LECTURES ON MIXED MOTIVES

SPENCER BLOCH

0. Introduction

0.1. What follows are notes on lectures I gave on mixed motives at the AMS summer conference in Santa Cruz. My experience with these lectures suggests that motives are like onions; they are complicated, multi-layered objects, and any attempt to cut too quickly to the heart of the matter can leave the audience in tears. A rigorous construction of The category of motives seems to founder on the rocks of Grothendieck’s currently inaccessible “standard conjectures” about algebraic cycles [29]. (For surprising progress, however, see [28].) Instead of precision and realism, we have to settle for a sort of cubist approach which attempts to see the subject from many different viewpoints simultaneously. Here is a casual list of some properties one expects for the category of motives. For a more careful discussion, the reader should see [35], [30], and [21]:

0.1.1. Algebraic varieties give motives: X ↦ h(X). More precisely, one fixes a ground field F and one deals with motives associated to varieties over F. The category of motives is a ℚ-linear abelian tensor category, and

h(X × Y) = h(X) ⊗ h(Y)

0.1.2. When X and Y are smooth and projective, a morphism h(X) → h(Y) is given by an algebraic cycle on X × Y of dimension dim Y. Rationally equivalent cycles give the same morphism. The question of what the appropriate equivalence relation on cycles is to get a good category of motives is subtle, and we shall avoid it. Roughly speaking, one can take a fine equivalence relation like rational equivalence, leading to the theory of Chow motives, or one can take a coarser relation like homological or numerical equivalence. Whatever equivalence relation is chosen, one introduces for X smooth, projective, and p on X ×X a projector (i.e. a cycle satisfying p^{2} = p in the correspondence ring), a formal image motive h(X, p). In particular, one of the standard conjectures (known for varieties over a finite field, [29]) says the diagonal Δ_{X} for a smooth, projective variety X is homologically equivalent to a sum of projectors (Kunneth decomposition). The resulting motivic decomposition is written

1991 Mathematics Subject Classification. 14C25, 14F99; Secondary 11G40.

Key words and phrases. motives, algebraic cycles, polylogarithms.

The author is supported in part by the NSF.

h(X) = ⊕_{i=0}^{2 dim X}h^{i}(X)

0.1.3. The motive of a point h^{0}(pt.) is written ℚ(0), and the Tate motive ℚ(−1) is defined to be h^{2}(ℙ^{1}). In other words, ℚ(−1) = h(ℙ^{1}, ℙ^{1} × pt.) One introduces a formal tensor inverse ℚ(1) such that ℚ(−1) ⊗ ℚ(1) ≅ ℚ(0). One then has ℚ(n) defined for all n ∈ ℤ by tensoring. We write

h(X)(n) := h(X) ⊗ ℚ(n).

0.1.4. Motives have realizations like H_{Betti} (Betti realization), H_{DR} (de Rham realization), H_{ét} (étale realization), and H_{crys} (H_{cris} in French) (Crystalline (French cristalline) realization). Realizations are functors from the category of motives to a category of vector spaces.

0.1.5. For X smooth, projective over a field F, h^{i}(X) is a pure motive of weight i. The realization functors preserve weights. The category of pure motives is conjectured to be semi-simple (see [28]). More generally, one expects a larger abelian category of mixed motives. Mixed motives M have increasing weight filtrations W▪M which are preserved by the realization functors. The category of mixed motives itself should be the core (heart) of a larger triangulated category D(F), the derived category of mixed motives over F.

0.1.6. More generally, for any base scheme S, one should define the derived category D(S) of motivic sheaves over S. As in the case of sheaves, given p : X → S, one expects various functors relating D(X) and D(S). For p : X → Spec (F) smooth and projective, we should have (cf. [16])

Rp_{∗}ℚ(0)_{X} ≅ ⨁ h^{i}(X)[−i].

Examples of motivic sheaves are the polylogarithm sheaves in [4] and [5], and the elliptic polylogarithm sheaves in [6]. It would be very interesting to understand the relationship between the Chow polylogarithms of Goncharov, to be discussed below, and motivic sheaves on the Chow variety.

0.1.7. The category of mixed Tate motives or mixed Tate motivic sheaves is the full subcategory of motives over S generated by the ℚ(n)_{S}. A very beautiful observation of Beilinson is that the functor M ↦ ⊕Hom(ℚ(n), gr_{−2n}^{W}M) is a fibre functor from the category of mixed Tate motives to the category of ℤ-graded vector spaces. This means that one may hope to write the category of mixed Tate motives as the category of finite dimensional graded representations of a graded Lie algebra. One candidate for such a Lie algebra will be discussed below in section 2. One may hope also that this point of view will lead to new insights about motives, just as galois groups give a powerful tool to understand number fields.

0.1.8. Let X be a smooth, projective variety over a field F, and let Z be a codimension r algebraic cycle on X. The pullback Z×(ℙ^{1})^{r} on X× (ℙ^{1})^{r} has dimension equal dim X and so induces a morphism of motives. It is straightforward to check this factors through the projector defining h^{2}(ℙ^{1}), so Z defines a map ℚ(−r) → h(X). Applying a Kunneth projector leads to a cycle class in Hom(ℚ, h^{2r}(X)(r)). With notation as in 0.1.6, let CH^{r}(X) denote the Chow group of codimension r algebraic cycles modulo rational equivalence. One expects a map (isomorphism?)

CH^{r}(X) ⊗ ℚ → Hom_{D(F)}(ℚ, Rp_{∗}ℚ(r)_{X}[2r])

and hence a filtration F^{∗}CH^{r}(X) ⊗ ℚ with

gr_{F}^{i} CH^{r}(X) ⊗ ℚ ≅^{?} Ext_{D(F)}^{i}(ℚ, h^{2r−i}(X)(r))

For results on this filtration cf. [36], [28].

Algebraic cycles also give classes in other Ext groups. Namely, we may introduce “simplices” Δ^{n} := Spec (F [t_{0}, …, t_{n}]/(∑ t_{i} − 1)) and consider cycles of codimension r on X × Δ^{n} meeting all faces properly. One has a cosimplicial object Δ▪ : Δ^{0}\overrightarrow{→}Δ^{1} ⋯ and hence by pullback of cycles a simplicial abelian group Z^{r}(X, ▪) of codimension r cycles on X × Δ▪. One defines the higher Chow groups CH^{r}(X, ▪) := H▪(Z^{r}(X, ▪)). Known candidates for the derived category of motives include cycle maps

CH^{r}(X, n) → Ext_{D(F)}^{2r−n}(ℚ(0), Rp_{∗}ℚ_{X}(r))

If we define the motivic cohomology

(0.1.8.1) H_{M}^{i}(X, ℚ(j)) := Ext_{D(X)}^{i}(ℚ(0)_{X}, ℚ(j)_{X})

≅ Ext_{D(F)}^{i}(ℚ(0)_{F} , Rp_{∗}(ℚ(j)_{X})) we obtain maps

CH^{r}(X, n) ⊗ ℚ → H_{M}^{2r−n}(X, ℚ(r)) (0.1.8.2)

which one may hope are isomorphisms.

For motives M and N over a number field F, one expects

Ext_{D(F)}^{i}(M, N) = (0), i ≥ 2.

Since Hom(ℚ, h^{2r−n}(X)(r)) = (0) for n ≥ 1 because of the weights, we conclude in the number field case

H_{M}^{2r−n}(X, ℚ(r)) ≅ Ext^{1}(ℚ, h^{2r−n−1}(X)(r)). (0.1.8.3)

This suggests a natural interpretation of the map 0.1.8.2. Let Z be a codimension r cycle on X × Δ^{n} representing a class in CH^{r}(X, n). Let ∂ ⊂ Δ^{n} be the union of the faces t_{i} = 0. Mirroring the situation for étale or Betti realizations one may imagine a Gysin sequence of motives

(0.1.8.4) 0 → h^{2r−1}(X × Δ^{n}, X × ∂))(r) → h^{2r−1}(X × Δ^{n} − |Z|, X × ∂))(r)

→ h_{2 dim(Z)}(|Z|, |Z| ∩ ∂)(− dim(Z)) →^{a} h^{2r}(X × Δ^{n}, X × ∂))(r)

Because the cycle Z pulls back to zero on X × Δ^{n−1}, we should get a cycle map [Z] : ℚ(0) → h_{2 dim(Z)}(|Z|, |Z| ∩ ∂)(− dim(Z)). For weight reasons, the composition a ∘ [Z] = 0, so we may pull back 0.1.8.4 along [Z]. If we assume (again thinking of the familiar realizations) h^{2r−1}(X × Δ^{n}, X × ∂))(r) ≅ h^{2r−n−1}(X)(r) we get an extension

0 → h^{2r−n−1}(X)(r) → E_{Z} → ℚ(0) → 0 (0.1.8.5)

realizing the class 0.1.8.3 associated to Z.

0.1.9. Of course, motives are related to arithmetic in many ways. Classically, the motive of a variety over a finite field determines the characteristic polynomial of Frobenius acting on cohomology, which in turn gives the zeta function. From the point of view of mixed motives and motivic cohomology, which is central to these lectures, an important link is provided by the Beilinson conjectures. Let X be smooth and projective over a number field, and let 𝔛 be a regular model of X. Fix integers r, n ≥ 0 with n + 1 − 2r ≤ −2. The Beilinson conjectures equate the rank of H_{M}^{n+1}(𝔛, ℚ(r)) with the order of vanishing of the Hasse-Weil L-function L(H_{ét}^{n}(X\overline{ℚ}, ℚ_{ℓ}), s) at the point s = n + 1 − r. (The actual conjectures are much more precise. They also determine the order of vanishing for weights n + 1 − 2r = −1, 0, and they relate the first non-vanishing term in the Taylor expansion of L to the determinant of a regulator map on motivic cohomology.)

