{−1}ν)^{o} = ν^{o}f^{o}.

Let F̃^{×♭} be the super extension of F̃^{×} induced from O^{♭}(W); for g ∈ F̃^{×} we denote the corresponding super line by λ_{g}. In particular, we have a super line λ_{ω(F)^{×}} on ω(F)^{×}.

Below we consider ω(F)^{×} as an F^{×}-torsor with respect to the action f, ν ↦ f^{−1}ν. The multiplication in F^{×♭} lifts this action to an action of F^{×♭} on λ_{ω(F)^{×}}.

Notice that over ω(F)^{×} the tensor square of λ is canonically trivialized: one has λ_{ν^{o}}^{⊗2} = O_{ν} since ν^{o2} = 1. Therefore we have defined a super μ_{2}-torsor μ_{ω(F)^{×}} on ω(F)^{×}.

The groupoid that corresponds to the F^{×♭}-action on ω(F)^{×} is a super extension G_{ω(F)^{×}}^{♭} of the simple transitive groupoid G_{ω(F)^{×}}. Its action on λ_{ω(F)^{×}} identifies G_{ω(F)^{×}}^{♭} with the λ_{ω(F)^{×}}-split super extension (see A2). Since λ_{ω(F)^{×}} comes from the super μ_{2}-torsor μ_{ω(F)^{×}} we see that G_{ω(F)^{×}}^{♭} comes from the μ_{ω(F)^{×}}-split super μ_{2}-extension G_{ω(F)^{×}}^{μ}. Explicitly, G_{ω(F)^{×}}^{μ} ⊂ G_{ω(F)^{×}}^{♭} is the super μ_{2}-subextension whose action preserves μ_{ω(F)^{×}} ⊂ λ_{ω(F)^{×}}.

If 1/2 ∈ R then μ_{ω(F)^{×}} is an étale torsor and G_{ω(F)^{×}}^{μ} is an étale groupoid extension of G_{ω(F)^{×}}. Therefore G_{ω(F)^{×}}^{μ} splits canonically over the formal completion^{68} Ĝ_{ω(F)^{×}} of G_{ω(F)^{×}}. The corresponding splitting of G_{ω(F)^{×}}^{♭} is called the canonical formal rigidification of G_{ω(F)^{×}}^{♭}.

Remark. The canonical formal rigidification of G_{ω(F)^{×}}^{♭} is completely determined by the connection on λ_{ω(F)^{×}} defined by the μ_{2}-structure. The horizontal leaves of this connection are just the orbits of the adjoint action of F^{×} on λ_{ω(F)^{×}} ⊂ F̃^{×♭}.^{69}

The above picture is compatible with the standard splittings of the Heisenberg group from 3.4. Namely, suppose we have O, A ⊂ F as in 3.4. Set O^{∘} := O^{⊥}, A^{∘} := A^{⊥}. So O^{∘} ⊂ ω(F) = F^{∗} is an Osubmodule which is a c-lattice, A^{∘} ⊂ ω(F) is an A-submodule which is a d-lattice. Suppose that O^{∘} is generated, as an O-module, by some element of ω(F)^{×}, and similarly, A^{∘} is generated, as an A-module, by some element of ω(F)^{×}. Let O^{∘×}, A^{∘×} ⊂ ω(F)^{×} be the sets of such generators. Then Õ^{×} := O^{×} ⊔ O^{∘×}, Ã^{×} := A^{×} ⊔ A^{∘×} are subgroups of F̃^{×}. Since Õ^{×} ⊂ O(W, O_{W}), Ã^{×} ⊂ O(W, A_{W}) (see Remark (ii) in 2.15 and Remarks (i), (ii) in 2.14 for notation) we have the standard sections

67We identify ν ∈ ω(F)^{×} with an F -linear isomorphism F →^{∼} ω(F), f ↦ fν.

68An R′-point of Ĝ_{ω(F)^{×}} is a pair of R′-points of ω(F)^{×} that coincide modulo a nilpotent ideal of R′.

69Notice that this action lifts the F^{×}-action f, ν ↦ f^{−2}ν on ω(F)^{×}.

(3.5.2)  s_{O_{W}}^{c} : Õ^{×} → F̃^{×♭}, s_{A_{W}}^{d} : Ã^{×} → F̃^{×♭}.

On O^{×}, A^{×} ⊂ F_{ω}^{×} these are splittings s_{O}^{c}, s_{A}^{d} from (3.4.1) (see Remark (ii) in 2.15). The restriction of (3.5.2) to O^{∘×}, A^{∘×} ⊂ ω(F)^{×} trivialize the restrictions to O^{∘×}, A^{∘×} of the μ_{2}-torsor μ_{ω(F)^{×}}. The corresponding splittings of the groupoid super extensions G_{O^{∘×}}^{♭}, G_{A^{∘×}}^{♭} come from sections s_{O}^{c}, s_{A}^{d}.

Remark. With ω(F) replaced by F^{∗} the above constructions remain valid for every F such that F^{∗} is a free F -module of rank 1. E.g. one can take for F an algebra F_{D} from 3.4; the divisor D need not be étale over Spec R.

3.6. Remarks. We assume that 1/2 ∈ R.

(i) Take any ν ∈ ω(F)^{×}. Then the canonical formal rigidification of G_{ω(F)^{×}}^{♭} provides a section ∇_{ν}^{♭} : F → F_{ω}^{♭}. As folows from the first Remark in 3.5 one has ∇_{f^{−2}ν}^{♭} = Ad_{f}∇_{ν}^{♭}, i.e., by (3.1.6),

(3.6.1)  ∇_{fν}^{♭}(a) = ∇_{ν}^{♭}(a) − \frac{1}{2}Res(ad log f).

Therefore the map ν ↦ ∇_{ν}^{♭} identifies the ω(F) = F^{∗}-torsor of sections F → F^{♭} with the push-out of the F^{×}-torsor ω(F)^{×} by the homomorphism −\frac{1}{2}d log : F^{×} → ω(F).

(ii) According to Remark (iii) in 2.16 the above ∇_{ν}^{♭} can be also described as follows. Consider the involution 𝔰_{ν} :=Ad_{ν^{o}} of O(W) and O^{♭}(W). It preserves F^{♭} ⊂ 𝔬^{♭}(W) and acts on F ⊂ 𝔬(W) as multiplication by −1. Our ∇_{ν}^{♭} is an 𝔰_{ν}-invariant splitting F → F^{♭}, and it is uniquely characterized by this property.^{70}

(iii) Set O := R[[t]] ⊂ F = R((t)). Consider the splitting s_{O}^{c} : O → F^{♭} from (3.4.1). Then for every a ∈ O and f ∈ F^{×} one has

(3.6.2)  ∇_{fdt}^{♭}(a) = s_{O}^{c}(a) − \frac{1}{2}Res(ad log f).

70Indeed, if there were two such, the difference would be a map F → O which is 𝔰_{ν}-equivariant, where 𝔰_{ν} acts by −1 on F and by +1 on O.

Indeed, ∇_{dt}^{♭}(a) = s_{O}^{c}(a) by the discussion at the end of 3.5; the general case follows from (3.6.1).

(iv) Let us describe the super μ_{2}-torsor μ_{ω(F)^{×}} for F = R((t)) explicitly. We can assume that R is reduced. On even connected components (those of the forms t^{2n}dt) our μ is even, and it can be trivialized choosing L = t^{−n}R[[t]] as in 2.17(i). On odd components (those of the forms t^{2n+1}dt) our μ is odd. For a form ν = rt^{2n+1}dt+.. our μ_{ν} is the μ_{2}-torsor of square roots of r as seen from (2.17.1) applied to L = t^{−n}R[[t]].

We see that the restriction of G_{ω(F)^{×}}^{μ} to even components is canonically trivialized. Therefore the restriction of G_{ω(F)^{×}}^{♭} to even components is also canonically trivialized. The restriction of G_{ω(F)^{×}}^{μ} to odd components has non-trivial monodromy.

3.7. Some twists. Let R be a commutative algebra, F_{R} a topological R-algebra. Assume that F_{R} is isomorphic to R((t)), so ω(F) = F^{∗}. Let V_{R} be a finitely generated projective F_{R}-module. Then V_{R} is a Tate R-module with respect to a natural topology whose base is formed by sub R-modules ΣI · v_{α} where {v_{α}} is a finite set of F_{R}-generators of V, I ⊂ F_{R} an open R-submodule.^{71}

Consider the action of F^{×} on V by homotheties F^{×} → GL(V). Let F_{(V)}^{×♭} be the pull-back of the Tate super extension GL(V)^{♭}. Its Lie algebra F_{(V)}^{♭} is a central extension of F by O.

Remark. Super O^{×}-extensions of F^{×} form a Picard groupoid with respect to Baer product (see A3). As follows from 2.10(iii), V ↦ F_{(V)}^{×♭} is a symmetric monoidal functor (with respect to direct sum of V ’s).

The following proposition-construction is crucial for the definition of ε-connection. The reader who is willing to admit that V is a free F -module can use instead a much shorter equivalent construction from 3.9(i) below.

Denote by n the rank of V_{R} (as of an F_{R}-module). We can consider n as a locally constant function on Spec R. Let F^{n♭} be the Baer nmultiple of the extension F^{♭}.

3.8. Proposition. Every R-relative connection 𝔡 on the F_{R}-line det_{F_{R}} V_{R} yields a canonical isomorphism of the Lie algebra extensions

(3.8.1)  ξ^{𝔡} : F^{n♭} →^{∼} F_{(V)}^{♭}

such that for η ∈ ω(F) and a^{♭} ∈ F^{n♭} lifting a ∈ F one has

(3.8.2)  ξ^{𝔡+η}(a^{♭}) − ξ^{𝔡}(a^{♭}) = −Res(aη) ∈ O ⊂ F_{(V)}^{♭}.

71We already met this topology in Example (ii) of 2.11.

Proof. (a) It is convenient to use a slightly different format. Namely, for every R-relative connection ∇ on V we will define a canonical isomorphism of the Lie algebra extensions

(3.8.3)  ξ^{∇} : F^{n♭} →^{∼} F_{(V)}^{♭}

such that for any ρ ∈ ω(F) ⊗_{F} End_{F} (V) one has

(3.8.4)  ξ^{∇+ρ}(a^{♭}) = ξ^{∇}(a^{♭}) − Res tr(aρ).

Such ξ^{∇} amounts to a datum of (3.8.1) subject to (3.8.2). Namely, ξ^{𝔡} that corresponds to ξ^{∇} is determined by the condition ξ^{tr∇} = ξ^{∇} where tr∇ is the connection on det_{F} V defined by ∇.^{72}

(b) To construct ξ^{∇} we use a “geometric” interpretation of F_{(V)}^{♭} from [BS] which we briefly recall now. The picture below is compatible with base change, so we skip R from the notation (replacing it by O when necessary, see 2.10).

Set V′ := Hom_{F} (V, F), V^{∘} := Hom_{F} (V, ω(F)) = ω(F) ⊗_{F} V′. These are finitely generated projective F -modules. As a Tate O-module V^{∘} is canonically isomorphic to the dual Tate O-module to V: the pairing V^{∘} × V → O is v^{∘}, v ↦ Res(v^{∘}(v)). One has a usual identification V ⊗_{F} V′ = End_{F} V.

The diagonal embedding Δ : Spec F ↪ Spec (F ˆ⊗_{O}F) is a Cartier divisor.^{73} Consider an exact sequence of F ˆ⊗_{O}F -modules

(3.8.5)  0 → V ˆ⊗_{O}V^{∘} → V ˆ⊗_{O}V^{∘}(Δ) → End_{F} V → 0

where the right arrow is the residue around the diagonal along the second variable. Let

(3.8.6)  0 → O → F_{(V)}^{♭′} → F → 0

be its pull-back by F ↪ End_{F} V, f ↦ f · id_{V}, and push-forward by V ˆ⊗_{O}V^{∘} → O, v ⊗ v^{∘} ↦ −Res(v^{∘}(v)). Notice that F_{(V)}^{♭′} inherits from V ˆ⊗_{R}V^{∘}(Δ) an F -bimodule structure. It defines a Lie bracket on F_{(V)}^{♭′} with the adjoint action of f^{♭′} coming from the commutator with f with respect to the F -bimodule structure. So F_{(V)}^{♭′} is a central Lie algebra extension of the commutative Lie algebra F.