0.1.10. Finally, although we have treated the categories of motives as ℚ-linear categories, the fact that cycles and Chow groups are integrally defined suggests one may consider motives with ℤ/nℤ-coefficients.A good portion of this paper will be devoted to an exposition of the Suslin-Voevodsky theorem relating cycles mod n to étale cohomology.

0.2. The content of these lectures is as follows (by section): 1. The derived category of motives over a field.

2. The Tannakian category of mixed Tate motives over a field.

3. Motives with ℤ/nℤ-coefficients and étale cohomology.

4. Chow polylogarithms.

1. The derived category of mixed motives

1.1. My purpose in this section is to expose Voevodsky’s work constructing a candidate for the derived category of mixed motives over a field. The references are [44],[45], and [46]. They, as well as a number of other relevant papers by Voevodsky are currently available from the K-theory web site maintained by Dan Grayson at UIUC. Competing theories by Hanamura and Levine were not as well documented in the literature at the time of the lectures, though now details of both Levine’s and Hanamura’s work are available ([32],[33], [23], [24]). It seems plausible that all three candidates are equivalent, but this has not been checked. Note that the construction of the derived category does not depend on any conjectures. Even in the case of mixed Tate motives, the existence of an abelian subcategory forming a core for the derived category seems to depend on a conjecture of Beilinson and Soulé [1],[39] which basically says there is no motivic cohomology in negative degrees. In terms of Chow groups, the conjecture says that for X smooth and complete over a field,

CH^{p}(X, q) ⊗ ℚ = (0), 2p − q ≤ 0. (1.1.0.1)

The relation between this and motivic cohomology is give by 0.1.8.2.

1.2. Voevodsky’s constructions involve a considerable amount of abstract category theory. Although it will not be possible to give detailed justifications in this survey, I will try when I can to give references in the literature where the reader can find more detail on these points.

1.2.1. Let k be a perfect field. We first define an additive category SmCor(k) by taking as objects smooth schemes of finite type over k and as morphisms X → Y the free abelian group on closed subvarieties Z ⊂ X × Y such that pr_{1} : Z → X is finite and surjective onto a connected component of X. Roughly speaking, we may think of Hom(X, Y) as the group associated to the monoid of morphisms of schemes

X → ∐_{n} Sym^{n}(Y).

(When X is non-normal, this is not correct. For a precise statement, cf. [41].) Note for U → V a morphism of smooth schemes, the graph provides a morphism U → V in SmCor(k).

Remark 1.2.1. The alert reader will have remarked that the variance here is the opposit of that in the introduction; i.e. these are homological motives.

1.2.2. We define H^{b}(SmCor(k)) to be the homotopy category of bounded complexes of objects in SmCor(k). Recall a homotopy of complexes h : C → D is a collection of maps h^{n} : C^{n} → D^{n−1}. Maps

d_{D} ∘ h^{n} − h^{n+1} ∘ d_{C} : C → D

for a homotopy h are said to be homotopic to zero. The homotopy category of complexes of objects in an additive category is obtained from the category of complexes by identifying morphisms whose difference is homotopic to zero, [27].

A triangulated category is a category C with an invertible shift operator C ↦ C[1] (inverse C ↦ C[−1]) and a collection of diagrams of the form A → B → C → A[1] (distinguished triangles) satisfying certain axioms (op. cit.), the most important of which imply that for an object D of C and a distinguished triangle as above, one gets long exact sequences of abelian groups

… Hom(D, C[−1]) → Hom(D, A) → Hom(D, B) → Hom(D, C) → Hom(D, A[1]) → …

… Hom(A[1], D) → Hom(C, D) → Hom(B, D) → Hom(A, D) → Hom(C[−1], D) …

The standard example is the homotopy category of complexes (possibly with some boundedness condition) over an additive category. C[1] denotes the complex C shifted to the left one step, and distinguished triangles are diagrams isomorphic to a diagram of the form

C →^{f} D → Cone(f) → C[1].

Here Cone(f) denotes the complex

Cone^{n} := D^{n} ⊕ C^{n+1}

with differential ∂ : Cone^{n−1} → Cone^{n} given by the matrix

(\begin{matrix}−d_{C}^{n} & f^{n} \\ 0 & d_{D}^{n−1} \end{matrix}) .

1.2.3. The next step is to impose on H^{b}(SmCor(k)) some basic isomorphisms and long-exact sequences. Voevodsky does this by inverting certain morphisms in H^{b}(SmCor(k)). For X over k we write [X] for the corresponding object in H^{b}(SmCor(k)). He first inverts all morphisms [X × 𝔸^{1}] → [X] induced by projections. Next, given an open covering W = U ∪ V, one obtains in an obvious way a complex

[U ∩ V] →^{f} [U] ⊕ [V] →^{g} [W],

and hence a morphism γ : Cone(f) → [W]. Inverting γ forces distinguished triangles

[U ∩ V] →^{f} [U] ⊕ [V] →^{g} [W] → [U ∩ V][1].

The actual business of localizing categories with respect to sets of morphisms is quite technical. We refer the reader to [27] and [37] for more information. The category DM_{gm}^{eff}(k) is defined to be the Karoubian envelope of this localized category, i.e. DM_{gm}^{eff}(k) is obtained from the localized category by formally adjoining kernels and images of projectors (morphisms p such that p^{2} = p).

1.2.4. The association X ↦ [X] defines a functor from the category of smooth varieties over k to DM_{gm}^{eff}(k). The latter category has a tensor structure, and [X × Y] = [X] ⊗ [Y]. One defines the Tate object ℤ(1) to be the complex [ℙ_{k}^{1}] → [Spec (k)] placed in degrees 2 and 3. The category DM_{gm}(k) is defined by formally inverting the functor ? ⊗ ℤ(1) on DM_{gm}^{eff}(k). Thus, DM_{gm}(k) contains objects ℤ(n) for all n ∈ ℤ. To show we have not totally lost the way, one has for X and Y smooth projective, and assuming varieties over k admit resolutions of singularities, that

Hom_{DM_{gm}(k)}([X], [Y]) ≅ CH_{dim X}(X × Y).

In fact, assuming resolution of singularities the link between DM_{gm}(k) and algebraic cycles is exactly what one would expect. For example, the higher Chow groups define the Borel-Moore homology of the theory. That is, for X a variety of finite type which is equidimensional of dimension n, one defines a compactified object [X]^{c} in DM_{gm}(k), and

Hom(ℤ(i)[j], [X]^{c}) ≅ CH^{n−i}(X, j − 2i).

There are similar links to Suslin homology [42], motivic cohomology [7], and bivariant cycle homology [17].

1.2.5. Sheaf-theoretic methods play an important role in understanding the structure of DM_{gm}(k). For X a smooth variety over k, let L(X) be the functor of points in SmCor(k); i.e. for Y irreducible, L(X)(Y) is the free abelian group on closed subvarieties Z ⊂ Y ×X finite surjective over Y. Recall the Nisnevich topology on a variety X has as coverings étale coverings U → X such that for every field K, U(K) ↠ X(K). L(X) is a sheaf for any reasonable topology, but a crucial property of the Nisnevich topology (also true for the étale topology, but false for the Zariski topology) is the following ([46], prop. 3.1.3):

Proposition 1.2.2. Let U → X be a Nisnevich covering. Then the sequence of sheaves of abelian groups

… → L(U ×_{X} U) → L(U) → L(X) → 0

is exact.

A contravariant functor from SmCor(k) to abelian groups is called a presheaf with transfers. The subcategory of preseaves with transfers whose underlying presheaf is a Nisnevich sheaf is denoted

Shv_{Nis}(SmCor)(k).

For example, L(X) is an object in Shv_{Nis}(SmCor)(k). Let

D = D^{−}(Shv_{Nis}(SmCor)(k))

denote the derived category of bounded above complexes. We define a functor \underline{C}_{n} on presheaves with transfer by

\underline{C}_{n}(F)(X) := F (Δ^{n} × X),

where Δ^{n} is as in 0.1.8. This extends in an obvious way to a functor to complexes: F ↦ \underline{C}▪(F) and then to a functor

R\underline{C}▪ : D → D.

A presheaf with transfers F is said to be homotopy invariant if the morphism X × 𝔸^{1} → X induces an isomorphism

F (X) → F (X × 𝔸^{1}).

DM^{eff} is defined to be the full subcategory of D of objects whose homology sheaves are homotopy invariant. One shows that the functor R\underline{C}▪ factors through DM^{eff} ⊂ D, and one gets a diagram

H^{b}(SmCor(k)) ⟶^{L} D
↓  ↓R\underline{C}
DM_{gm}^{eff}(k)  −−→^{i} DM^{eff}(k)

The crucial point is the existence of the dashed arrow i. This embeds the motive category DM_{gm}^{eff}(k) (which should be thought of as very mysterious) as a full subcategory of the derived category of bounded above Nisnevich sheaves with transfer and homotopy invariant cohomology. What’s more, the image of i is big, in the sense that given any object A ∈ DM^{eff}(k) there exists a distinguished triangle A → B → C → A[1] such that B and C are direct sums of objects in the image of i.