Now there is a canonical isomorphisms of central O-extensions

(3.8.7)  F_{(V)}^{♭′} →^{∼} F_{(V)}^{♭}.

73For F_{R} →^{∼} R((t)) one has F ˆ⊗_{R}F →^{∼} R[[t_{1}, t_{2}]][t_{1}^{−1}, t_{2}^{−1}].

72Notice that ∇ form an ω(F) ⊗_{F} End_{F} V -torsor Conn(V), and ∇ ↦ tr∇ identifies the ω(F)-torsor of 𝔡’s with the ω(F)-torsor induced from Conn(V) by the map id_{ω(F)}⊗tr: ω(F) ⊗_{F} End_{F} V → ω(F).

Namely, for f^{♭′} ∈ F_{(V)}^{♭′} the corresponding f^{♭} ∈ F_{(V)}^{♭} is defined as follows. Choose k = k(t_{1}, t_{2})dt_{2} ∈ V ⊠ V^{∘}(Δ) that lifts f^{♭′}. It defines an “integral oprator”^{74}

(3.8.8)  A_{k} : V → V, A_{k}(v)(t_{1}) := −Res_{t_{2}=0}k(t_{1}, t_{2})v(t_{2})dt_{2}.

It is easy to see that A_{k} ∈ 𝔤𝔩_{d}(V) and A_{k}^{∞} = f^{∞}. As was explained in the end of 2.13, such A_{k} defines a lifting of f to F_{(V)}^{♭}; this is our f^{♭}. The independence of the auxiliary choices is immediate.

(c) Now we are ready to define (3.8.3) . Let F^{♭′} be the central extension (3.8.6) for V = F. Its Baer n-multiple F^{n♭′} is the pushforward of the exact sequence 0 → F ˆ⊗_{O}ω(F) → F ˆ⊗_{O}ω(F)(Δ) → F → 0 by the map F ˆ⊗_{O}ω(F) → O, f ⊗ ν ↦ −nRes(fν). According to (3.8.7) we can rewrite (3.8.3) as an isomorphism ξ^{∇} : F^{n♭′} →^{∼} F_{(V)}^{♭′}.

Our ξ^{∇} will come from certain morphism of F ˆ⊗_{O}F -modules Ξ^{∇} : F ˆ⊗_{O}ω(F)(Δ) → V ˆ⊗_{O}V^{∘}(Δ). Since F ˆ⊗_{O}ω(F)(Δ) is a free F ˆ⊗_{O}F - module of rank 1, such Ξ^{∇} is multiplication by a section γ^{∇} ∈ V ˆ⊗_{O}V′. This γ^{∇} must satisfy the condition γ^{∇}|_{Δ} = id_{V} ∈ V ⊗_{F} V′ = End_{F} V in order to assure that ξ^{∇} is well-defined. Notice that ξ^{∇} depends only on the restriction of γ^{∇} to the first infinitesimal neighbourhood Δ^{(1)} of Δ.

Let ∇′ be the connection on V′ dual to ∇. Denote by ˜∇ the Orelative connection on the F ˆ⊗_{O}F -module V ˆ⊗_{O}V′ defined by ∇ and ∇′. The restriction of the relative Kähler differentials of F ˆ⊗_{O}F to Δ equals ω(F) ⊕ ω(F), so the composition of ˜∇ with the restriction to Δ is a morphism ¯∇ : V ˆ⊗_{O}V′ → (ω(F) ⊕ ω(F)) ⊗_{F} V ⊗_{F} V′.

Now our γ^{∇} ∈ V ˆ⊗_{O}V′ is any section killed by ¯∇ whose restriction to Δ equals id_{V}. Such γ^{∇} exists since id_{V} is a horizontal section of End_{F} V = V ⊗_{F} V′, and its restriction to Δ^{(1)} is uniquely defined. Thus we defined ξ^{∇}. Formula (3.8.4) holds since for ρ = ρ(t)dt one has γ^{∇+ρ} = γ^{∇} + ρ(t_{1})(t_{2} − t_{1}) on Δ^{(1)}. □

3.9. Remarks. (i) Isomorphisms (3.8.1) and (3.8.3) are compatible with direct sums of V ’s. Precisely, both F^{n♭} and F_{(V)}^{♭} transform direct sum of V ’s into the Baer sum of extensions (this is true on the group level; see 2.10(iii) for F_{(V)}^{×♭}). With respect to these identifications ξ^{⊕∇_{α}} equals the Baer sum of ξ^{∇_{α}}.

Therefore if V is a free F -module then isomorphisms (3.8.1) or (3.8.3) can be described directly as follows. As we mentioned above, 2.10(iii) yields a canonical identification of super extensions F_{(F^{n})}^{×♭} = F^{×n♭}.^{75} So every F -basis ϕ : F^{n} →^{∼} V yields an isomorphism

74Here k(t_{1}, t_{2})v(t_{2})dt_{2} ∈ V ⊠ ω(Δ).

(3.9.1)  ξ^{ϕ} : F^{×n♭} →^{∼} F_{(V)}^{×♭}.

The corresponding isomorphism of Lie algebras is ξ^{∇_{ϕ}} of (3.8.3) where ∇_{ϕ} is the connection that corresponds to the trivialization ϕ. You define then ξ^{∇} for arbitrary ∇ using ξ^{∇_{ϕ}} and (3.8.4).

(ii) Isomorphisms ξ^{ϕ} of (3.9.1) satisfy the property

(3.9.2)  ξ^{ϕg}(f^{♭}) = ξ^{ϕ}(f^{♭}){det g, f}^{♭}.

Here f ∈ F^{×}, f^{♭} ∈ F^{×n♭} is a lifting of f, g ∈ GL(n, F) is an invertible matrix, det g ∈ F^{×} its determinant, and { , }^{♭} is (3.1.2) . Indeed, we can rewrite our statement as the equality {g, f_{V} }^{♭} = {det g, f}^{♭} where the l.h.s. is the Tate commutator pairing in GL(n, F) ⊂ GL_{R}(F^{n}) (and f there means f · id_{F^{n}}) . This property is clear for the diagonal and unipotent matrices g from 2.10(iii),(iv). This proves our statement in case when R is a local Artinian ring (then F = R((t)) is also local Artinian, so every g is a product of unipotent and diagonalizable matrices). The general case reduces to that one as follows.

Notice that the pairing GL_{n}(F) × F^{×} → R^{×}, g, f ↦ {g, f}^{♭}, is continuous. Precisely, for every g, f there exists N ≫ 0 such that for every β ∈ t^{N}Mat_{n}(R[[t]]), α ∈ t^{N}R[[t]] one has g+β ∈ GL_{n}(F), f +α ∈ F^{×}, and {g, f}^{♭} = {g + β, f + α}^{♭}. Same is true for {det g, f}^{♭}. So we can assume that all but finitely many coefficients of f and entries of g are non-zero. Replacing R by its subring generated by these coefficients and the inverses of the top non-nilpotent coefficients of det g and f we can assume that R is finitely generated over ℤ, hence R is Noetherian. We want to prove that two elements of R^{×} are equal. It suffices to check this on every infinitesimal neighbourhood of every point of Spec R, and we are done.

(iii) The involution 𝔰_{ν} from 3.6(ii) can be easily described in terms of the identification F^{♭} = F^{♭′} (case V = F of (3.8.7)). Namely, for ν = ν(t)dt 𝔰_{ν} comes from the involution of F ˆ⊗_{R}ω(F)(Δ) which sends f(t_{1}, t_{2})ν(t_{2})dt_{2} to f(t_{2}, t_{1})ν(t_{2})dt_{2}. Therefore the 𝔰_{ν}-fixed section ∇_{ν}^{♭} : F → F^{♭′} from 3.6(i) is

(3.9.3)  f(t) ↦ \frac{f(t_{1})dt_{2}}{t_{2} − t_{1}} + \frac{1}{2}(f′(t_{1}) + f(t_{1})\frac{ν′(t_{1})}{ν(t_{1})})dt_{2}.

75The latter extension is the Baer n-multiple of F^{×♭}.

3.10. The canonical flat connections. Let us rephrase the material of 3.5 and 3.8 in a format to be used in sect. 4.

Our input is a triple (R, F, V) where R is a ℚ-algebra, F = F_{R} a topological R-algebra, V = V (F) a finitely generated projective F - module. We assume that locally in the flat topology of Spec R our F_{R} becomes a finite direct product ΠF_{i} where every F_{i} is isomorphic to R((t)). All our constructions are Spec R-local, so we can assume that the above decomposition holds on Spec R itself. Then V = ΠV_{i} where V_{i} is a finitely generated projective F_{i}-module.

By 3.3(iii) one has a canonical identification F^{∗} = ω(F) = Πω(F_{i}). Recall that F^{×} = ΠF_{i}^{×} is a formally smooth group ind-scheme. We have an F^{×}-torsor F^{∗×} = ω(F)^{×} = Πω(F_{i})^{×}.

As in 3.7 we notice that V is a Tate R-module in a natural way, so the F^{×}-action on V by homotheties F^{×} ↪ GL(V) defines^{76} the super extension F_{(V)}^{×♭} of F^{×}. The embeddings F_{i}^{×} ↪ F^{×} lift canonically to mutually commuting embeddings F_{i(V_{i})}^{×♭} ↪ F_{(V)}^{×♭} (see 2.10(iii)). For

f ∈ F^{×} we denote its super line in F_{(V)}^{×♭} by λ_{f}^{(V)}, so λ_{(f_{i})}^{(V)} = ⊗λ_{f_{i}}^{(V_{i})}.

Let E be any super line bundle E on ω(F)^{×} equipped with an action of F_{(V)}^{×♭} which lifts the action f, ν ↦ f^{−1}ν of F^{×} on ω(F)^{×}.^{77} We are not interested in its nature at the moment. Notice that an F_{(V)}^{×♭}-action on E amounts to a collection of mutually commuting actions of F_{i(V_{i})}^{×♭} lifting the F_{i}^{×}-action along ω(F_{i})^{×}.

The key structure on E that arises automatically is:

a rule that assigns to every connection 𝔡 (relative to R) on det V a flat connection ∇^{𝔡} on E.

Let us define ∇^{𝔡} assuming that F is isomorphic to R((t)). In general case this will define ∇^{𝔡} separately in every ω(F_{i})^{×}-direction. Since ω(F)^{×} = Πω(F_{i})^{×} this determines our connection; its flatness is immediate from the construction.

Denote by F̂^{×} the formal completion of F^{×}. Let F̂^{×♭}, F̂_{(V)}^{×♭} be the restrictions of the super extensions F^{×♭}, F_{(V)}^{×♭} to F̂^{×}; these are central O^{×}-extensions of F̂^{×}. Since we are in characteristic 0, the isomorphism of Lie algebras (3.8.1) amounts to an isomorphism ξ^{𝔡} : F̂^{×n♭} →^{∼} F̂_{(V)}^{×♭}.

76Pulling back the Tate extension GL^{♭}(V).

77By definition, such E is a rule that assigns: (i) to every point ν ∈ ω(F)^{×}(R′) a super R′-line E_{ν}; (ii) to every f ∈ F^{×}(R) an isomorphism λ_{f}^{(V)} · E_{ν} →^{∼} E_{f^{−1}ν}; (iii) to every morphism r : R′ → R′′ an isomorphism E_{rν} = E_{ν} ⊗_{R′} R′′. This datum should satisfy the obvious compatibilities.

The F_{(V)}^{×♭}-action on E yields, via ξ^{𝔡}, an action of F̂^{×n♭} on E which is the same as an action of the corresponding groupoid Ĝ_{ω(F)^{×}}^{n♭} (see 3.5). Now the canonical formal rigidification Ĝ_{ω(F)^{×}} → Ĝ_{ω(F)^{×}}^{♭} from 3.5 provides an Ĝ_{ω(F)^{×}}-action on E. According to Grothendieck [Gr], such action amounts to a flat connection. This is our ∇^{𝔡}.

The construction of ∇^{𝔡} is compatible with base change.

Remarks. (i) The connection ∇^{𝔡} acts on E by the structure F_{(V)}^{×♭}- action via a family of splittings ∇_{ν}^{𝔡} : F → F_{(V)}^{♭}, ν ∈ ω(F)^{×}, where

(3.10.1)  ∇_{ν}^{𝔡} := ξ^{𝔡}∇_{ν}^{♭}.