For X a smooth variety over k, the composition \underline{C}▪(X) := RC∘L(X) is the Suslin complex [42]. One has for X and Y smooth schemes over k,

Hom_{DM_{gm}^{eff}(k)}([X], [Y][p]) ≅ H_{Nis}^{p}(X, \underline{C}▪(Y)).

2. The Tannakian Category of Mixed Tate Motives

2.1. In this section, I want to describe a program to construct the abelian category of mixed Tate motives over a field k. The work in question (in large measure joint with I. Kriz) is published in [9] and [11], but I should also acknowledge considerable inspiration from private conversations with A. Beilinson and P. Deligne. A general references for tannakian categories is [15].

2.1.1. What we are after is a category of what might be thought of as motivic sheaves over Spec (k). We consider only those sheaves which admit a weight filtration such that the successive quotients are (direct sums of) Tate objects ℚ(n). Our objects will have realizations, including ℓ-adic galois representations for ℓ prime to the characteristic of k, and mixed Tate Hodge structures if k ↪ ℂ. Presumably, when k is a function field over ℂ, one has a realization as a variation of mixed Tate Hodge structure, but that theory has not been worked out.

2.1.2. Our category will be the abelian category of corepresentations of a certain Hopf algebra H := H^{0}(B(𝔑▪)). Here 𝔑▪ is a differential graded algebra built from algebraic cycles on Δ▪_{k} and B denotes the bar (or cobar, depending on your point of view) construction. The fact that these corepresentations have realizations stems from the existence of algebraic cycle classes in ℓ-adic and Betti cohomology. It is very hard (we tried!) to envision a procedure for generating (open, relative,simplicial,...) varieties whose cohomology groups might give all mixed Tate motives. This seemingly chaotic problem is reinterpreted via the Tannakian formalism as a problem of constructing a pro-algebraic group over ℚ. Because of the weight filtration, this proalgebraic group will be unipotent, so the problem can be linearized.

Our Hopf algebra is essentially the coordinate ring of (our candidate for) this group.

2.1.3. This program for constructing motives can be criticized on several grounds. First, we only consider mixed Tate motives. This is not so serious. In fact, the weight grading functor from mixed motives to pure motives should have the properties of a fibre functor (basically, exactness, compatibility with ⊗ and faithfulness) so one may try to construct a Lie algebra or Hopf algebra in the category of pure motives whose representations give all mixed motives.

Second, and more seriously, the construction of an abelian category of motives seems to be deeper than the construction of a triangulated “derived category”. We are force to assume a deep conjecture that our cycle complex 𝔑▪ is a K(π, 1) in the sense of Sullivan. This is necessary to insure that the motivic cohomology (which coincides with the cohomology of 𝔑▪) agrees with the cohomology of the abelian category of corepresentations we construct.

Finally, and in some sense most serious of all, is the question whether our methods construct “all” mixed Tate motives. For example, in our category, Ext^{1}(ℚ, ℚ(1)) = k^{×}, i.e. all extensions of ℚ by ℚ(1) are Kummer. However, in the real world, one has the following construction. Let V be a smooth, projective variety of dimension 3 over k. Let A = ∑ a_{i}A_{i} and B = ∑ b_{j}B_{j} be algebraic 1-cycles on V. Assume A and B are homologous to 0 and have disjoint support, |A| ∩ |B| = ∅. It is an easy exercise in ℓ-adic or Betti cohomology to construct a subquotient H^{3}(X − A, B) (depending on the coefficients a_{i} and b_{j}) of H^{3}(X − |A|, |B|) whose weight graded pieces are ℚ(1), H^{3}(V)(2), and ℚ. It seems reasonable to interpret H^{3}(X − A, B) as a motive. If H^{3}(V) = (0), the middle piece disappears, and we get an extension of ℚ by ℚ(1). If either A or B is algebraically equivalent to 0, one can show this extension is Kummer in the sense that there exists x ∈ k^{×} such that the image of x in the ℓ-adic completion of k^{×} classifies the extension for all ℓ (resp. the image of x ∈ ℂ^{×} classifies the extension of Hodge structures if k ⊂ ℂ). However, if neither A nor B is algebraically equivalent to 0, I know of no way of proving this.

On the other hand, the Tannakian viewpoint is enormously powerful. Think of the central role of Galois groups in number theory or fundamental groups in topology!

2.2. We want to construct a category

𝔐𝔗𝔐 = 𝔐𝔗𝔐(k)

with the following properties 1. 𝔐𝔗𝔐 is an abelian category.

2. Objects in 𝔐𝔗𝔐 have finite length. The simple objects are objects ℚ(n), for n ∈ ℤ.

3. Hom(ℚ(m), ℚ(n)) = \begin{cases}ℚ & m = n \\ 0 & m ≠ n \end{cases}.

4. Objects M have functorial increasing weight filtrations W▪M. Thus, ℚ(n) has weight −2n, and in general, gr_{2n}^{W}M ≅ ⊕_{finite}ℚ(−n).

The beautiful remark of Beilinson is that, given such a category, one gets a fibre functor

ϕ : 𝔐𝔗𝔐 → {ℤ-Graded ℚ- Vector Spaces}

by defining

ϕ_{−n}(M) := Hom(ℚ(n), gr_{−2n}^{W}M).

(It would be better to double the index and call this ϕ_{−2n}, but we will follow established usage. The reader should bear in mind that this grading never involves signs.) For example if M has a two step filtration 0 → ℚ(1) → M → ℚ → 0, ϕ(M) = ℚ_{−1} ⊕ ℚ_{0}.

One then deduces from Tannaka theory [2], [15] the existence of a graded pro-Lie algebra over ℚ,

L = L_{−1} ⊕ L_{−2} ⊕ ⋯

and an identification

𝔐𝔗𝔐 = {finite dim. graded ℚ-representations of L}.

The grading corresponds to the weights and will be referred to as the Adams grading to distinguish it from the cohomological grading.

2.3. Examples. We assume for a while we have the category 𝔐𝔗𝔐 and see what can be said about L. We expect, by Kummer theory,

Ext_{𝔐𝔗𝔐}^{1}(ℚ, ℚ(1)) ≅ k^{×} ⊗ ℚ. (2.3.0.1)

Such an extension corresponds under the fibre functor as above to a representation

L → End(ℚ_{−1} ⊕ ℚ_{0}), ℓ ↦ (\begin{matrix}0 & ρ(ℓ) \\ 0 & 0 \end{matrix}) .

Here ρ is a homomorphism of graded Lie algebras

L → ℚ_{−1} = Hom(ℚ, ℚ_{−1}),

i.e. an element of degree +1 in the linear dual space to L. Note that [L, L] ⊂ ⊕_{n≤−2}L_{n}, so any element of degree +1 in the dual will be a homomorphism of Lie algebras with target ℚ with the trivial Lie structure. Write M := L^{∨} (dual as pro-object), so (2.3.0.1) implies

M_{1} ≅ k^{×} ⊗ ℚ.

Before moving on, it is perhaps worth saying a few words about realizations in this simple example. There is a simple dictionary between extensions of ℤ by a group object G in a category and G-torseurs. Given an extension, 0 → G → E →^{ϕ} ℤ → 0, (the object represented by) ϕ^{−1}(1) is a G-torseur. Conversely, given a G-torseur T, one easily constructs an extension of ℤ such that T is identified with the inverse image of 1. Thus, to realize an extension of ℚ by ℚ(1) corresponding to an element x ∈ k^{×}, it suffices to realize the corresponding torseur T_{x} := ϕ_{x}^{−1}(1). These realizations are the obvious ones:

Hodge realization T_{x} := {2πin log(x) | n ∈ ℤ}
ℓ-adic realization T_{x} := {(x = x_{1}, x_{2}, …) | x_{n}^{ℓ} = x_{n−1}}.

Notice the Hodge realization is a torseur under 2πiℤ and the ℓ-adic realization is a torseur under ℤ_{ℓ}(1). To obtain vector spaces, one simply tensors with ℚ.

2.3.1. Consider next a 3-dimensional representation V = ℚ_{0} ⊕ ℚ_{−1} ⊕ ℚ_{−2} of L given by a matrix

ρ := (\begin{matrix}0 & x & z \\ 0 & 0 & y \\ 0 & 0 & 0 \end{matrix})

with x, y ∈ M_{1} and z ∈ M_{2}. The dual of the Lie algebra bracket [ ] : L ∧ L → L gives a graded map

d : M → ∧^{2}M.

We extend d to a map d : ⋀▪ M → ⋀^{▪+1} M by d(x_{1} ∧ … ∧ x_{n}) = ∑(−1)^{i}x_{1} ∧ … x̂_{i} ∧ … x_{n}. The dual of the Jacobi identity is the statement d^{2} = 0, so ⋀▪ M becomes a differential graded algebra.

For a, b ∈ L the condition that ρ be a representation of Lie algebras is

[ρ(a), ρ(b)] = ρ([a, b]),

where the bracket on the left is the commutator bracket in End(V). Both sides are zero unless a and b have degree −1. In this case, the identity to be checked is

z([a, b]) = x(a)y(b) − x(b)y(a)

i.e.

(x ∧ y)(a ∧ b) = dz(a ∧ b)

i.e.

(2.3.1.1)  dz = x ∧ y.