Here ∇_{ν}^{♭} : F → F^{♭} → F^{n♭} was defined in 3.6(i).

(ii) For χ ∈ ω(F) and a ∈ F one has (see (3.8.2))

(3.10.2)  ∇_{ν}^{𝔡+χ}(a) = ∇_{ν}^{𝔡}(a) − Res(aχ).

(iii) Suppose that F = R((t)) and V = V_{R}((t)) where V_{R} is a projective R-module of rank n. Set O := R[[t]] ⊂ F, V (O) := V_{R}[[t]] ⊂ V. According to (2.13.3) we have a section s_{V (O)}^{c} : O^{×} → F_{(V)}^{×♭}, so E is an O^{×}-equivariant bundle. Set χ^{𝔡} := 𝔡(γ)/γ ∈ ω(F) where γ is any trivialization^{78} of det V_{R}. Then for a ∈ O, f ∈ F^{×} one has

(3.10.3)  ∇_{fdt}^{𝔡}(a) = s_{V (O)}^{c}(a) − Res (aχ^{𝔡} + \frac{n}{2}a d log f).

Indeed, if 𝔡 came from a trivialization of V_{R} then it is formula (3.6.2); the general case follows from (3.10.2).

In particular, if 𝔡 is non-singular with respect to det V (O) ⊂ det V (i.e., χ^{𝔡} ∈ ω(O)) then the restriction of ∇^{𝔡} to the O^{×}-torsor ω(O)^{×} ⊂ ω(F)^{×} coincides with the “constant” connection defined by the O^{×}- action on E.

(iv) Of course, to define ∇^{𝔡} we need only the action of F̂^{×♭} on E. The whole F^{×♭}-action will be used in the next subsection.

3.11. The global picture. Now we have X, D, U as in 3.4. We assume that D is étale over Spec R. Let V be a vector bundle on U, i.e., we have a finitely generated projective O(U)-module V (U). Then F := F_{D} and V (F) := F ⊗_{O(U)} V (U) satisfy the assumptions of 3.10.

Set F_{(V)}^{×♭} := F_{(V (F))}^{×♭}. Since V (U) ⊂ V (F) is a d-lattice, (2.13.3) lifts the embedding O(U)^{×} ↪ F^{×} to a splitting^{79}

(3.11.1)  s_{V (U)}^{d} : O(U)^{×} → F_{(V)}^{×♭}.

78γ exists locally on Spec R.

79For V = O_{U} this is (3.4.2).

Consider any E as in 3.10. Then O(U)^{×} acts on E via the splitting s̄_{V (U)}^{d} : O(U)^{×} → F_{(V)}^{×♭} defined by (3.11.1).

Let ω = ω_{U/R} be the canonical bundle. We assume that ω(U) is a free O(U)-module (of rank 1), i.e., the set of generators ω(U)^{×} ⊂ ω(U) is non-empty. Then ω(U)^{×} ⊂ ω(F)^{×} is an O(U)^{×}-torsor.

Consider the restriction E_{ω(U)^{×}} of E to ω(U)^{×}. The O(U)^{×}-action yields a “constant” flat connection on E_{ω(U)^{×}} which we denote by ∇^{0}.

Suppose we have a connection 𝔡 on the line bundle det V on U. Let ∇^{𝔡_{F}} be the connection on E that corresponds, by the above, to the restriction 𝔡_{F} of 𝔡 to F.

3.12. Lemma. The restriction of ∇^{𝔡_{F}} to ω(U)^{×} is equal to ∇^{0}.

Proof. (a) By (3.10.2) and the residue formula the restriction of ∇^{𝔡_{F}} to ω(U)^{×} does not depend on 𝔡.

(b) Assume that V is a trivial vector bundle on U. Fix an isomorphism O_{U}^{n} →^{∼} V and let 𝔡 be the connection that corresponds to the trivialization det ϕ of det V. Then ϕ yields an identification ξ^{ϕ} : F^{×n♭} →^{∼} F_{(V)}^{×♭} of (3.9.1), hence we get an F^{×n♭}-action on E. By 3.9(i) our ∇^{𝔡_{F}} comes from this action via the canonical formal rigidification (see 3.5). We are done by the discussion at the end of 3.5, since ξ^{ϕ} identifies sections s_{O(U)^{∘}}^{d} and s_{V (U)^{∘}}^{d}.

(c) Let us reduce the general situation to the case of trivial V. Our X, D, V, some ν ∈ ω(U)^{×}, and E_{ν} are defined over a finitely generated subring of R. Since E is an equivariant bundle over a torsor, we see that the whole our datum is defined over this subring. Therefore we can assume that R is a finitely generated ℚ-algebra. Then in order to check that two connections are the same it suffices to do this on ifinitesimal neighbourhoods of points in Spec R. So, by base change, we can assume that R is a local Artinian ℚ-algebra. Then one can choose D′ = D ⊔ D′′ such that V is trivial on U′ := X ∖ D′ ⊂ U. It remains to show that our statement for (U, V_{U}) follows from the one for (U′, V_{U′}).

Set F = F_{D}, F′ = F_{D′}, F′′ = F_{D′′}; let O′′ ⊂ F′′ be the product of local rings, so we have V (O′′) ⊂ V (F′′). The projection V (F′) ↠ V (F)× V (F′′)/V (O′′) is preserved by F^{×} × O^{′′×}-action, so the restriction of F_{(V)}^{′′×♭} to F^{×} × O^{′′×} equals the pull-back of F_{(V)}^{×♭}.^{80} The exact sequence 0 → V (U) → V (U′) → V (F′′)/V (O′′) → 0 shows that this isomorphism identifies the restriction of s_{V (U′)}^{d} to O(U)^{×} ⊂ O(U′)^{×} with s_{O(U)}^{d}. Let E′ be an F_{(V)}^{′×♭}-equivariant super line on ω(F′)^{×} = ω(F)^{×} × ω(F′′)^{×} whose restriction to ω(F)^{×} × ω(O′′)^{×} coincides with the pull-back of E. Now on ω(F)^{×} × ω(O′′)^{×} the O(U)^{×}-action on E′ (which comes from the O(U′)^{×}-action) coincides with the O(U)^{×}-action on E, and for every connection 𝔡 on det V the connection ∇^{𝔡_{F′′}} on E′ coincides with the connection ∇^{𝔡_{F}} on E. We are done. □

80i.e., we trivialize F_{(V)}^{′′×♭} over O^{′′×} by means of s_{V (O)}^{c}, see Remark (iii) in 3.10.

4. The ε-factors

From now on we work over a base field k of characteristic 0. “Absolute connection” means a connection relative to k.

4.1. The ε-lines. Let K be a field, F := K((t)), O := K[[t]] ⊂ F; we consider F as a topological K-algebra. Let ω = ω(F) := ω(F/K) be the 1-dimensional F -vector space of Kähler differentials, ω = F dt, Der(F/K) = Hom_{F} (ω, F) = F ∂_{t} the one of vector fields.

Suppose V is an F -vector space of dimension n equipped with a connection ∇ : V → ω ⊗_{F} V. For a non-zero ν ∈ ω let τ_{ν} := ν^{−1} be the corresponding vector field. Consider V as a Tate K-vector space (see 2.11), so we have a continuous operator τ_{ν} = ∇(τ_{ν}) : V → V. It is well-known^{81} that the corresponding asymptotic operator τ_{ν}^{∞} acting on V^{∞} (see 2.13) is invertible. So we have the ℤ-graded super K-line

(4.1.1)  E(V, ∇)_{ν} = E(V)_{ν} := det(V^{∞}, τ_{ν}^{∞})

called the ε-line of (V, ∇) at ν.

Remark. The determinant super K-line of the de Rham cohomology det H_{dR}^{·}(V) is canonically trivialized (see 5.9(a)(iv)).

Sometimes it is convenient to compute E(V)_{ν} as follows. Set τ = τ_{ν}. Denote by D_{F} the associative K-algebra of differential operators; let D_{F}^{≤i} be its standard filtration by degree. So F and V are left D_{F} - modules. It is well-known (see e.g. [D1] p.42 or [M] III 1.1) that V is generated by a single vector e. Then e, τe, .., τ^{n−1}e is an F -base in V, so τ^{n}e = a_{n−1}τ^{n−1}e + .. + a_{0}e for certain a_{i} = a_{i}(t) ∈ F. Set A_{τ} := (−1)^{n−1}τ^{n} + ∑_{i≤n−1}(−1)^{i}τ^{i}a_{i} ∈ D_{F}. This is a non-zero differential

operator, so the corresponding asymptotic operator^{82} is invertible, and we have the ℤ-graded super K-line det(F^{∞}, A_{τ}^{∞}).

4.2. Lemma. There is a natural isomorphism of ℤ-graded super lines

(4.2.1)  E(V, ∇)_{ν} = det(F^{∞}, A_{τ}^{∞}).

81and follows from the fact that V is generated by a single vector, see below.

82We consider F as a Tate K-vector space.

Proof. By definition, E(V)_{ν} = det(V/L_{V} , V/L_{V} , τ^{∞}) where L_{V} ⊂ V is any c-lattice. Take for L_{V} the sum L_{F} e + τ(L_{F} e) + .. + τ^{n−1}(L_{F} e) where L_{F} ⊂ F is any c-lattice in F. To compute our determinant line consider the Tate submodule V′ := F e + τ(F e) + .. + τ^{n−2}(F e). Since τ induces isomorphisms V′ →^{∼} τ(V′) and L_{V} ∩ V′ →^{∼} L_{V} ∩ τ(V′) we see that^{83} E(V)_{ν} = det(V/(L_{V} ⊕ τ(V′)), V/(L_{V} + V′), τ^{∞}). The

identifications F →^{∼} V/V′, f ↦ τ^{n−1}(fe), and F →^{∼} V/τ(V′), f ↦

fe, are compatible with c-lattices, and τ(τ^{n−1}(fe)) is equal to A_{τ}(f)e modulo τ(V′). Therefore E(V)_{ν} = det(F/L_{F} , F/L_{F} , A_{τ}^{∞}), which yields (4.2.1). We leave it to the reader to check that our identification does not depend on the choice of L_{F}. □

4.3. Corollary. The degree of the ℤ-graded super line E(V, ∇)_{ν} is equal to i(∇) + (v(ν) + 1)n where i(∇) is the irregularity of ∇,^{84} and v(ν) the valuation of ν.^{85}

Proof. Write ν = fdt/t, so τ_{ν} = f^{−1}t∂_{t}. The degree of E(V)_{ν} is the index of the operator τ^{∞} acting on V^{∞}. The index of (f^{−1})^{∞} acting on V^{∞} is equal to v(f)n. Thus it suffices to check 4.3 for ν = dt/t.

By 4.2 the degree of E(V)_{dt/t} is equal to the index of the operator A_{t∂_{t}}^{∞}. On the other hand, acording to [D1] p.110 or [M] IV(4.6), one has i(∇) = sup(0, −v(a_{i})) where a_{i} ∈ F are coefficients of A_{t∂_{t}}. Notice that A_{t∂_{t}} shifts the filtration t^{j}K[[t]] on F by i(∇), and the corresponding operator between the associated graded quotients t^{j}K → t^{j−i(∇)}K is non-zero for almost all j’s. So the index of A_{t∂_{t}}^{∞} equals i(∇), q.e.d. □

4.4. Families of ε-lines. Let us consider the picture of 4.1 depending on parameters. So we have (R, F, V, ∇) where (R, F, V) are as in 3.10, ∇ is an R-relative connection on V.

For ν ∈ ω(F)^{×} (it exists flat locally on Spec R) let τ_{ν} := ν^{−1} ∈ Der(F/R) be the corresponding vector field, so we have ∇(τ_{ν}) : V → V. We say that our Spec R-family is ε-nice if the asymptotic operator ∇(τ_{ν})^{∞} : V^{∞} → V^{∞} is invertible (here we consider V as a Tate Rmodule, see 2.11, 2.13). This property does not depend on the choice of ν and it is local with respect to flat topology of Spec R.

In ε-nice situation for ν ∈ ω(F)^{×} we have ℤ-graded super R-line

(4.4.1)  E(V, ∇)_{ν} = E(V)_{ν} := det(V^{∞}, ∇(τ_{ν})^{∞})

83Use (2.9.1) for M = N = V/L_{V}, f^{∞} = τ^{∞}, and 1-term filtrations N_{1} := V′/L_{V} ∩ V′ ⊂ N, M_{1} := ∂_{t}(V′)/L_{V} ∩ τ(V′) ⊂ M; then apply (2.6.2) to trivialize the line det(M_{1}, N_{1}, τ^{∞}|_{N_{1}}) by det(τ|_{N_{1}}).

84See e.g. [D1] or [M] IV, sect. 4.

85i.e., ν = ct^{v(ν)}dt+ higher order terms, c ≠ 0.

called the ε-line. The base change of an ε-nice family is ε-nice, and ε-lines are compatible with the base change.

Remark. The above construction is compatible with disjoint union of families of formal discs. Namely, assume that we have a finite collection of families (F_{αR}, V, ∇_{α}). Set F_{R} := ΠF_{αR}, V := ΠV_{α}, ∇ := Π∇_{α}. Then ω(F)^{×} = Πω(F_{α})^{×}, (F, V, ∇) is ε-nice if and only if such is every (F_{α}, V_{α}, ∇_{α}), and, by (2.6.3), there is a canonical isomorphism

(4.4.2)  E(V, ∇) = ⊠E(V_{α}, ∇_{α}).

How can one determine if a family is ε-nice? We know (see 4.1) that this is always true if R is a field, hence if R is an Artinian algebra (see Remark (b) in 2.5). Thus every family is ε-nice at the generic point.

According to 2.5(ii), a Spec R-family is ε-nice if and only if for a c-lattice^{86} L ⊂ V the quotient R-module V/(L + ∇(τ_{ν})(V)) is finitely generated, and the irregularity function i : Spec R → ℤ, x ↦ i(∇_{x}), is locally constant (see 4.3, 2.4). The latter condition is superfluous if R is Noetherian (see Remark (c) in 2.5). The absence of jumps of irregularity alone does not imply the family is ε-nice.^{87}

Remarks. (i) If R is a local ring then we are not aware of any example of a family with constant irregularity which is not ε-nice.

(ii) If ∇ comes from an absolute connection on V then it looks probable that the absence of jumps of irregularity implies that the family is ε-nice.

4.5. The ε-connection. Assume that we are in situation of 4.4 and our family is ε-nice. The compatibility with base change shows that E(V)_{ν} form a ℤ-graded super line bundle over the F^{×}-torsor ω(F)^{×}. Let us show that E(V) is canonically rigidified with respect to infinitesimal variations of ν, i.e., E(V) carries a canonical flat connection ∇^{ε} relative to Spec R called the ε-connection.

Our problem is local with respect to flat topology of Spec R, so we can assume that F_{R} is isomorphic to a product of several copies of R((t)). Consider the super extension F_{(V)}^{×♭} of F^{×} (see 3.7). Now the F^{×}-action on ω(F)^{×}, f, ν ↦ f^{−1}ν, lifts canonically to an F_{(V)}^{×♭}-action on E(V). Namely, for f ∈ F^{×}, ν ∈ ω(F)^{×} one has τ_{f^{−1}ν} = fτ_{ν}, hence ∇(τ_{f^{−1}ν}) = f∇(τ_{ν}). Our action is the product map

(4.5.1) det(V^{∞}, f^{∞}) · det(V^{∞}, ∇(τ_{ν})^{∞}) →^{∼} det(V^{∞}, ∇(τ_{f^{−1}ν})^{∞}).

86We consider V as a Tate R-module.

87E.g., consider R = Spec k[x], F = R((t)), V = F, ∇(∂_{t}) = ∂_{t} + x/t.

So our E fits into the setting of 3.10. Let 𝔡 :=tr∇ be the connection on det V defined by ∇. Our ∇^{ε} is the connection ∇^{𝔡} from 3.10.

Remark. ∇^{ε} is functorial with respect to isomorphisms of (F, V, ∇), and compatible with base change and disjoint sum identification (4.4.2).

Consider the group ind-scheme Aut(F) on Spec R, Aut(F)(R′) := Aut(F_{R′}) (see 3.2). Let Aut(F)ˆ be its formal completion; this is a formal group whose Lie algebra is the Lie algebra of vector fields Θ(F) := Der(F_{R}/R). The group Aut(F) acts on the ind-scheme ω(F)^{×}.

This action can be lifted to an action of the formal completion Aut(F)ˆ on E(V) in the following two ways:

(a) Using ∇ one lifts the action of Aut(F)ˆ on F to V. This action is compatible with ∇, so Aut(F)ˆ acts on E(V) by transport of structure.

(b) The ε-connection ∇^{ε} lifts the Aut(F)ˆ -action on ω(F)^{×} to E(V).

4.6. Proposition. The above two Aut(F)ˆ -actions on E(V) coincide.

Proof. It suffices to check that the two actions of the Lie algebra Θ(F) on E(V) coincide. For θ ∈ Θ(F) let θ^{(a)}, θ^{(b)} be the corresponding actions of θ on E(V). Then κ(θ) := θ^{(a)} − θ^{(b)} is an O-linear endomorphism of E(V), i.e., a function on ω(F)^{×}. We want to show that κ : Θ(F) → O(ω(F)^{×}) vanishes.

Action (a) preserves ∇^{ε} (see Remark in 4.5), so for every θ, θ′ ∈ Θ(F) one has [θ^{(a)}, θ^{′(b)}] = [θ, θ′]^{(b)} which is [θ^{(b)}, θ^{′(b)}]. Hence θ′(κ(θ)) = 0, i.e., the image of κ belongs to the subspace of Θ(F)-invariant functions.

Therefore the two actions coincide on the commutator [Θ(F), Θ(F)]. We are done since Θ(F) is a perfect topological Lie algebra. □

Remark. Using definition (a) we see that the Aut(F)ˆ -action on E(V) is compatible with the F̂_{(V)}^{×♭}-action from 4.5, i.e., E(V) carries a natural action of the semi-direct product of Aut(F)ˆ and F̂_{(V)}^{×♭}.^{88}

4.7. More on the Θ(F)-action on E(V). First let us discuss the Θ(F)-action on ω(F)^{×}. Consider a formally smooth morphism

(4.7.1)  𝔯 : ω(F)^{×} → 𝔸_{R}^{1}, 𝔯(ν) := Res(ν),

invariant with respect to the Aut(F)-action.

Take any ν ∈ ω(F)^{×}. The tangent space to ν equals F.^{89} So the Θ(F)-action yields a morphism ζ_{ν} : Θ(F) → F.

88Aut(F)ˆ acts on F̂_{(V)}^{×♭} since it acts on V via ∇.

89a ∈ F corresponds to a derivative ϕ ↦ ∂_{ϵ}ϕ((1 + ϵa)ν).

Lemma. (i) The image of ζ_{ν} equals τ_{ν}(F) ⊂ F. It coincides with the tangent space to the fiber of 𝔯 at ν, hence Aut(F)ˆ acts formally transitively along the fibers of 𝔯.

(ii) The kernel of ζ_{ν}, i.e., the stabilizer of ν in Θ(F), is Oτ_{ν} ⊂ Θ(F).

Proof. The differential to 𝔯 at ν equals the functional r_{ν} : F → R, f ↦ Res(fν), so the tangent space to the fiber of 𝔯 at ν equals Kerr_{ν}. For θ ∈ Θ(F) one has ζ_{ν}(θ) = Lie_{θ}(ν)/ν = −d(νθ)/ν. Therefore the multiplication by ν isomorphism identifies ζ_{ν} with the de Rham differential F → ω(F). This implies our assertions. □

Let us describe the Θ(F)-action on E(V) directly in terms of the F_{(V)}^{♭}-action on E (see (4.5.1)). Since E(V) is a F_{(V)}^{×♭}-torsor, the Θ(F)- action can be considered as a rule that assigns to ν ∈ ω(F)^{×} a lifting ζ_{ν}^{♭} : Θ(F) → F_{(V)}^{♭} of ζ_{ν} : Θ(F) → F.

Lemma. The kernel of ζ_{ν}^{♭} coincides with the kernel of ζ_{ν}, and its image coincides with image of τ_{ν} acting on F_{(V)}^{♭}. Thus ζ_{ν}^{♭} is the composition of ζ_{ν} and the inverse to the isomorphism τ_{ν}(F_{(V)}^{♭}) →^{∼} τ_{ν}(F).

Proof. Consider the section ∇_{ν}^{ε} : F → F_{(V)}^{♭}. According to 4.6 one has ζ_{ν}^{♭} = ∇_{ν}^{ε}ζ_{ν}, so Kerζ_{ν}^{♭} = Kerζ_{ν}. Since ∇_{ν}^{ε} is τ_{ν}-invariant (see Remark in 4.6) and τ_{ν} kills O ⊂ F_{(V)}^{♭}, one has τ_{ν}(F_{(V)}^{♭}) = ∇_{ν}^{ε}(τ_{ν}(F)). Since ζ_{ν}(Θ(F)) = τ_{ν}(F) by the previous Lemma, we are done. □

4.8. The absolute ε-connection. Suppose that V carries an absolute flat connection ∇^{abs} that extends ∇. Let us show that the super line bundle E(V, ∇) on ω(F)^{×} acquires then an absolute flat connection which extends the relative ε-connection.

So let T be a test k-algebra, I ⊂ T a nilpotent ideal, and t, t′ ∈ ω(F)^{×}(T) two T -points that coincide mod I. We want to define a canonical isomorphism

(4.8.1)  E(V, ∇)_{t} →^{∼} E(V, ∇)_{t′}

which is the identity mod I, satisfies a transitivity property, and is compatible with the base change (see [Gr]).

Let t_{R}, t′_{R} : R → T be the corresponding T -points of Spec R. Let F_{T} := F ˆ⊗_{t_{R}}T, V_{T} , ∇_{T} be the t_{R}-pull-back of F, V, ∇; this is an ε-nice T -family. Define F′_{T}, V′_{T}, ∇′_{T} using t′_{R} instead of t_{R}. The two families coincide mod I. Since F is formally smooth the identification F_{T/I} = F′_{T/I} can be extended to an isomorphism of topological T - algebras ϕ : F_{T} →^{∼} F′_{T}. Now the absolute connection ∇^{abs} lifts ϕ to an isomorphism ϕ_{V} : V_{T} →^{∼} V′_{T} which extends the identity isomorphism V_{T/I} = V′_{T/I}. It is automatically compatible with ∇_{T}, ∇′_{T}.

Let ϕ_{ω} : ω_{F_{T}}^{×} →^{∼} ω_{F′_{T}}^{×} be the isomorphism of T -ind-schemes defined by ϕ. We have the ϕ_{ω}-isomorphism of ε-lines defined by ϕ_{V}

(4.8.2)  ϕ_{V}^{ε} : E(V_{T} , ∇_{T}) →^{∼} E(V′_{T} , ∇′_{T})

which extends the identity isomorphism mod I and is compatible with the ε-connections. We can consider t, t′ as T -points of ω_{F_{T}}^{×}, ω_{F′_{T}}^{×}. They coincide mod I, so ϕ_{V}^{ε} and ∇^{ε} yield an identification

(4.8.3)  E(V_{T} , ∇_{T})_{t} →^{∼} E(V′_{T} , ∇′_{T})_{ϕ_{ω}(t)} →^{∼} E(V′_{T} , ∇′_{T})_{t′}.

The super lines from (4.8.3) equal the super lines from (4.8.1) by base change, so we have defined the promised isomorphism (4.8.1). It does not depend on the auxiliary choice of ϕ: indeed, various ϕ are Aut(F_{T})ˆ - conjugate, so the assertion follows from 4.6. The transitivity and base change properties are clear.

4.9. The standard isomorphisms. Here is a list (cf. 1.1):

(i) Direct sums. Let F_{R} be as in 4.5, and (V_{α}, ∇_{α}) is a finite collection of F -modules with relative connection as in 4.5 such that every every (V_{α}, ∇_{α}) is ε-nice in the sense of loc. cit. Then (V, ∇) := (⊕V_{α}, ⊕∇_{α}) is ε-nice, and, by (2.6.3), we have a canonical isomorphism of ℤ-graded super lines

(4.9.1)  E(V, ∇) = ⊗E(V_{α}, ∇_{α})

on ω(F)^{×} compatible with ε-connections. This isomorphism is compatible with the constraints, so E is a symmetric monoidal functor from the category of (V, ∇) with respect to direct sums to that of super lines with connection on ω(F)^{×}. If our ∇’s come from absolute connections then (4.9.1) is compatible with the absolute ε-connections.