2.3.2. The cohomology of the Lie algebra L is given by the cohomology of the complex (∧▪M, d) (with ∧^{i} in degree i) and is supposed to be the motivic cohomology H_{M}^{q}(Spec (k), ℚ(p)). The latter has been conjecturally identified by Beilinson with the γ-graded pieces of algebraic K-theory. In sum, we expect

H^{q}(^{⋀▪} M, d)_{deg. p} = H_{M}^{q}(Spec (k), ℚ(p)) ≅ gr_{γ}^{p}K_{2p−q}(k) ⊗ ℚ.

In particular, since gr_{γ}^{2}K_{2}(k) ⊗ ℚ = K_{2}(k) ⊗ ℚ, we look for an exact sequence

M_{2} →^{d} ⋀^{2} M_{1} →^{s} K_{2}(k) ⊗ ℚ → 0.

The map labeled s should be the symbol map. In particular, for x ∈ k^{×} −1 one has the Steinberg relation s(x∧(1−x)) = 0, so there should be an element

θ_{2}(x) ∈ M_{2}; d(θ_{2}(x)) = x ∧ (1 − x) (2.3.2.1)

As a consequence of (2.3.1.1) and (2.3.2.1), the Steinberg relation in K_{2} should correspond to a family of mixed Tate motives of dimension 3 parametrized by ℙ^{1}−{0, 1, ∞}, with weight graded pieces ℚ, ℚ(1), ℚ(2) and underlying Kummer extensions of ℚ by ℚ(1) and ℚ(1) by ℚ(2) given by x and 1 − x respectively. Here x is the standard parameter on ℙ^{1}. These are the dilogarithm motives. More generally, one looks for elements θ_{n}(x) ∈ M_{n} satisfying

dθ_{n}(x) = θ_{n−1}(x) ⊗ x. (2.3.2.2)

The corresponding polylogarithm motives have been constructed [5],[9]. They are intimately related to the Zagier conjecture about values of zeta functions of number fields [47].

2.4. Construction. Here is an outline of the construction of L proposed in [11]. We construct an augmented Hopf algebra χ_{mot} over ℚ, and M := I/I^{2} where I ⊂ χ is the augmentation ideal. Note that M and χ_{mot} should be graded (Adams grading, cf. 2.2). The construction is in several steps:

1. Using algebraic cycles we construct an augmented, graded, differential graded algebra over ℚ

A▪ = ⊕_{n≥0}A▪(n) (2.4.0.3)

… → A^{−1} → A^{0} → A^{1} → … (2.4.0.4)

0 → A^{+} → A → ℚ → 0 (2.4.0.5)

A^{p}(m) ⊗ A^{q}(n) → A^{p+q}(m + n) (2.4.0.6)

2. We define the bar construction B(A▪).

3. We define χ_{mot} := H^{0}(B(A)).

2.4.1. Construction of A. Write □^{1} = ℙ_{k}^{1} − {1}, □^{n} = (□^{1})^{n}. Each □^{n} has faces of all codimensions defined by setting various coordinates = 0, ∞. We define groups

G_{n} := S_{n} × {±1}^{n} semi-direct product

where S_{n} is the symmetric group, acting by permutation on {±1}^{n}. The group G_{n} acts on □^{n} with S_{n} acting by permutation of coordinates and the i-th {±1} acting by sending the i-th coodinate function to its inverse. Also, we have a homomorphism

Alt : G_{n} → {±1}

which is the usual alternating map on S_{n} and the non-trivial character on each factor {±1}. The group A^{i}(r) is defined by applying the Altprojector to the ℚ-vector space spanned by all irreducible, codimension r closed subvarieties V ⊂ □^{2r−i} such that V meets properly (i.e. in the correct codimension) all faces of □^{2r−i}. For example A^{1}(2) is spanned by irreducible curves in □^{3} not meeting any of the loci where two coordinates are not invertible and not contained in the set where any single coordinate is not invertible.

One has face maps δ_{j}^{k} : □^{2r−i} → □^{2r−(i−1)} for 1 ≤ j ≤ 2r − i + 1 and k = 0, ∞. The alternating sum of the resulting pullback maps on cycles induces a map

∂ : A^{i−1}(r) → A^{i}(r).

As usual, ∂^{2} = 0, so ⊕_{r≥0}A▪(r) becomes a graded complex. We have

□^{2r−i} × □^{2s−j} = □^{2(r+s)−(i+j)}

The resulting exterior product on cycles Z, W ↦ Z×W composed with the Alt projection gives ⊕_{r≥0}A▪(r) the structure of a graded differential graded algebra,

A^{i}(r) ⊗ A^{j}(s) ↦ A^{i+j}(r + s).

The augmentation A▪ → ℚ is zero on A^{i}(r) unless i = r = 0 (cod. 0 cycles on □^{0} = Spec (k)), in which case it is the obvious isomorphism. The kernel of the augmentation is denoted A^{+} = ⊕A^{+i}.

Concerning A▪, the main result is [7]

H^{q}(A▪(r)) ≅ gr_{γ}^{r}K_{2r−q}(k) ⊗ ℚ := H_{M}^{q}(Spec (k), ℚ(r)). (2.4.1.1)

Note that the complex A▪ is not connected, i.e. there are non-trivial terms in negative degrees.

Conjecture 2.4.1. (Soulé-Beilinson) H^{q}(A^{+▪}) = (0) for q ≤ 0.

The discussion below suggests a stronger conjecture:

Conjecture 2.4.2. The differential graded algebra A▪ is a K(π, 1) in the sense of Sullivan [40].

Recall Sullivan’s theory associates to a differential graded algebra A▪ another d.g.a. called the 1-minimal model

𝔐▪ →^{ϕ} A▪.

As an algebra,

𝔐▪ ≅ ∧▪𝔐^{1}.

As mentioned in 2.3.1 above, the dual 𝔏 of 𝔐^{1} inherits the structure of a (pro-)Lie algebra, with the bracket coming from the (pro-)dual of the differential d : 𝔐^{1} → ∧^{2}𝔐^{1}. When A▪ is the differential graded algebra of forms on a manifold, one recovers from 𝔐▪ the Lie algebra associated to the nilpotent quotient of the fundamental group. The algebra A▪ is said to be a K(π, 1) if it is quasi-isomorphic to its 1-minimal model. The idea is that in our case, the Lie algebra 𝔏 should be the Lie algebra of the Tannaka group of the category of mixed motives over k. For this to fit with established dogma (viz. the definition of motivic cohomology (2.4.1.1)) it should be the case that the Lie algebra cohomology of 𝔏 coincides with the cohomology of A▪. This is the content of 2.4.2.

2.4.2. Bar construction. We write A▪ for a (not necessarily reduced) augmented d.g.a. over ℚ as above. A^{+▪} ⊂ A▪ is the kernel of the augmentation. The bar complex B(A) is a double complex as indicated below:

⋮  ⋮  ⋮
∂↑  ...  ∂↑  ∂↑
(A^{+⊗3})^{2}  ⟶^{δ} (A^{+} ⊗ A^{+})^{2} ⟶^{δ} A^{2} ⟶ 0
∂↑  ∂↑  ...  ∂↑
(A^{+⊗3})^{1}  ⟶^{δ} (A^{+} ⊗ A^{+})^{1} ⟶^{δ} A^{1} ⟶ 0
∂↑  ∂↑  ∂↑ ...
(A^{+⊗3})^{0}  ⟶^{δ} (A^{+} ⊗ A^{+})^{0} ⟶^{δ} A^{+0} ⟶^{0} ℚ
∂↑  ∂↑  ∂↑
⋮  ⋮  ⋮

The complex is numbered so the diagonal indicated with dots represents total degree 0, and we define our Hopf algebra

χ_{mot} := H^{0}(B(A)).

Technically, our bar complex is the two-sided bar complex B(A) = B(ℚ, A, ℚ). One writes | rather than ⊗, and the differential δ is defined by (a_{i} ∈ A^{+})

δ[a_{1}|a_{2}| … |a_{r}] = ∑_{i=1}^{r−1}(−1)^{i}δ_{i}[a_{1}|a_{2}| … |a_{r}]

δ_{i}[a_{1}|a_{2}| … |a_{r}] := [a_{1}|a_{2}| … |a_{i} · a_{i+1}| … |a_{r}]

(The maps δ_{0} and δ_{r} are both zero in this case.) The map ∂ is defined in the obvious way extending the differential on A▪, and the total differential d is the sum with appropriate signs of δ and ∂. For more details see [11], §2.

Theorem 2.4.3. Let C be the category of finite dimensional graded co-represntations of χ_{mot}.

(i) Objects of C admit mixed Tate Hodge and ℓ-adic realizations. These arise from homomorphisms of Hopf algebras

χ_{mot} → χ_{ét} (resp. χ_{betti})

where χ on the right is the Hopf algebra associated to the Tannakian category of mixed Tate ℓ-adic (resp. Hodge) realizations. (In the ℓ-adic case, one should assume that k contains only finitely many ℓ-th power roots of 1 in order to control Hom(ℚ_{ℓ}(m), ℚ_{ℓ}(n)).) (ii) Ext_{C}^{1}(ℚ, ℚ(1)) ≅ k^{×} ⊗ ℚ.

(iii) C contains polylogarithm corepresentations corresponding to (2.3.2.2). The Hodge realizations of these corepresentations are the Hodge polylogarithms as given in [5].