(ii) Filtrations. Let (F_{R}, V, ∇) be an ε-nice family and assume we have a filtration on V compatible with ∇ such that (F_{R}, grV, gr∇) is ε-nice.^{90} Then (2.9.1) yields a canonical isomorphism

(4.9.2)  E(V, ∇) = E(grV, gr∇)

compatible with ε-connections. If our ∇’s come from absolute connections then (4.9.2) is compatible with the absolute ε-connections.

(iii) Induction. Let (F′_{R}, V′, ∇′) be an ε-nice family, F_{R} another topological R-algebra as in 4.5, and F_{R} → F′_{R} a morphism of topological R-algebras. It is automatically étale. Denote by V our V′ considered as an F -module; it carries the induced connection ∇. One has an embedding ω(F)^{×} ↪ ω(F′)^{×}. Now there is a canonical isomorphism^{91}

90Probably, the latter condition is automatic.

(4.9.3)  E(V′, ∇′)|_{ω(F)^{×}} →^{∼} E(V, ∇)

compatible with ε-connections. If ∇′ comes from an absolute connection, then it defines an absolute connection on V, and (4.9.3) is compatible with the absolute ε-connections.

(iv) Non-singular situation. Assume that F_{R} is isomorphic to R((t)) and (V, ∇) is non-singular, i.e., F ⊗_{R} V^{∇} →^{∼} V. Let ω(F)^{0} ⊂ ω(F)^{×} be the 0^{th} connected component, i.e., the component of dt. Then there is a canonical isomorphism of ℤ-graded super lines

(4.9.4)  E(V, ∇)|_{ω(F)^{0}} = (det V^{∇})_{ω(F)^{0}}

compatible with the relative connections (the ε- and the trivial one respectively). If ∇ comes from an absolute connection then (4.9.4) is compatible with the absolute connections.

To establish (4.9.4) we choose an isomorphism R((t)) →^{∼} F_{R}; let O ⊂ F be the image of R[[t]]. Set V (O) := O ⊗_{R} V^{∇} ⊂ V. Then ∇(∂_{t}) preserves V (O), the induced operator ∇(∂_{t}) on V/V (O) is injective,

and V^{∇} →^{∼} t^{−1}V (O)/V (O) →^{∼} Coker(∂_{t} : V/V (O) → V/V (O)) where

the left arrow is multiplication by t^{−1}. The composition yields

(4.9.5)  E(V, ∇)_{dt} = det(V/V (O), ∇(∂_{t})^{∞}) →^{∼} det V^{∇}.

Our connection comes^{92} from a trivialization of V. By 3.9(i) and discussion at the end of 3.5^{93} the connection ∇^{ε} is “constant” along the fibers of ω(F)^{0} → Spec R. We define (4.9.4) as the horizontal morphism which is (4.9.5) at dt.^{94}

Let us check that our isomorphism does not depend on the auxiliary choice of t. Since ∇ is non-singular, the action of Aut(F) on F lifts to an action on V = F ⊗_{R} V^{∇} which preserves ∇. Thus Aut(F) acts by transport of structure on E(V, ∇). This action preserves ∇^{ε}, and the induced action on E(V, ∇)_{ω(F)^{0}}^{∇^{ε}} is trivial by 4.6.^{95} Since Aut(F) acts transitively on the set of t’s, and the action of g ∈ Aut(F) on E(V) sends the isomorphism (4.9.4) defined by means of t to that defined by means of g(t), we are done.

91Take any ν ∈ ω(F)^{×} ⊂ ω(F′)^{×}; let τ_{ν} ∈ Der(F/R), τ′_{ν} ∈ Der(F′/R) be the corresponding vector fields. Then ∇(τ_{ν}) = ∇′(τ′_{ν}), hence the ε-lines are the same.

92Locally on Spec R.

93Notice that ω(F)_{red}^{0} = ω(O)_{red}.

94The fibers of ω(F)^{0} → Spec R are connected, so this defines (4.9.4) uniquely.

95Recall that Aut(F) is connected.

(v) The product formula. Assume we are in a global situation as discussed in 3.11, so we have a family X of smooth projective curves over Spec R, a relative divisor D ⊂ X such that the projection D → Spec R is étale and surjective, and a vector bundle V on U := X ∖ D. Let ∇ be a (relative) connection on V. Consider the morphism ∇ : V (U) → (ω ⊗ V)(U) of projective R-modules.^{96} We say that our situation is ε-nice if ∇ is a Fredholm morphism. Then one has the corresponding ℤ-graded super R-line

(4.9.6)  det(RΓ_{dR}(U, V)[1]) := det((ω ⊗ V)(U), V (U), ∇^{∞}).

Since V (U) ⊂ V (F) is a d-lattice, i.e., V (U)^{∞} →^{∼} V (F)^{∞}, we see that (U, V, ∇) is ε-nice if and only if (F, V (F), ∇) is ε-nice in the sense of 4.4. Below we assume this.

4.10. Proposition. There is a canonical isomorphism of ℤ-graded super lines on ω(U)^{×} (the product formula)

(4.10.1)  E(V (F), ∇)|_{ω(U)^{×}} →^{∼} det(RΓ_{dR}(U, V)[1])_{ω(U)^{×}}

where the r.h.s. is the pull-back of the determinant R-line to ω(U)^{×}. This isomorphism is compatible with the relative connections (the l.h.s. carries the ε-connection, the r.h.s. the trivial one).

If V carries an absolute integrable connection which extends ∇ then (4.10.1) is compatible with the corresponding absolute connections (the ε- and the Gauß-Manin one). Thus (4.10.1) describes the determinant of the Gauss-Manin connection in terms of the absolute ε-connections.

Proof. A trivialization ν ∈ ω(U)^{×} identifies ∇ : V (U) → (ω ⊗ V)(U) with ∇(τ_{ν}) : V (U) → V (U) where τ_{ν} := ν^{−1} ∈ Der(U/R), so we get det(RΓ_{dR}(U, V)[1])_{ν} = det(V (U), V (U), ∇(τ_{ν})^{∞}). The latter super line is identified with E(V (F), ∇)_{ν} via the asymptotic isomorphism

V (U)^{∞} →^{∼} V (F)^{∞}.

The compatibility with relative connections follows from 3.12. It implies immediately the compatibility with absolute connections. □

4.11. Remark. The above standard isomorphisms are mutually compatible in the obvious sense; they are compatible with the disjoint sum identifications (4.4.2). In particular, if D is a disjoint union of sections x_{i} then F = ΠF_{i} (the product of local fields at x_{i}) and E(V (F), ∇)_{ν} = ⊗E(V (F_{i}), ∇)_{ν}, hence the name “product formula” for (4.10.1).

96The R-projectivity follows since O(U) is a projective R-module and V (U) is a projective O(U)-module.

4.12. Duality. Suppose that we have a family (F, V, ∇) such that H_{dR}^{·}(V, ∇) = 0, i.e., ∇ : V → ω ⊗ V is an isomorphism. Let V′ be the dual vector bundle equipped with the dual connection ∇′. It satisfies the same property. For ν ∈ ω(F)^{×} the Tate R-module V′ identifies with the dual to V via the pairing v, v′ ↦ Res(v′(v)ν), and ∇′(τ_{ν}) is the operator adjoint to −∇(τ_{ν}) = ∇(τ_{−ν}). So (2.13.5) yields a canonical identification of super lines

(4.12.1)  E(V′, ∇′)_{ν} = E(V, ∇)_{−ν}^{−1}.

We leave it to the reader to check that it is compatible with the εconnections.

Suppose now we are in the global situation of 4.9(v) and the local de Rham cohomology vanish, i.e., ∇ : V (F) →^{∼} (ω ⊗ V)(F). Thus H_{dR}^{0}(U, V) = H_{dR}^{0}(U, V′) = 0 and the global Poincaré duality identifies H_{dR}^{1}(U, V′) with the dual to H_{dR}^{1}(U, V). Passing to determinants we get a canonical identification det RΓ_{dR}(U, V′) = det RΓ_{dR}(U, V)^{−1}. The product formula is compatible with these identifications.

Remark. We have formulared compatibility (4.12.1) in the situation when the local de Rham cohomology H_{dR}^{·}(V, F) vanish. If the base ring R is a field (or, more generally, an Artinian algebra) then (4.12.1) holds for arbitrary (V, ∇) by (2.13.6) since the local super line det(V, ∇(τ_{ν})) = det H_{dR}^{·}(V) is canonically trivialized (see 5.9(iv)).^{97}

4.13. Questions. (a) What happens when singular points merge together, i.e., we do not assume that D is étale over Spec R?

(b) Is it true that (E, ∇^{ε}) is uniquely determined by compatibilities (4.4.2), (4.9.1)–(4.9.4), (4.10.1)?^{98}

5. Some formula (with apologies to Charlotte)^{99}

In this section we assume, for technical reasons, that our parameter space^{100} is Spec K where K is a field. The case of an arbitrary smooth parameter space reduces to this situation (since a line bundle with connection on a smooth variety is determined by its restriction to the generic point).

97Probably, this is true for arbitrary R under the assumption that both ∇ and ∇′ are ε-nice. It also looks probable that ∇ is ε-nice iff such is ∇′.

98One can show that the connection on our E (defined in (4.4.1)) is uniquely determined by the condition that it is compatible with the standard isomorphisms.

99The authors refer to “Charlotte’s Web”, an acclaimed novel by E. B. White ruminating over the problems of a person in quest for tenure.

100This is Spec R of the previous section.

5.1. Dependence on ν. Let F be a local K-field, V_{F} an F -vector space of dimension n equipped with an absolute connection ∇. We have the ε-line E = E(F, V_{F}); this is a super line with an absolute integrable connection ∇^{ε} on the ind-scheme of invertible relative 1forms ω(F)^{×}. Our aim is to compute its fiber E_{ν} = E(F, V_{F})_{ν} over a K-point ν of ω(F)^{×} which is a super K-line equipped with an integrable connection.

When doing computations it is convenient to choose ν in a special way, e.g. to make it fixed by horizontal vector fields. One can pass then to arbitrary ν using Proposition below.

Choose a parameter t ∈ F. Then F = K′((t)) where a finite extension K′/K is the integral closure of K in F. Set O := K′[[t]] ⊂ F. Below we make vector fields on K act on F so that for θ ∈ Θ_{K} =DerK/k one has θ(t) = 0. Hence Θ_{K} acts on ω(F)^{×}. Differential 1-forms and vector fields on Spec F and ω(F)^{×} decompose into “horizontal” and “vertical” components accordingly. For 1-forms on Spec F these components are denoted by lower indices x and t. E.g., for f ∈ F we write df = d_{x}f + d_{t}f where d_{x}f ∈ F ⊗ Ω_{K/k}^{1}, d_{t}f ∈ ω(F) = Ω_{F/K}^{1}.

Choose an O-lattice V_{O} ⊂ V_{F}. It yields a section s_{V_{O}}^{c} : O^{×} → F_{(V_{F})}^{×♭} (see (2.13.3), 3.7). Since E is an F_{(V_{F})}^{×♭}-equivariant line bundle (see 4.5), s_{V_{O}}^{c} defines an O^{×}-equivariant structure on E.^{101}

Assume we have ν, ν′ ∈ ω(F)(K) such that v(ν) = v(ν′) =: ℓ. Then u := ν′/ν ∈ O^{×}, i.e., u = u_{0} + u_{1}t + .., u_{i} ∈ K′, u_{0} ≠ 0. So the equivariant structure provides an isomorphism

(5.1.1)  α^{u} = α_{V_{O}}^{u} := s_{V_{O}}^{c}(u) : E_{ν′} →^{∼} E_{ν}.

We want to see to what extent α^{u} is horizontal with respect to connections ∇^{ε}, i.e., to compute ∇^{ε} log α^{u} := (∇^{ε}α^{u})/α^{u} ∈ Ω_{K/k}^{1}.

Set χ = χ_{x} + χ_{t} := (tr∇) log γ ∈ Ω_{F/k}^{1} where γ is any generator of det V_{O}. This is a closed 1-form; its class modulo Ω_{O/k}^{1} does not depend on the choice of γ.

5.2. Proposition. One has

(5.2.1)  ∇^{ε} log α^{u} = tr_{K′/K}(\frac{nℓ}{2}d log u_{0} + Res(d log u ∧ χ)).

Proof. By base change we can assume that K′ = K.

Take any a ∈ K[[t]] = O ⊂ F, a = a_{0} + a_{1}t + ... Consider a as a vertical vector field on ω(F)^{×} (the corresponding derivative is ϕ ↦ aϕ, aϕ(ν) := ∂_{ϵ}ϕ((1 + ϵa)ν)). It acts on E according to ∇^{ε}: namely, ∇^{ε}(a) is the action of ∇^{𝔡}(a) ∈ F_{(V)}^{♭} on E (see 4.5). Our a also acts on E according to the O^{×}-equivariant structure, i.e., as s_{V (O)}^{c}(a) ∈ F_{(V)}^{♭}.

101Recall that the O^{×}-action on E lifts the O^{×}-action u, ν ↦ u^{−1}ν on ω(F)^{×}.

From now on we work on the connected component ω(F)_{red}^{ℓ} of the reduced scheme ω(F)_{red}^{×} (the action of a preserves ω(F)_{red}^{ℓ}). By (3.10.3) and 4.5 one has^{102}

(5.2.2)  ∇^{ε}(a) − s_{V (O)}^{c}(a) = −\frac{nℓ}{2}a_{0} − Res(aχ_{t}).

Let r be a non-zero O^{×}-invariant section of E. Then r_{ν}, r_{ν′} are trivializations of, respectively, E_{ν} and E_{ν′} which are identified by the isomorphism α^{u}. Set ψ := ∇^{ε} log r. One has ∇^{ε} log α^{u} = ψ_{ν} − ψ_{ν′}.

Our ψ is a closed 1-form on the O^{×}-torsor ω(F)_{red}^{ℓ}. By (5.2.2) the vertical part of ψ is an O^{×}-invariant form whose value on the vertical vector field a ∈ O equals −\frac{nℓ}{2}a_{0} − Res(aχ_{t}). If ψ′ is any other 1-form with these properties then ψ′−ψ is the pull-back of a 1-form on Spec K, hence ∇^{ε} log α^{u} = ψ′_{ν} − ψ′_{ν′}.

Let us construct such ψ′. Fix ν′ and consider ν as a variable point of ω(F)_{red}^{ℓ}. The scheme ω(F)_{red}^{ℓ} ˆ⊗_{K}O carries a canonical invertible function u: for ν ∈ ω(F)_{red}^{ℓ} the restriction of u to the fiber over ν is the function ν′/ν. The pull-back of χ from F to ω(F)_{red}^{ℓ} ˆ⊗_{K}F is a closed 1-form which we denote also by χ. Set ψ′ := \frac{nℓ}{2}d log u_{0}+Res(d log u∧χ) where Res is the residue along the fibers of ω(F)_{red}^{ℓ} ˆ⊗_{K}F → ω(F)_{red}^{ℓ}.

Incerting ν, ν′ into ψ′ we get (5.2.1). □

5.3. Case of regular singularity. We use notation from 5.1.

Suppose that (the vertical part of) ∇ has regular singularities. Then one can find an O-lattice V_{O} ⊂ V_{F} such that V_{O} is preserved by the action of ∇(t∂_{t}) and horizontal vector fields Θ_{K}.

Consider a K′-vector space V_{0} := V_{O}/tV_{O}. The action of horizontal vector fields define a connection ∇_{0} on V_{0}. The action of ∇(t∂_{t}) defines a ∇_{0}-horizontal endomorphism of V_{0} which we denote by κ.^{103}

Remark. Recall a standard way to construct such V_{O}. Let K̄′ be an algebraic closure of K′. Choose any (set-theoretic) section s : K̄′/ℤ → K̄′ that commutes with the Galois action. Let V_{K′} be the sum of generalized s(λ)-eigenspaces of ∇(t∂_{t}), λ ∈ K̄′/ℤ. Then V_{K′} is a K′- structure on V_{F}. Our lattice is V_{O} := V_{K′}[[t]].

102Here the l.h.s. is a (constant) function on ω(F)_{red}^{ℓ}, in the r.h.s. a is considered as an element of F.

103So κ is the residue of the polar part of ∇ with respect to V_{O}.

Consider a Fredholm endomorphism ∇(t∂_{t}) of V_{F} /V_{O}. Its determinant line is canonically trivialized. Namely, V_{F} /V_{O} decomposes into a direct sum of generalized eigenspaces of ∇(t∂_{t}); for each eigenvalue λ the corresponding subspace P_{λ} is finite-dimensional. So the embedding P_{0} ↪ V_{F} /V_{O} induces an isomorphism on cohomology hence an isomorphism of the determinant lines. And the determinant line for P_{0} is trivialized since dim P_{0} < ∞.

Take any ν ∈ ω(F)^{×}(K), v(ν) =: ℓ. So ν = ut^{ℓ}dt, u ∈ K′[[t]]^{×}, u = u_{0} + u_{1}t + .., u_{i} ∈ K′, u_{0} ≠ 0. We want to compute E_{ν} = E(F, V)_{ν} in terms of V_{O}.

There is a natural isomorphism of super lines

(5.3.1)  β_{ν} : E_{ν} →^{∼} (det_{K}V_{0})^{⊗ℓ+1}.

Namely, recall that E_{ν} = det(V_{F}^{∞}, ∇(τ_{ν})^{∞}) = det(V_{F} /V_{O}, ∇(τ_{ν})^{∞}) where τ_{ν} := ν^{−1} = u^{−1}t^{−ℓ}∂_{t}. Let us present ∇(τ_{ν})^{∞} as a composi-

tion V_{F} /V_{O} ⟶^{∇(t∂_{t})} V_{F} /V_{O} ⟶^{t^{−ℓ−1}} V_{F} /t^{−ℓ−1}V_{O} ⟶^{u^{−1}} V_{F} /t^{−ℓ−1}V_{O} ⟶^{id^{∞}} V_{F} /V_{O}.

We already trivialized the determinant line of the first operator; that of the middle operators is trivialized since they are isomorphisms. Thus E_{ν} = det(V_{F} /V_{O}, V_{F} /t^{−ℓ−1}V_{O}, id^{∞}) = (det V_{0})^{⊗ℓ+1}.^{104}

The r.h.s. of (5.3.1) carries a connection defined by ∇_{0}. It yields, via β_{ν}, a connection on E_{ν} which we also denote by ∇_{0}.

5.4. Proposition. ∇^{ε} − ∇_{0} = −tr_{K′/K}(Trκ − n(ℓ/2 + 1))d log u_{0}).

Proof. For ν = t^{ℓ}dt, i.e., u = 1, the formula is clear. Indeed, any horizontal vector field θ commutes with τ_{ν}, so ∇(θ) acts on the determinant line of τ_{ν} by transport of structure, and ∇^{ε}(θ) coincides with this action (see 4.8).

For arbitrary ν notice that the isomorphism β_{ν}^{−1}β_{t^{ℓ}dt} : E_{t^{ℓ}dt} →^{∼} E_{ν} is inverse to the isomorphism α^{u} of (5.1.1) defined by the lattice t^{−ℓ−1}V_{O}. The corresponding form χ from 5.1 is (Trκ − n(ℓ + 1))dt/t. Now use (5.2.1). □

Remark. Choose a k-structure V_{k} ⊂ V_{F} such that V_{O} is the O-module generated by V_{k}. Let ∇^{0} be the absolute connection on V_{F} , V_{K′} defined by V_{k}, ∇^{0}(V_{k}) = 0. Then ∇ − ∇^{0} = η + t^{−1}gdt where η ∈ Ω_{K′/k}^{1} ⊗ EndV_{K′}[[t]], g ∈ EndV_{K′}[[t]]. Write η = η_{0} + η_{1}t + .., g = g_{0} + g_{1}t + .., η_{i} ∈ Ω_{K′/k}^{1} ⊗ EndV_{K′}, g_{i} ∈ EndV_{K′}. Then V_{0} = V_{K′}, ∇_{0} = ∇^{0} + η_{0}, κ = g_{0}, so we can rewrite formula 5.4 as

(5.4.1) ∇^{ε} − ∇^{0} = tr_{K′/K}((ℓ + 1)Trη_{0} − (Trg_{0} − n(ℓ/2 + 1))d log u_{0}).

104The latter identification comes from isomorphisms t^{−i} : t^{i}V_{O}/t^{i+1}V_{O} →^{∼} V_{0}.

5.5. Case of an irregular admissible connection. As above, we follow notation from 5.1.

Suppose that there is an O-lattice V_{O} ⊂ V_{F} such that ∇ is madmissible with respect to V_{O} for some m ≥ 2 (see [BE3]). This means that the vertical part of ∇ has pole of “exact” order m, i.e., t^{m}∇(∂_{t}) : V_{O} →^{∼} V_{O}, and the horizontal part of ∇ has pole of order ≤ m − 1.

Choose a k-structure V_{k} ⊂ V_{F} such that V_{O} is the O-submodule generated by V_{k}. Let ∇^{0} be the corresponding absolute connection on V_{F}, ∇^{0}(V_{k}) = 0. Set ∇ − ∇^{0} = A = A_{t} + A_{x} ∈ Ω_{F/k}^{1} ⊗ EndV. Write A_{t} = t^{−m}gdt, A_{x} = t^{−m+1}η. Admissibility means that g ∈ AutV_{K′}[[t]], η ∈ Ω_{K′/k}^{1} ⊗ EndV_{K′}[[t]]. So g = g_{0} + g_{1}t + .., η = η_{0} + η_{1}t + .. where g_{i} ∈ EndV_{K′}, η_{i} ∈ Ω_{K′/k}^{1} ⊗ EndV_{K′}, g_{0} is invertible.

Consider a 1-form ν = ut^{ℓ}dt ∈ ω(F)^{×}, u ∈ O^{×}, and the corresponding E_{ν} = E(F, V_{F})_{ν}. We have a natural isomorphism of super lines

(5.5.1)  β_{ν} : E_{ν} →^{∼} (det_{K}(V_{K′}))^{⊗ℓ+m}

similar to (5.3.1). Namely, notice that the operator ∇(τ_{ν}) yields an isomorphism V_{F} /V_{O} →^{∼} V_{F} /t^{−ℓ−m}V_{O} which trivializes the super line det(V_{F} /t^{−ℓ−m}V_{O}, V_{F} /V_{O}, ∇(τ)^{∞}). Our identification is the composition E_{ν} = det(V_{F} /V_{O}, V_{F} /t^{−ℓ−m}V_{O}, id^{∞})·det(V_{F} /t^{−ℓ−m}V_{O}, V_{F} /V_{O}, ∇(τ)^{∞}) = det(V_{F} /V_{O}, V_{F} /t^{−ℓ−m}V_{O}, id^{∞}) = (det_{K}(V_{K′}))^{⊗ℓ+m}.

The k-structure V_{k} on V_{K′} defines an integrable connection on V_{K′}, hence on the r.h.s. of (5.5.1). Denote by ∇_{E}^{0} the corresponding connection on E_{ν}.

5.6. Theorem. One has ∇^{ε} − ∇_{E}^{0} = tr_{K′/K}(ϕ_{1} + ϕ_{2} + ϕ_{3}) where ϕ_{i} ∈ Ω_{K′/K}^{1} are:

ϕ_{1} = ResTr(g^{−1}dg ∧ A_{x}) − \frac{m}{2}d log det(g_{0})

ϕ_{2} = (ℓ + m)ResTr(A_{x} ∧ dt/t)

ϕ_{3} = n(ℓ/2 + m)d log u_{0} − Res(d log u ∧ TrA).

Proof. (a) By base change we can assume that K′ = K, so F = K((t)).

(b) It suffices to treat the case of ν = t^{ℓ}dt. Indeed, the formula for arbitrary ν follows then from (5.2.1) since β_{ν}^{−1}β_{t^{ℓ}dt} : E_{t^{ℓ}dt} →^{∼} E_{ν} is inverse to the isomorphism α^{u} of (5.1.1) for the lattice t^{−ℓ−m}V_{O}. The form χ from 5.1 is TrA − (ℓ + m)ndt/t. We get the term ϕ_{3}.