Example 2.4.4. The dilogarithm corepresentation associated to a ∈ k − {0, 1} may be described as follows. Define

θ_{2}(a) = Alt({(t, 1 − t, 1 − at^{−1}) | t ∈ ℙ^{1}} ∩ □^{3}) ∈ A^{1}(2).

Then (θ_{2}(a), (a) ⊗ (1 − a)) ∈ A^{1}(2) ⊕ (A^{1}(1) ⊗ A^{1}(1)) is a cocycle representing a class Li_{2}(a) in χ_{mot} = H^{0}(B(A)). The comultiplication map ψ : χ_{mot} → χ_{mot} ⊗χ_{mot} (cf. [11], (2.1.4)) maps Li_{2}(a) to (a)∪(1− a). This is the condition to get a rank 3 corepresentation as desired.

3. Motives with ℤ/nℤ-coefficients and étale cohomology

3.1. A beautiful link between motivic cohomology and étale cohomology has been established by Suslin and Voevodsky [42], [43]. There are two main theorems. The first, Theorem 3.1.1 below, links étale cohomology to the derived category of mixed motives as constructed in §1. The second, Theorem 3.1.2, relates étale cohomology and higher Chow groups.

Let k be an algebraically closed field. Results in this section depend on resolution of singularities, so we will eventually need to assume Char(k) = 0. Recall we have defined 0.1.8 a cosimplicial variety Δ▪. For F : (Sch/k)^{0} → (ab) we have a complex 1.2.5

C▪(F), C_{n}(F) := F (Δ^{n}).

In particular, for F = L(X) as in 1.2.5, we have

C_{n}(X) := C_{n}(L(X)) = {free abel. grp. on Z ⊂ X × Δ^{n} | Z → Δ^{n} finite surjective}.

For X/S there is also a relative version C▪(X/S) generated by cycles Z such that Z → S × Δ^{n} is finite surjective. The algebraic singular homology of a variety X/k with ℤ/nℤ-coefficients is defined by

H_{∗}^{sing}(X, ℤ/nℤ) := H_{∗}(C▪(X) ⊗ ℤ/nℤ). (3.1.0.1)

Theorem 3.1.1. Let X be a variety over an algebraically closed field k of characteristic 0. Then

H_{∗}^{sing}(X, ℤ/nℤ) ≅ H_{∗}^{ét}(X, ℤ/nℤ).

One way to calculate the étale homology on the right is to embed k ↪ ℂ and compute the singular homology of X(ℂ). From this perspective, the isomorphism arises from a map (+ indicates group completion of the indicated algebraic or topological monoid)

C▪(X) ≅ Hom_{algebraic}(Δ▪, ∐_{d=0}^{∞} Sym^{d}(X))^{+} →

Hom_{topological}(Δ▪_{top.}, ∐_{d=0}^{∞} Sym^{d}(X)_{top.})^{+}

The higher chow groups with finite coefficients are given by

CH^{r}(X, n; ℤ/mℤ) := H_{n}(Z^{r}(X, ▪) ⊗ ℤ/mℤ) (3.1.0.2)

where Z^{r}(X, ▪) is defined in 0.1.8.

Theorem 3.1.2. Let X be an equi-dimensional quasiprojective variety over an algebraically closed field of characteristic 0. Let i ≥ d = dim X. Then

CH^{i}(X, n; ℤ/mℤ) ≅ H_{ét,c}^{2(d−i)+n}(X, ℤ/mℤ(d − i))^{∗}.

(The group on the right is the dual of the cohomology with compact supports.) When X is smooth, this can be rewritten

CH^{i}(X, n; ℤ/mℤ) ≅ H_{ét}^{2i−n}(X, ℤ/mℤ(i)).

Remark 3.1.3. (i) Note the essential stability hypothesis i ≥ dim X.

Without this, the theorem is false, even when n = 0. (ii) Although the proofs of these results require some delicate work with

exotic Grothendieck topologies, the results themselves are totally “motivic”. The singular homology (with finite coefficients) is calculated wholly in terms of algebraic cycles.

The remainder of this section is devoted to a sketch of the proof of theorem 3.1.1.

3.2. Rigidity. We begin with a simple algebraic singular homology calculation.

Proposition 3.2.1. Consider a diagram of schemes

\begin{aligned}X & ↪ & \overline{X} \\ ↘ & & ↙ \\ & S \end{aligned}

Assume X is a dense affine open subscheme of \overline{X} smooth over S with fibre dimension 1. Let Y = \overline{X} − X. Assume S is affine and normal, \overline{X} is normal and proper over S, and that there exists an affine open U ⊂ \overline{X} with Y ⊂ U. Define Pic(\overline{X}, Y) to be the group of line bundles on \overline{X} trivialized on Y. Then

H_{0}^{sing}(X/S) ≅ Pic(\overline{X}, Y)
H_{i}^{sing}(X/S) = (0); i > 0.

sketch of proof. One has by normality

Pic(\overline{X}, Y) ≅ Pic(\overline{X} × Δ^{n}, Y × Δ^{n}).

Consider the sheaf (1+I_{n})^{×} ⊂ O_{\overline{X}×Δ^{n}} of invertible functions ≡ 1 along Y × Δ^{n}. Let A▪ be the simplicial abelian group of global sections of (1 + I▪)^{×}. Let G▪ be the corresponding simplicial group of sections defined on some neighborhood of Y × Δ▪. One gets an exact sequence

0 → A▪ → G▪ → C▪(X/S) → Pic(\overline{X} × Δ▪, Y × Δ▪) → 0.

The proposition follows from an argument (omitted) showing G▪ is acyclic.

Recall we have the notion (1.2.5) of a homotopy-invariant presheaf with transfers Φ : (Sch/k)^{0} → (ab).

Theorem 3.2.2 (Rigidity). Let Φ be a homotopy-invariant presheaf with transfers. Let n be an integer prime to the characteristic of k such that n · Φ = (0). Let S = Spec(Λ), where Λ is the henselization of 𝔸_{k}^{ℓ} at the origin. Then Φ(S) ≅ Φ(k). (Here S is interpreted as a pro-scheme, S = \underleftarrow{lim}_{α}S_{α}, and Φ(S) := \underrightarrow{lim} Φ(S_{α}).)

Sketch of proof. The proof is by induction on ℓ, the case ℓ = 0 being trivial. Let T be the henselization at 0 of 𝔸^{ℓ−1}. S can be thought of as a limit over smooth T -pointed affine curves X over T. We consider the resulting fibre diagram:

\begin{matrix}X_{S} & → & X \\ ↓↑ & τ↗ & ↓↑ σ \\ S & →^{ν} & T. \end{matrix}

It will suffice to show the two resulting pullback maps

τ^{∗}, ν^{∗} ∘ σ^{∗} : Φ(X) → Φ(S) (3.2.0.3)

coincide.

Because Φ has transfers, we can define a map

C_{0}(X/T) ⊗ Φ(X) → Φ(T)

associating to a generator Z ⊂ X of C_{0}(X/T) the composition

Φ(X) → Φ(Z) →^{Tr} Φ(T)

Because Φ is homotopy invariant, this factors through a pairing

Pic(\overline{X}, \overline{X} − X) ⊗ Φ(X) ≅ H_{0}^{sing}(X/T) ⊗ Φ(X) → Φ(T).

Let a be a section of X/T. Then a represents a class [a] ∈ Pic(\overline{X}, \overline{X} − X), and the map induced by [a] from the above equals the pullback a^{∗} : Φ(X) → Φ(T). If a and b are sections of X/T which agree at the closed point of T, standard properties of Pic show that [a] − [b] is divisible by n, so we conclude

a^{∗} = b^{∗} : Φ(X) → Φ(T).

We apply this to X_{S}/S and the two sections S → X_{S} induced by τ and σ ∘ ν to get the desired equality (3.2.0.3).

3.3. Topologies. One of the really new ideas in the proof of theorem 3.1.1 involves the systematic use of very fine topologies. (So fine, in fact, that functors of points of representable objects are not sheaves.) A morphism f : Y → X is a topological epimorphism if U ⊂ X open ⟺ f^{−1}(U) ⊂ Y open.(Note this implies f is surjective.) f is a universal topological epimorphism if this condition continues to hold under pullback. The h-topology has as coverings all universal topological epimorphisms. Coverings for the qfh-topology are maps ϕ which are quasi-finite and h. A crucial property of qfhcovers Y → X of connected normal schemes X is that taking Y′ ⊂ Y to be the union of the irreducible components of Y which dominate X, Y′ → X is a qfh-cover. Thus the following is not a qfh-cover.

↓^{}

This means that a qfh-cover of a strictly henselian local ring admits as a refinement another strictly henselian local ring. It follows without much difficulty that writing

(Sch/k)_{qfh} →^{β} (Sch/k)_{ét}

for the morphism of sites, one has

(3.3.0.4) (ℤ/nℤ)_{ét} ≅ β_{∗}(ℤ/nℤ)_{qfh} ≅ Rβ_{∗}(ℤ/nℤ)_{qfh}

Example 3.3.1. Recall in section 1.2.5 we have defined L(X) to be the functor of points of a scheme X in the category SmCor(k). Let ℤ′ := ℤ[1/p] with p = Char(k). Define ℤ′[T] for a set T to be the free ℤ′-module on T. Let ℤ′[X]_{qfh} denote the qfh-sheaf associated to the presheaf

S ↦ ℤ′[Map_{Sch}(S, X)].

Then, for S normal and connected

Γ(S, ℤ′[X]_{qfh}) ≅ L(X)(S).

3.4. Homological algebra. In some sense, the crucial point of the proof is the following:

Theorem 3.4.1. Let k = k̄ be an algebraically closed field of characteristic 0. Let Φ be a homotopy-invariant presheaf with transfers on the category of schemes over k. Write Φ_{ét}, Φ_{qfh}, Φ_{h} for the corresponding sheaves on the big sites for the indicated topologies on schemes over k. Then

(3.4.0.5) Ext^{∗}(Φ_{ét}, ℤ/nℤ) ≅ Ext^{∗}(Φ_{qfh}, ℤ/nℤ) ≅ Ext^{∗}(Φ_{h}, ℤ/nℤ)
≅ Ext_{Ab}(Φ(Spec (k)), ℤ/nℤ).

Sketch of proof. The first three isomorphisms follow from (3.3.0.4) and related ideas. For the last isomorphism, we work with h-sheaves throughout and so we surpress the subscript h. Let Φ_{0} be the h-sheaf associated to the constant presheaf (on the big site) X ↦ Φ(Spec (k)). Define Φ′ by the exact sequence of h-sheaves

0 → Φ_{0} → Φ → Φ′ → 0.

Let _{n}Φ_{0} ⊂ Φ_{0} and _{n}Φ ⊂ Φ denote the kernels of multiplication by n. The rigidity theorem 3.2.2 implies for S smooth and strictly henselian

nΦ_{0}(S) ≅ _{n}Φ(S); (Φ_{0}/nΦ_{0})(S) ≅ (Φ/nΦ)(S)

However, resolution of singularities is an h-covering, so we conclude that multiplication by n is an isomorphism on the h-sheaf Φ′, and

Ext_{h}^{∗}(Φ, ℤ/nℤ) ≅ Ext_{h}^{∗}(Φ_{0}, ℤ/nℤ).

Finally, we need to understand the Ext group on the right. We have

H^{∗}(X_{h}, ℤ/nℤ) ≅ Ext^{∗}(ℤ[X]_{h}, ℤ/nℤ), (3.4.0.6)

since both sides calculate the derived functors of global sections. Let A = Φ(Spec (k)). Also Φ_{0} ≅ A[Spec (k)]_{h}, since both sides are associated to the same presheaf. Using H^{∗}(Spec (k)_{h}, ℤ/nℤ) = (0), it is now easy to see

Ext_{h}^{∗}(Φ_{0}, ℤ/nℤ) ≅ Ext_{Ab}^{∗}(Φ(Spec (k), ℤ/nℤ).

3.5. Spectral Sequences. We continue to work in the category of algebraic k-schemes, where k is an algebraically closed field of characteristic 0. Let F : (Sch/k)^{0} → (Ab) be a sheaf (h, qfh, or ét). We have a surjection

⨁_{(S,f)} ℤ[S]_{t} ↠ F

where the subscript t denotes sheafification for the appropriate topology, and the sum runs over pairs S, f with S a k-scheme and f ∈ F (S). This construction may be iterated to get a presentation of F in terms of direct sums of ℤ[S]_{t}.

Proposition 3.5.1. With notation as above (all functors are sheafified for the t topology, so we drop the t), the natural map ℤ[𝔸^{1}] → ℤ induces isomorphisms

Ext^{∗}(F, ℤ/nℤ) ≅ Ext^{∗}(F ⊗ ℤ[𝔸^{1}], ℤ/nℤ). (3.5.0.7)

Proof. Suppose first F = ℤ[X]. Then F ⊗ℤ[𝔸^{1}] is the t-sheaf associated to the presheaf

S ↦ ℤ[Map(S, X)] ⊗ ℤ[Map(S, 𝔸^{1})] ≅ ℤ[Map(S, X × 𝔸^{1})]

from which it follows that ℤ[X] ⊗ ℤ[𝔸^{1}] ≅ ℤ[X × 𝔸^{1}]. In this case, by (3.4.0.6), we must show

H^{∗}(X_{t}, ℤ/nℤ) ≅ H^{∗}((X × 𝔸^{1})_{t}, ℤ/nℤ).

This is well known for t = ét. In the other cases it follows from theorem 3.4.1. For general F, we use the above construction to write an exact sequence of sheaves

0 → D → E → F → 0,

where the assertion of the proposition is true for E. Note in general the map in (3.5.0.7) is injective (it has a section). The long-exact Ext-sequence and induction on i gives

Ext^{i−1}(D, ℤ/nℤ)  →  Ext^{i}(F, ℤ/nℤ)  → · → ·
∥  ∩  ∥  ∩
Ext^{i−1}(D ⊗ ℤ[𝔸^{1}], ℤ/nℤ) → Ext^{i}(F ⊗ ℤ[𝔸^{1}], ℤ/nℤ) → · → ·

The assertion of the proposition follows from a diagram chase.

3.5.1. Contractible sheaves. A sheaf F is said to be contractible if there exists a map F ⊗ ℤ[𝔸^{1}] →^{ϕ} F such that ϕ ∘ i_{0} = 0 and ϕ ∘ i_{1} = Id. Here i_{j} : ℤ → ℤ[𝔸^{1}] correspond to the points j = 0, 1. As a consequence of (3.5.0.7), F contractible implies Ext^{∗}(F, ℤ/nℤ) = (0).

Given a sheaf F and an integer q, we write F_{q} for the sheaf

X ↦ F (X × 𝔸^{q}).

Let F̃_{q} := Coker(F → F_{q}).

Proposition 3.5.2. The sheaf F̃_{q} is contractible. In particular,

Ext^{∗}(F, ℤ/nℤ) ≅ Ext^{∗}(F_{q}, ℤ/nℤ) (3.5.1.1)

Proof. Let μ : 𝔸^{1} × 𝔸^{q} → 𝔸^{q} be the map (t, u) ↦ tu. We define a map ψ : F_{q} ⊗ ℤ[𝔸^{1}] → F_{q} by associating to a section s ∈ F (X × 𝔸^{q}) and a map f : X → 𝔸^{1} the section g^{∗}(s) where g : X × 𝔸^{q} → X × 𝔸^{q} is the map

X × 𝔸^{q} ⟶^{f×Id} 𝔸^{1} × X × 𝔸^{q} ⟶^{Id_{X}×μ} X × 𝔸^{q}.

It is straightforward to check that ψ induces a contraction map

F̃_{q} ⊗ ℤ[𝔸^{1}] → F̃_{q}

with the required properties.

3.5.2. Let F be a sheaf as above. Recall we have defined in 1.2.5 a homological complex of sheaves

\underline{C}▪(F) := ⋯ → F_{2} → F_{1} → F.

Define the hyperext groups

E^{∗} := Ext^{∗}(\underline{C}▪(F), ℤ/nℤ).

There are three spectral sequences

I_{1}^{p,q} := Ext^{q}(F_{p}, ℤ/nℤ) ⇒ E^{p+q} (3.5.2.1)

II_{2}^{p,q} := Ext^{p}(H^{q}(\underline{C}▪(F)), ℤ/nℤ) ⇒ E^{p+q} (3.5.2.2) Ext_{Ab}^{p}(H^{q}(C▪(F)), ℤ/nℤ) ⇒ (3.5.2.3)

Ext_{Ab}^{p+q}(C▪(F), ℤ/nℤ) := H_{sing}^{∗}(F, ℤ/nℤ)

The first spectral sequence degenerates by virtue of (3.5.1.1), so

E^{p} ≅ Ext^{p}(F, ℤ/nℤ). (3.5.2.4)

Now we assume the sheaf F is endowed with transfer maps as in 1.2.5. The homology groups H_{q}(\underline{C}▪) are then shown to be homotopy invariant sheaves with transfer, so by theorem 3.4.1 the spectral sequences (3.5.2.2) and (3.5.2) coincide at E_{2}. Since there are maps from the one to the other, we conclude with notation as in (3.5.2)

(3.5.2.5) Ext^{p}(F, ℤ/nℤ) ≅ E^{∗} ≅ H_{sing}^{∗}(F, ℤ/nℤ)

It suffices finally to take F = ℤ[X]_{ét} and substitute in the above, getting

H_{ét}^{p}(X, ℤ/nℤ) ≅ H_{sing}^{∗}(X, ℤ/nℤ).

This completes the proof of theorem 3.1.1. (We leave to the reader the minor business of translating the cohomological formula above into the homological statement of the theorem.)

4. Polylogarithms and Chow Regulators

In this final section I will explain the work of Beilinson and Deligne on polylogarithms [5], and some recent generalizations, due to Goncharov [18], involving Chow varieties.

4.1. Let k be a field. As has already been mentioned briefly (2.3.2.2) the polylogarithm Lie algebra is the graded pro-Lie algebra

L^{pl} = L_{−1}^{pl} ⊕ L_{−2}^{pl} ⊕ …

with dual M^{pl} given by

M_{1}^{pl} = k^{×} ⊗ ℚ
M_{n}^{pl} = some quotient of k − {0, 1}; z ↦ θ_{n}(z)

The dual of the bracket is a differential d : M^{pl} → ∧^{2}M^{pl} defined by

dθ_{n}(z) = (z) ∧ θ_{n−1}(z).

(Here θ_{1}(z) := 1 − z.) For N ≥ 2 given, the N-polylogarithm representation 𝔐_{z}^{N} associated to z ∈ k − {0, 1} has the following shape. For N = 1, it is the Kummer extension

0 → ℚ(1) → 𝔐_{z}^{1} → ℚ → 0.