(c) From now on ν = t^{ℓ}dt. Set h := 1 + t^{m}g^{−1}∇^{0}(∂_{t}) ∈ GL(V_{F}). Our E_{ν} is the determinant line of the invertible operator

(5.6.1)  ∇(τ_{ν}) = t^{−ℓ−m}gh

acting on a Tate K-vector space V_{F} = V_{K}((t)). Notice that both factors g and h preserve the lattice V_{O}, so we have g̃ := s_{V_{O}}^{c}(g), h̃ := s_{V_{O}}^{c}(h) ∈ F_{(V_{F})}^{×♭} (see (2.13.3)). The operator t is defined over k, i.e., t ∈ GL(V_{k}((t))) ⊂ GL(V_{K}((t))). Let t̃ ∈ GL^{♭}(V_{k}((t))) ⊂ GL^{♭}(V_{K}((t))) be any lifting of t which is also defined over k. So we have

(5.6.2)  t̃^{−ℓ−m}g̃h̃ ∈ E_{ν}.

It follows from the construction of ∇_{E}^{0} that

(5.6.3)  ∇_{E}^{0}(t̃^{−ℓ−m}g̃h̃) = 0.

It remains to compute ϕ := ∇^{ε} log(t̃^{−ℓ−m}g̃h̃) ∈ Ω_{K/k}^{1}.

(d) Since τ_{ν} is preserved by the action of horizontal vector fields our ∇^{ε} coincides with the ∇-action of horizontal vector fields (see 4.8).

Explicitly, for θ ∈ Θ_{K} ⊂ Θ_{F} the infinitesimal automorphism 1 + ϵ∇(θ) acts on the Tate K-vector space F = K((t)) hence on GL(V_{F}) and its super extension GL^{♭}(F). This action fixes ∇(τ_{ν}) ∈ GL(V_{F}); the action on its super line E_{ν} ⊂ GL^{♭}(V_{F}) is r ↦ r + ϵ∇^{ε}(θ)r.

Notice that the ∇-action of horizontal vector fields preserves, in fact, both t and gh ∈ GL(V_{F}). So (1 + ϵ∇(θ))t̃^{−ℓ−m} = (1 + ϵϕ_{2}(θ))t̃^{−ℓ−m} and (1 + ϵ∇(θ))(g̃h̃) = (1 + ϵϕ_{1}(θ))g̃h̃ for some ϕ_{i}(θ) ∈ K ⊂ 𝔤𝔩(V_{F}). Thus

(5.6.4)  ϕ = ϕ_{1} + ϕ_{2}.

We will compute ϕ_{i} separately:

(e) Since ∇^{0}(θ)t̃ = 0 one has 1 + ϵϕ_{2}(θ) = Ad_{1+ϵA_{x}(θ)}(t̃^{−ℓ−m})/t̃^{−ℓ−m} or ϕ_{2}(θ) = −(ℓ + m)(A_{x}(θ)^{♭} − Adt̃A_{x}(θ)^{♭}) where A_{x}(θ)^{♭} is any lifting of A_{x}(θ) ∈ End_{F} (V_{F}) ⊂ 𝔤𝔩(V_{F}) to 𝔤𝔩^{♭}(V_{F}). By (3.1.6), 3.3(iii) one has ϕ_{2}(θ) = (ℓ + m)ResTr(A_{x}(θ)dt/t) or

(5.6.5)  ϕ_{2} = (ℓ + m)ResTr(A_{x} ∧ dt/t).

(f) Since ∇^{0}(θ)(V_{O}) ⊂ V_{O} the action of 1 + ϵ∇^{0}(θ) preserves the subgroup GL(V_{F} , V_{O}) ⊂ GL(V_{F}) and commutes with s_{V_{O}}^{c} (see (2.13.3)). The adjoint action of 1 + ϵA_{x}(θ) does not have this property. However it sends both g and h to GL(V_{F} , V_{O}): this is clear for h, and for g you notice that, due to integrability of ∇, one has

(5.6.6) [A_{x}(θ), g] = t∂_{t}(η(θ)) − (m − 1)η(θ) − θ(g) ∈ EndV_{K}[[t]].

So Ad_{1+ϵA_{x}(θ)}g̃ = (1 + ϵϕ′(θ))s_{V_{O}}^{c}(Ad_{1+ϵA_{x}(θ)}g) for certain ϕ′(θ) ∈ K. We have defined ϕ′ ∈ Ω_{K/k}^{1}; replacing g by h we get ϕ′′ ∈ Ω_{K/k}^{1}.

We see that (1 + ϵ∇(θ))g̃ = (1 + ϵϕ′(θ))s_{V_{O}}^{c}(Ad_{1+ϵ∇(θ)}g); same for g replaced by h and ϕ′ by ϕ′′. The product of these identities gives

(5.6.7)  ϕ_{1} = ϕ′ + ϕ′′.

(g) Let us rewrite ϕ′, ϕ′′ in the following general manner:

Assume we have a ∈ 𝔤𝔩(V_{F}) and q ∈ GL(V_{F}) such that q(V_{O}) = V_{O} and [a, q](V_{O}) ⊂ V_{O}. Set b := a − Ad_{q}a, b^{♭} := a^{♭} − Ad_{q}a^{♭} where a^{♭} ∈ 𝔤𝔩^{♭}(V_{F}) is any lifting of a. Notice that b^{♭} is well-defined^{105} and b(V_{O}) ⊂ V_{O}. Set c(a, q) := b^{♭} − s_{V_{O}}^{c}b ∈ K ⊂ 𝔤𝔩^{♭}(V_{F}). We have

(5.6.8)  ϕ′(θ) = c(A_{x}(θ), g), ϕ′′(θ) = c(A_{x}(θ), h).

The following simple lemma helps to compute c:

Lemma. c(a, q) = Tr(V_{O}, (a^{♯} − Ad_{q}a^{♯})|_{V_{O}}) where a^{♯} ∈ 𝔤𝔩(V_{F}) is any endomorphism with open kernel such that (a − a^{♯})(V_{O}) ⊂ V_{O}.

Proof of Lemma. Notice that c(a, q) vanishes if a(V_{O}) ⊂ V_{O}.^{106} So we can assume that a = a^{♯}. Then b^{♭} = s_{d}(b) (see the end of 2.13), and we are done since s_{d}b − s_{V_{O}}^{c}b = Tr(V_{O}, b|_{V_{O}}) for every b ∈ 𝔤𝔩(V_{F}) ∩ 𝔤𝔩(V_{F} , V_{O}). □

To compute ϕ′, ϕ′′ we apply the lemma to q = g, h and a^{♯} = A_{x}Π where Π : V_{F} → V_{F} is a projector with kernel t^{m−1}V_{K}[[t]] and image t^{m−2}V_{K}[t^{−1}].

(h) Let us begin with ϕ′. One has ϕ′ = Tr(V_{O}, A_{x}Π − gA_{x}Πg^{−1}) = Tr(V_{O}, (A_{x} − gA_{x}g^{−1})Π) + Tr(V_{O}, gA_{x}(g^{−1}Π − Πg^{−1})).

The first summand vanishes. Indeed, B := A_{x}−gA_{x}g^{−1} ∈ EndV_{K}[[t]]; our summand is (m − 1)Tr(B_{0}). But Tr(B_{0}) = (TrB)_{0}, and TrB = 0.

The second summand can be rewritten as Tr(V_{O}, A_{x}g^{−1}(Πg−gΠ)) = −Tr(V_{O}, A_{x}g^{−1}(1 − Π)gΠ). An immediate calculation identifies it with −ResTr_{F} (gd_{t}(A_{x}g^{−1})) = −ResTr_{F} (A_{x}g^{−1}d_{t}g) = ResTr_{F} (g^{−1}d_{t}g ∧ A_{x}).

One has ϕ′′ = Tr(V_{O}, A_{x}Π − hA_{x}Πh^{−1}) = −Tr(V_{O}, t^{m}g^{−1}∂_{t}A_{x}Π) = \frac{m(m−1)}{2}Trg_{0}^{−1}η_{0}. As follows from (5.6.6)^{107} it equals −\frac{m}{2}Trg_{0}^{−1}dg_{0} = −\frac{m}{2}d log(det g_{0}). So, by (5.6.7),

(5.6.9)  ϕ_{1} = ResTr_{F} (g^{−1}d_{t}g ∧ A_{x}) − \frac{m}{2}d log(det g_{0}).

We are done. □

5.7. Example. Suppose that our (V_{F} , ∇) is such that the corresponding relative connection ∇_{/K} is a successive extension of a single irregular relative connection (L_{F} , ∇_{/K}) of rank 1.

Let us show that (V_{F} , ∇) is admissible. We need to construct an O-lattice V_{O} ⊂ V_{F} as in 5.5. Notice that the relative connection on L can be extended to an absolute integrable connection so that L^{⊗n} is isomorphic to det V as lines with absolute connection. Set P := V ⊗ L^{−1}. As a relative connection our P is a successive extension of trivial connections. There is a unique O-lattice P_{O} ⊂ P preserved by ∇(t∂_{t}) and such that ∇(t∂_{t}) acts on the fiber P_{0} := P_{O}/tP_{O} as a nilpotent matrix. Our P_{O} is automatically preserved by horizontal vector fields. Set V_{O} := P_{O} ⊗ L_{O} ⊂ V where L_{O} ⊂ L is any O-lattice.

105i.e., it does not depend on the choice of a^{♭}.

106To see this consider a^{♭} = s_{V_{O}}^{c}a.

107Indeed, by (5.6.6) one has (A_{x} − gA_{x}g^{−1})_{0} = −((m − 1)η_{0} + d_{x}g_{0})g_{0}^{−1}. Hence Tr((m − 1)η_{0} + d_{x}g_{0})g_{0}^{−1}) = 0.

To check the conditions from 5.5 choose a k-structure in V_{O}. Let ∇^{0} be the corresponding connection, so ∇ = ∇^{0} + A_{t} + A_{x}. Then A_{t} = fdt/t^{m} +hdt/t where f ∈ O^{×}, h = h_{0} +h_{1}t+.., h_{i} ∈ K′ ⊗EndV_{k}, h_{0} is nilpotent, and A_{x} = a_{q}t^{q} + a_{q+1}t^{q+1} + .., a_{i} ∈ Ω_{K′/k}^{1} ⊗ EndV_{k}, a_{q} ≠ 0. We want to show that q ≤ −m. Indeed, if q ≤ −m then the integrability condition implies that qa_{q} + [h_{0}, a_{q}] = 0 which contradicts nilpotency of h_{0}.

5.8. Lemma. There is an isomorphism of super lines with connection

(5.8.1)  E(F, V) →^{∼} E(F, L)^{⊗n}

where L is the F -line with an absolute connection defined above.

Proof. A straightforward comparison of formula 5.6 for both parts of (5.7.1) (notice that the term Tr(h_{0}a_{1−m}) for l.h.s. vanishes since a_{1−m} is a scalar matrix^{108} and h_{0} is nilpotent). □

5.9. A general digression. In the next subsection we will show that computation of E_{ν} for arbitrary (V_{F} , ∇) can be reduced, in principle, to the situation of 5.7. Before doing this let us list some (well-known) general properties of connections on F = K((t)). For a K-relative connection (V, ∇_{/K}) we denote by H_{dR}^{·}(V) the de Rham cohomology.

(a) For every (V, ∇_{/K}) as above H_{dR}^{0}(V) is the space of horizontal morphisms F → V, and H_{dR}^{1}(V) is dual to the space of horizontal morphisms V → F.

Some immediate corollaries:

(i) If (V, ∇_{/K}) is a non-trivial irreducible relative connection then H_{dR}^{·}(V) = 0. For the trivial connection F = (F, ∇_{/K}^{0}) one has H_{dR}^{0}(F) = H_{dR}^{1}(F) = F.

(ii) For arbitrary (V, ∇_{/K}) the Euler characteristic χ_{dR}(V) vanishes. (iii) Since Ext^{·}(V, V′) = H_{dR}^{·}(Hom(V, V′)) we see that every inde-

composable^{109} V = (V, ∇_{/K}) is isomorphic to V^{irr} ⊗ F^{(n)} where V^{irr} is an irreducible relative connection and F^{(n)} is the nilpotent Jordan block of length n.^{110}

108Indeed, one has [h_{0}, a_{1−m}] = (m − 1)a_{1−m} + d_{x}f_{0} due to integrability. Since h_{0} is nilpotent this implies that a_{1−m} = (1 − m)^{−1}d_{x}f_{0}.