For N ≥ 2 one gets an extension

0 → (Sym^{N−1}(𝔐_{z}^{1}))(1) → 𝔐_{z}^{N} → ℚ → 0.

4.1.1. The Betti and deRham realizations of this family of mixed motives give rise to a variation of mixed Tate Hodge structure over ℙ_{ℂ}^{1} − {0, 1, ∞}. To describe this, recall the polylogarithm functions

Li_{1}(z) := − log(1 − z); Li_{n}(z) := ∫_{0}^{z} Li_{n−1}(t)\frac{dt}{t}.

These are multivalued functions which are analytic away from {0, 1, ∞}. The principal branch of Li_{n}(z) is given by the power series

Li_{n}(z) = ∑_{p=1}^{∞} \frac{z^{p}}{p^{n}} .

Denote by L(z) (resp. L^{(N)}(z)) the infinite (resp. (N + 1) × (N + 1)) matrix (resp. top left hand corner matrix)

1  0  0 . . . . . .
−Li_{1}(z)  1  0 . . . . . .
−Li_{2}(z) log(z)  1 . . . . . .
−Li_{3}(z) \frac{(log(z))^{2}}{2!} log(z) 1 …
⋮  ⋮  ⋮  ⋮

Define

A(z) := L(z)τ(2πi); resp. A^{(N)}(z) := L^{(N)}(z)τ^{(N)}(2πi)

where τ(λ) (resp. τ^{(N)}(λ)) denotes the matrix

(\begin{matrix}1 & 0 & 0 & … \\ 0 & λ & 0 & … \\ 0 & 0 & λ^{2} & … \\ ⋮ & ⋮ & ⋮ & ... \end{matrix})

(resp. the evident (N + 1) × (N + 1)-truncation).

For R = ℚ, ℝ, ℂ write R^{[0,N]} for the space of all N +1-column vectors

( r_{0} ; ⋮ ; r_{N} )

with r_{i} ∈ R. For 0 ≤ a ≤ b ≤ N, let R^{[a,b]} ⊂ R^{[0,N]} be the subspace with r_{i} = 0 for r ∉ [a, b]. Write 𝔐^{(N)}(R) for the R-structure in the fibre of the variation of mixed Hodge structure associated to 𝔐^{(N)} over the point z ∈ ℙ^{1} − {0, 1, ∞}. We have

𝔐_{z}^{(N)}(ℚ) = A^{(N)}(z) · ℚ^{[0,N]}

W_{−2k}𝔐_{z}^{(N)}(ℚ) = A^{(N)}(z) · ℚ^{[k,N]} (weight filtration)

F^{−k}𝔐_{z}^{(N)}(ℂ) = ℂ^{[0,k]} (Hodge filtration)

4.1.2. Notice that even though the matrix A^{(N)}(z) is multiple-valued, the effect of analytic continuation around a loop is to replace it by A^{(N)}(z)▪B where B is lower triangular with 1’s on the diagonal and entries in ℚ. In particular, the ℚ-structure and weight filtration above are independent of the branch. This remark leads [5] to a very beautiful construction of single-valued ℝ-valued variants of the polylogarithm. As above, the image of A^{(N)}(z) in GL_{N+1}(ℂ)/GL_{N+1}(ℝ) is independent of the branch. Consider the matrix

B := τ(i)^{−1}A\overline{A}^{−1}τ(i)

Clearly, B is independent of the branch, and

\overline{B}^{−1} = τ(−1)Bτ(−1).

Since B is triangular with 1’s on the diagonal, we can take the logarithm, and

log(B) = −ad(τ(−1))(log(\overline{B})),

i.e.

log(B)_{r,s} ∈ \begin{cases}\sqrt{−1} ℝ & r ≡ s mod 2 \\ ℝ & r ≢ s mod 2. \end{cases}

An elegant matrix calculation leads to the expression

− log(B^{1/2}) = (\begin{matrix}0 & … \\ D_{1}(z) & … \\ D_{2}(z) & … \\ D_{3}(z) & … \\ ⋮ & ⋮ \end{matrix})

where the D_{k}(z) are given by

D_{k}(z) = \begin{cases}\sqrt{−1} ∑_{ℓ=0}^{k−1} \frac{b_{ℓ}}{ℓ!} (log(|z|^{2}))^{ℓ}Im(Li_{k−ℓ}(z)) & k ≡ 0 mod 2 \\ ∑_{ℓ=0}^{k−1} \frac{b_{ℓ}}{ℓ!} (log(|z|^{2}))^{ℓ}Re(Li_{k−ℓ}(z)) & k ≡ 1 mod 2 \end{cases} (4.1.2.1)

(Here the b_{ℓ} are Bernoulli numbers.) For example

D_{2}(z) = Im(Li_{2}(z)) + log(|z|)arg(1 − z).

To my knowledge, these particular single-valued variants of the polylogarithms were first written down in [47]. Related functions were defined in [13] and [38].

4.1.3. Exercise. Recall in 2.4.1 we constructed a graded differential graded algebra based on cycles on affine spaces □^{r} := (ℙ^{1} − {1})^{r} over a field k. The polylogarithm map

θ_{n} : k − {0, 1} → A^{1}(n) ⊂ {codim. n alg. cycles on □^{2n−1}}

is defined by taking θ_{n}(a) to be (−1)^{n}(n−1)/2 times the skew-symmetrization in the sense of 2.4.1 of the locus parametrized in non-homogeneous coordinates by

(x_{1} …, x_{n−1}, 1 − x_{1}, 1 − \frac{x_{2}}{x_{1}} , …, 1 − \frac{x_{n−1}}{x_{n−2}} , 1 − \frac{a}{x_{n−1}} ).

We define B_{n} to be the subgroup of A^{1}(n)/d(A^{0}(n)) generated by the θ_{n}(a) for a ∈ k, a ≠ 0, 1. Try to use the differential d on A▪ to define a map B_{n} → B_{n−1} ⊗ k^{×}. Note there is a problem. Try to write down a polylogarithm representation using the bar complex [11].

4.2. Chow regulators. This section is an introduction to the paper of Goncharov, [18].

4.2.1. Fix non-negative integers n = p + q with n ≥ 1. Let L = {L_{0}, …, L_{n}} be a set of hyperplanes in general position in ℙ_{ℂ}^{n}. By abuse of notation, we will sometimes write L = ⋃ L_{i}. If L_{i} : z_{i} = 0, then ℙ^{n} − L ≅ ℂ^{×n} with coordinates z_{i}/z_{0}. The differential form

Ω_{L} := (2πi)^{−n}d log(z_{1}/z_{0}) ∧ … ∧ d log(z_{n}/z_{0})

has integer periods, so ImΩ_{L} is exact. There is no canonical primitive for this form, but if we choose an auxillary hyperplane H in general position with respect to L we may scale the z_{i} so H : ∑ z_{i} = 0. This fixes the coordinate functions, and we may fix a primitive as follows. For any meromorphic functions f_{1}, …, f_{n}, we define (alt_{n} is the alternating projection for the action of the symmetric group on the f_{i}):

(4.2.1.1) r_{n}(f_{1}, …, f_{n}) := alt_{n}( ∑_{j≥0} \frac{1}{(2j + 1)!(n − 2j − 1)!}

log |f_{1}|d log |f_{2}| ∧ … ∧ d log |f_{2j+1}| ∧ d(i argf_{2j+2}) ∧ … ∧ d(i argf_{n})).

Then

(4.2.1.2) dr_{n} = Im((2πi)^{−n}d log f_{1} ∧ … ∧ d log f_{n})

In particular, writing r_{n}(L, H) := r_{n}(z_{1}/z_{0}, …, z_{n}/z_{0}), we have

dr_{n}(L, H) = ImΩ_{L}.

4.2.2. Let Z_{p}^{q}(L) be the open set in the Chow variety of codimension q algebraic cycles on ℙ^{n} consisting of cycles meeting all faces L_{I} = ∩_{i∈I}L_{i} properly, i.e. in the correct dimension. For example Z_{0}^{n}(L) = ℙ^{n} −L = ℂ^{×}n. Let

a_{i} : Z_{p}^{q}(L) → Z_{p−1}^{q}(L_{i}) (4.2.2.1)

b_{j} : Z_{p}^{q}(L) → Z_{p}^{q−1}(L_{j}) (4.2.2.2)

be defined by intersection (resp. projection) of cycles.

By definition, the space of p-currents on a manifold is the dual space to the space of C_{∞} p-forms with compact support. A current on a singular reduced analytic set is a current on the open set of smooth points. Goncharov defines a (q − p − 1)-current ω_{p}^{q}(L, H) on Z_{p}^{q}(L) with the following properties:

dω_{0}^{q}(L, H) = ImΩ_{L} (4.2.2.3)

ω_{p}^{q}(L, H) = ∑_{i=0}^{p+q}(−1)^{i}a_{i}^{∗}ω_{p−1}^{q}(L^{(i)}, H^{(i)}) (4.2.2.4)

∑_{j=0}^{p+q+1}(−1)^{j}b_{j}^{∗}ω_{p}^{q}(L, H) = 0 (4.2.2.5)

Here L^{(i)} and H^{(i)} are the evident restrictions to ℙ^{n−1} = L_{i}.

4.2.3. Let Γ_{p}^{q} be the incidence correspondence

Γ_{p}^{q}
π_{1}↙ ↘^{π_{2}}
ℙ^{n}  Z_{p}^{q}(L),

and define

ω_{p}^{q}(L, H) := π_{2∗}π_{1}^{∗}r_{n}(L, H).

Note π_{2} is proper, so pushforward of currents is defined. Actually, one has to be a bit careful, because the pullback ψ = π_{2}^{∗}(ϕ) of a C^{∞} function on Z_{p}^{q}(L) with compact support contained in the smooth locus does not necessarily have support contained in the smooth locus on Γ_{p}^{q}. However, ψ is continuous, and, viewing π_{1}^{∗}r_{n}(L, H) as being defined on an open set, the integral

∫_{Γ_{p}^{q}} ψπ_{1}^{∗}r_{n}(L, H)

is defined.

4.2.4. One verifies that ω_{p}^{q}(L, H) satisfies (4.2.2.3)-(4.2.2.5).

Theorem 4.2.1. (i) ω_{p}^{q} is a (q − p − 1)-current. It is a C^{∞}-form on

the open set

Ẑ_{p}^{q} ⊂ Z_{p}^{q}(L)

parametrizing smooth cycles in transverse position relative to L. In particular, ω_{p}^{p+1} is an L^{1}-function on Z_{p}^{p+1}(L).

(ii) Restriction of cycles to faces makes Ẑ_{▪}^{q} a simplicial variety:

Ẑ_{▪}^{q} : ⋯ Ẑ_{p+1}^{q}\underrightarrow{\overrightarrow{:}}Ẑ_{p}^{q} ⋯

Equations (4.2.2.3) and (4.2.2.4) imply that ω_{p}^{q} is a 2q-cocycle representing a class in the Deligne cohomology

[w_{p}^{q}] ∈ H_{D}^{2q}(Ẑ_{▪}^{q}, ℝ(q)).

(iii) For a ∈ ℂ − {0, 1}, let h_{a} denote the point in the Chow variety of codimension n cycles in ℙ^{2n−1} corresponding to the closure in ℙ^{2n−1} of the affine n − 1-plane defined in affine coordinates by

z_{n} = 1 − z_{1}; z_{n+i} = z_{i} − z_{i+1}; 1 ≤ i ≤ n − 2
z_{2n−1} = z_{n−1} − a

This cycle is not in good position, but the function ω_{n−1}^{n} is defined and continuous at h_{a}, and

ω_{n−1}^{n}(h_{a}) = (2πi)^{−n}D_{n}(a)

where D_{n}(a) is defined in (4.1.2.1).

Remark 4.2.2. (i) One would like to better understand the functions ω_{n−1}^{n}. Is there a family of motives over Ẑ_{n−1}^{n} giving rise to a mixed Tate variation of mixed Hodge structure?

(ii) Do the functions ω_{n−1}^{n}, or more generally the currents ω_{p}^{q}, admit some interpretation in terms of symplectic geometry for the group acting on ℙ^{p+q}?

References

[1] Beilinson, A.A., Higher Regulators and Values of L-functions, VINITI 24 (1984), 181-238

[2] Beilinson, A. A., MacPherson, R., and Schechtman, V. V., Notes on Motivic Cohomology, Duke Math. J. 54 (1987), 679-710

[3] Beilinson, A. A., Goncharov, A. B., Schechtman, V. V., and Varchenko, A. N., Aomoto Dilogarithms, Mixed Hodge Structures, and Motivic Cohomology of a Pair of Triangles in the Plane, The Grothendieck Festschrift, Vol 1, Progress in Math., Vol. 86, Birkhauser, Boston (1990), 131-172

[4] Beilinson, A. A., Polylogarithm and Cyclotomic Elements, (preprint)

[5] Beilinson, A. A.,Deligne, P., Interprétation motivique de la conjecture de Zagier, in Proc. Symp. Pure Math 55, part 2, AMS (1994)

[6] Beilinson, A.A., Levin A., The Elliptic Polylogarithm, in Proc. Symp. Pure Math. 55, part 2, AMS (1994)

[7] Bloch, S., Algebraic Cycles and Higher K-theory, Advances in Math. 61 (1986), 267-304

[8] Bloch, S., Algebraic Cycles and the Beilinson Conjectures, Contemp. Math. 58 (1986), no. 1, 65-79

[9] Bloch, S., Algebraic Cycles and the Lie Algebra of Mixed Tate Motives, JAMS, Vol. 4, no. 4 (1991)

[10] Bloch, S., The Moving Lemma for Higher Chow Groups, Journ. of Algebraic Geometry 3, (1994), 537-568

[11] Bloch, S., and Kriz, I., Mixed Tate Motives, Ann. Math. 140, (1994), 557-605 [12] Bloch, S., An Elementary Presentation for K-groups and Motivic Cohomology,

in Proc. Symp. Pure Math. 55, part 1, AMS (1994)

[13] Bloch, S., Higher Regulators, Algebraic K-theory, and Zeta Functions of Elliptic curves, preprint (1979)

[14] Deligne, P., Le Groupe Fondamental de la Droite Projective Moins Trois Points, in Galois Groups over ℝ, MSRI publications 16, Springer-Verlag (1989) 79-297

[15] Deligne, P., and Milne, J.S., Tannakian Categories, in Hodge Chcles, Motives, and Shimura Varieties, LNM 900, Springer-Verlag, Berlin, 1982

[16] Deligne, P., Décompositions dans la categorie dérivée, in Proc. Symp. Pure Math. 55, Part 1, AMS (1994)

[17] Friedlander, E., Voevodsky, V., Bivariant Cycle Cohomology, (preprint)

[18] Goncharov, A. B., Chow Polylogarithms and Regulators, Math. Res. Lett. 2 (1995), 95-112

[19] Goncharov, A. B., Geometry of Configurations, Polylogarithms, and Motivic Cohomology, Adv. in Math. (to appear)

[20] Goncharov, A. B., Polylogarithms and Motivic Galois Groups, Symp. in Pure Math. 55 (1994), 43-97

[21] Grothendieck, A., Standard Conjectures on Algebraic Cycles, in Algebraic Geometry - Bombay Colloquium 1968, Oxford (1969) 193-199

[22] Hain, R., Classical Polylogarithms, in Proc. Symp. Pure Math. 55, part 2, AMS (1994)

[23] Hanamura, M., Mixed Motives and Algebraic Cycles I, Math. Res. Lett. 2 (1955) no 6., 811-821

[24] Hanamura, M., Mixed Motives and Algebraic Cycles II, (preprint)

[25] Hanamura, M., Homological and Cohomological Motives of Algebraic Varieties, (preprint)

[26] Hanamura, M., MacPherson, R., Geometric Construction of Polylogarithms, Duke Math. J. 70 (1993), no. 3, 481-516

[27] Hartshorne, R., Residues and Duality, LNM 20, Springer-Verlag (1966)

[28] Jannsen, U., Motives, Numerical Equivalence, and Semi-simplicity, Inv. Math. 107 (1992) 447-452

[29] Kleiman, S., Algebraic Cycles and the Weil Conjectures, in Dix exposés sur la cohomologie des schémas North-Holland, Amsterdam (1968) 359-386

[30] Kleiman, S., Motives, in Algebraic Geometry - Oslo, 1970 Sijthoff and Noordhoff, Groningen (1970) 53-82

[31] Kleiman, S., The Standard Conjectures, in Proc. Symp Pure Math. 55, part 1, AMS (1994)

[32] Levine, M., Motivic Cohomology and Algebraic Cycles: a Categorical Approach, (preprint)

[33] Levine, M., On the Derived Motivic Category, (preprint)

[34] Lichtenbaum, S., Motivic Complexes, in Proc. Symp. Pure Math. 55, part 1, AMS (1994)

[35] Manin, Y., Correspondences, Motives, and Monoidal Transformations, Math. USSR Sb. 6 (1968) 439-470

[36] Murre, J., On a Conjectural Filtration on the Chow Groups of an Algebraic Variety, Indag. Mathem., N.S. 4(2) (1993) 177-188

[37] Neeman, A., Voevodsky, V., Book on Category Theory (to appear)

[38] Ramakrishnan, D., Analogues of the Bloch-Wigner function for higher Polylogarithms, in Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, part I, II, Contemp. Math. 55, AMS (1983) 371-376

[39] Soulé, C.,Opérations en K-théorie algébrique, Can. J. Math., Vol. XXXVII, no. 3 (1985), 488-550 [40] Sullivan, D., Infinitesimal Calculations in Topology, Publ. Math. IHES 47

(1977) 269-331

[41] Suslin, A., Voevodsky, V., Relative Cycles and Chow Sheaves, (preprint)

[42] Suslin, A., Voevodsky, V., Singular Homology of Abstract Algebraic Varieties (preprint)

[43] Suslin, A., Higher Chow Groups and Étale Cohomology, (preprint)

[44] Voevodsky, V., Homology of Schemes I, (preprint)

[45] Voevodsky, V., Cohomological Theory of Presheaves with Transfers, (preprint)

[46] Voevodsky, V., Triangulated Category of Motives over a Field, (preprint)

[47] Zagier, D., Polylogarithms, Dedekind Zeta Functions, and the Algebraic Ktheory of Fields, Proc. Texel Conf., Birkhäuser, Basel (1991)

[48] Zagier, D., The Bloch-Wigner-Ramakrishnan Polylogarithm Function, Math. Ann. 286 (1990)

Department of Mathematics, University of Chicago, Chicago, IL 60637

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