109i.e., V cannot be represented as a direct sum of connections of smaller rank.

In particular, every V = (V, ∇_{/K}) admits a canonical decomposition by ∇_{/K}-isotypical components

(5.9.1)  V = ⊕V^{L}.

This decomposition is labeled by isomorphism classes of relative irreducible connections L and characterized by property that every irreducible subquotient of V^{L} is isomorphic to L.

(iv) The ℤ-graded super line det H_{dR}^{·}(V) is canonically trivialized.^{111} This trivialization is uniquely characterized by two properties: it is

compatible with exact sequences of V, and for irreducible V it comes from (i) above. Here is a construction.

By (iii) V splits canonically into a direct sum V = V^{un} ⊕V^{nun} where V^{un} is the ∇_{/K}-isotypical component of the trivial connection in V. By (i) one has H_{dR}^{·}(V^{un}) = H_{dR}^{·}(V).

There is a unique K[[t]]-sublattice V_{O}^{un} ⊂ V^{un} preserved by ∇(t∂_{t}) and such that the operator κ induced by ∇(t∂_{t}) on the fiber V_{0}^{un} := V_{O}^{un}/tV_{O}^{un} is nilpotent. The de Rham complex V^{un} → ω ⊗ V^{un} is quasiisomorphic to its logarithmic subcomplex^{112} V_{O}^{un} → ω_{O log} ⊗ V_{O}^{un}. The latter projects quasiisomorphically^{113} to the complex V_{0}^{un} →^{κ} V_{0}^{un} whose determinant is det V_{0}^{un}/ det V_{0}^{un} = K. This is our trivialization.

Remark. If ∇_{/K} comes from an absolute connection then our trivialization is horizontal with respect to the Gauß-Manin connection.

(b) Let F′/F be a finite extension, V_{F} = (V_{F} , ∇_{/K}) a relative connection on F.

(i) V_{F} is semi-simple if and only if its pull-back V_{F′} to F′ is semisimple.

Indeed, to see that the pull-back of an irreducible V_{F} is semi-simple notice that the maximal semi-simple subobject of V_{F′} descends to a subobject of V_{F}, hence it equals V_{F′}. The rest follows from (a)(iii).

(ii) If V_{F} is irreducible and V_{F′} is a direct sum of several copies of the same rank 1 connection L_{F′} then V_{F} has rank 1.

110i.e., F^{(n)} is the connection ∇_{/K} on F^{n} with ∇_{/K} − ∇_{/K}^{0} = dt/t· the nilpotent Jordan block of length n.

111The argument below works also in case when our base ring is an arbitrary Artinian algebra.

112Here ω_{O log} = t^{−1}K[[t]]dt ⊂ ω.

113By the projections V_{O}^{un} → V_{0}^{un}, Res : ω_{O} → K.

Indeed, the class of L_{F′} is Gal(F′/F)-invariant, hence L_{F′} comes from a line with connection L_{F} on F.^{114} Replacing V_{F} by V_{F} ⊗ L_{F}^{−1} we are reduced to the situation when L_{F′} is a trivial connection. Then V_{F} has regular singularities and all eigenvalues of ∇_{/K}(t∂_{t}) are in ℚ. Every irreducible connection with these properties has rank 1.

(c) If (V_{F} , ∇_{/K}) is irreducible then one can find a finite extension F′/F such that the pull-back V_{F′} of V_{F} to F′ is isomorphic to a direct sum of connections of rank 1.

Indeed, we have the desired decomposition over F′′ = K̄((t^{1/e})), where K̄ is an algebraic closure of K, due to the Levelt-Turritin theorem (see e.g. [M] III(1.2)) and (b)(i). It is defined over some F′ = K′((t^{1/e})), K′ a finite extension of K, for the following reason. First, the rank 1 connections L_{F′′} that occur in the Levelt-Turritin decomposition are defined over some F′ as above. Since F′′ ⊗ Hom(L_{F′}, V_{F′}) →^{∼}

Hom(L_{F′′}, V_{F′′}), one has ⊕ L_{F′} ⊗ Hom(L_{F′}, V_{F′}) →^{∼} V_{F′}.

(d) Now let V_{F} be an F -vector space equipped with an absolute integrable connection ∇; denote by ∇_{/K} the corresponding relative connection. Let V_{F} = ⊕V_{F}^{α} be a decomposition which is ∇_{/K}-horizontal. Assume that for every V_{F}^{α} ≠ V_{F}^{α′} there is no non-trivial ∇_{/K}-horizontal morphisms V_{F}^{α} → V_{F}^{α′}. Then our decomposition is ∇-horizontal.^{115}

Therefore for every (V_{F} , ∇) the decomposition (5.9.1) by ∇_{/K}-isotypical components is ∇-horizontal. Thus every indecomposable (V_{F} , ∇) is ∇_{/K}-isotypical.

5.10. A reduction to the rank 1 case. We use the notation from 5.1, so F = K′((t)), etc.

Let us show that computation of the ε-connection for arbitrary (V_{F} , ∇) can be reduced, in principle, to the situation considered in 5.7, hence, by (5.8.1), to the rank 1 situation. Due to the multiplicativity property of E we can, and will, assume that (V_{F} , ∇) is indecomposable. Furthemore, we assume that ∇ does not have regular singularities (as a relative connection, see 5.3).

Proposition. For such (V_{F} , ∇) there exists a finite extension F′/F such that (V_{F} , ∇) is isomorphic to the push-forward of a connection (V′_{F′}, ∇′) on Spec F′ of type considered in 5.7.

114Isomorphism classes of lines with connection on F′ form the group ω(F′)/d log(F′^{×}) = (ω(F′)/ω(O′))/ℤ. Every Galois invariant there lifts to the one in ω(F′), i.e., comes from ω(F).

115Indeed, for every horizontal vector field θ the (α, α′)-component of ∇(θ) is a ∇_{/K}-horizontal morphism V_{F}^{α} → V_{F}^{α′}.

One has E(F, V_{F})_{ν} = E(F′, V′_{F′})_{ν} by (4.9.4); this is the promised reduction.

Proof. Below, as in 5.9, we denote by ∇_{/K} the relative connection that corresponds to ∇ (the vertical part of ∇).

By 5.9 (d) (V, ∇_{_{/}K}) is ∇_{/K}-isotypical. By 5.9 (c) applied to the irreducible constituent of (V, ∇_{/K}) there is a finite Galois covering F′′/F such that every irreducible subqoutient of (V_{F′′}, ∇_{/K}) has rank 1. Consider the ∇_{/K}-isotypical decomposition V_{F′′} = ⊕V_{F′′}^{L}; its components are labeled by irreducible L = (L_{F′′}, ∇_{/K})’s of rank 1.

The Galois group Gal(F′′/F) action on V_{F′′} permutes components V_{F′′}^{L}. The action on the set of components is transitive since V_{F} is indecomposable.

Pick one of L’s that occur in our decomposition; denote V_{F′′}^{L} by V′_{F′′} and its connection by ∇′. Let F′ ⊂ F′′ be the invariant field of the stabilizer of V_{F′′}^{L} in Gal(F′′/F). We get an F′-vector space V′_{F′} := (V′_{F′′})^{Gal(F′′/F′)} with connection ∇′. Its push-forward to F is identified with (V_{F} , ∇) in the obvious way. Thus (V′_{F′}, ∇′) is indecomposable, hence ∇_{/K}-isotypical. By 5.9 (b)(ii) applied to the irreducible constituent of (V′_{F′}, ∇′_{/K}) our (V′_{F′}, ∇′) is of type considered in 5.7. We are done. □

References

[ACK] E. Arbarello, C. De Concini, V. Kac. The infinite wedge representation and the reciprocity law for algebraic curves. Theta Functions – Bowdoin 1987,Proc. Symp. Pure Math. 49, Part 1, 171-190, AMS (1989)

[Ba] H. Bass. Big projective modules are free. Illinois J. of Math., 7 (1963), 24-31.

[BD] A. Beilinson, V. Drinfeld. Quantization of Hitchin’s integrable system and hecke eigensheaves. Unposted preprint (1998).

[BS] A. Beilinson, V. Schechtman. Determinant bundles and Virasoro algebras. Commun. in Math. Phys. 118 (1988), 651-701.

[BDM] P. Deligne, J. Morgan. Notes on supersymmetry (following Joseph Bernstein). Quantum fields and strings: a course for mathematicians, part 1, AMS (1999), 41–98.

[BE1] S. Bloch, H. Esnault. A Riemann-Roch theorem for flat bundles, with values in the algebraic Chern-Simons theory. Annals of Math. 151 (2000), 1025-1070.

[BE2] S. Bloch, H. Esnault. Gauß-Manin determinants for rank 1 irregular connections on curves, Math. Ann. 321 (2001), 15-87, with an addendum: the letter of P. Deligne to J.-P. Serre (Feb. 1974) on ϵ-factors, 65-87.

[BE3] S. Bloch, H. Esnault. A formula for Gauß-Manin determinants. math.AG/0004079, 51 pages.

[C] C. E. Contou-Carrère. Jacobienne locale, groupe de bivecteurs de Witt universel et symbole local modéré. C. R. Acad. Sci. Paris, t. 318, Série I (1994), 743-746.

[DKJM] E. Date, M. Kashiwara, M. Jimbo, T. Miwa. Transformation groups for soliton equations. Non-linear Integrable Systems - Classical Theory and Quantum Theory (Kyoto 1981), 39-119, World Sci. Publishing, Singapore, 1983.

[D1] P. Deligne. Équations Différentielles à Points Singuliers Réguliers. Lecture Notes in Mathematics 163 (1970), Springer Verlag.

[D2] P. Deligne. Les constantes des equations fonctionelles des fonctions L, Lecture Notes in Mathematics 349 (1973), 501-597, Springer Verlag.

[D3] P. Deligne. Le symbole modéré, Publications Mathématiques de l’IHES, 73 (1991), 147-181.

[Dr] V. Drinfeld. Almost projective modules and families of Tate spaces. Preprint (1999)

[GK] N. Katz. Local-to global extensions of representations of fundamental groups. Ann. Inst. Fourier (Grenoble). 36 (1986), no.4, 69-106.

[Gr] A. Grothendieck. Crystals and the De Rham cohomology of schemes, in Dix Exposés sur la Cohomologie des Schémas. North-Holland, 1968, 306-358.

[K] I. Kaplansky. Projective modules, Annals of Math. 68 (1958), 372-377. Reprinted in Selected Papers by I. Kaplansky (Springer-Verlag, 1995).

[KM] F. Knudsen, D. Mumford. The projectivity of the moduli space of stable curves. I. Preliminaries on ”det” and ”Div”. Math. Scand. 39 (1976), no. 1, 19–55.

[L] G. Laumon. Transformation de Fourier, constantes d’équations fonctionnelles, et conjecture de Weil, Publications Mathématiques de l’IHES 65 (1987), 131-210.

[M] B. Malgrange. Equations Différentielles à Coefficients Polynomiaux. Progress in Math. 96, Birkhäuser, Boston, 1991.

[Mu] D. Mumford. Lectures on Curves on an Algebraic Surface. Annals of Math. Studies 59 (1966), Princeton University Press.

[PS] A. Pressley, G. Segal. Loop Groups. Oxford Mathematical Monographs, (1986).

[SW] G. Segal, G. Wilson. Loop groups and equations of KdV type. Publ. IHES 61 (1985), 5-65.

[SGA 4] Théorie des topos et cohomologie étale de schémas, t. III. Lecture Notes

in Mathematics 305 (1972), Springer Verlag.

[T1] J. Tate. Fourier analysis in number fields and Hecke’s zeta functions. in: Algebraic Number Theory, edited by Cassels and Fröhlich, Academic Press, (1967).

[T2] J. Tate. Residues of differentials on curves. Ann. Sci. École Norm. Sup., sér. 4, 1 (1968), 149–159.

Dept. of Mathematics, University of Chicago, Chicago, IL 60637, USA

E-mail address: sasha@math.uchicago.edu

Dept. of Mathematics, University of Chicago, Chicago, IL 60637, USA

E-mail address: bloch@math.uchicago.edu

Universität Essen, FB6, Mathematik, 45117 Essen, Germany

E-mail address: esnault@uni-essen.de