Draft: May 19, 2008

ON A CLASS OF COHERENT RINGS, WITH APPLICATIONS TO THE SMOOTH REPRESENTATION THEORY OF GL_{2}(ℚ_{p}) IN

CHARACTERISTIC p

MATTHEW EMERTON

In this short note, we introduce a certain class of coherent rings, namely noncommutative polynomial rings of the form A[F], where A is a commutative Noetherian ring and F is a variable satisfying the commutation relation F · a = F (a)F, where we use the same notation F to stand for some given flat endomorphism of A. (See Proposition 1.3 for the proof that such a ring is coherent.)

If we take A to be a smooth finite type k-algebra, where k is a field of characteristic p, and F to be the absolute Frobenius endomorphism of A, then (sheafified versions of) the ring A[F] play a prominent role in the paper [7]; in particular, it follows from the results of this note that the sheaf of rings O_{F,X} on a smooth finite type k-scheme X, introduced in [7], is coherent. (See Example 1.4 below.) This sheds a new light on some of the results of [7].

Suppose instead that we take A = k[[t]], where k is a finite field of characteristic p, and take F to be the relative Frobenius endomorphism of A over k. If we let P denote the Borel subgroup of GL_{2}(ℚ_{p}), then any smooth P -representation over k is naturally an A[F]-module. (See Section 4 below.) In particular, this is true of a smooth GL_{2}(ℚ_{p})-representation over k. Suppose that V is a finitely generated admissible smooth GL_{2}(ℚ_{p})-representation over k, and let V_{0} be finite-dimensional k-subspace invariant under KZ (where, as usual, K := GL_{2}(ℤ_{p}) and Z denotes the centre of GL_{2}(ℚ_{p})) which generates V over GL_{2}(ℚ_{p}). We show that the A[F]- submodule of V generated by V_{0} is finitely presented over A[F], and is admissible as an A-module – i.e. is Pontrjagin dual to a finitely generated A-module. (See Theorem 4.7 below.) This result is essentially due to Colmez (see [4] and Remark 4.8 below). However, his proof is more involved than ours, and relies in particular on the classification of irreducible admissible smooth GL_{2}(ℚ_{p})-representations over k due to Barthel-Livné [1] and Breuil [3]. Our proof is independent of this classification, and indeed can be used to rederive it. (See Section 5 below.)

Acknowledgments. I would like to thank Toby Gee for his close reading of an earlier draft of this note.

1. A class of coherent rings

Let A be a Noetherian commutative ring, and let F : A → A be a flat endomorphism. We let A[F] denote the non-commutative polynomial ring in the variable “F ” over A, with the commutation relation

F · a = F (a)F.

If M is an A-module, we write F^{∗}M := A ⊗_{A} M, the tensor product being taken with respect to the map F : A → A. If M is an A[F]-module, then the action of the element F on M induces an A-linear map ϕ_{M} : F^{∗}M → M, defined via ϕ_{M} : a ⊗ m ↦ aF · m. Conversely, if M is an A-module, and ϕ_{M} : F^{∗}M → M is an A-linear map, then there is a unique extension of the A-module structure on M to an A[F]-module structure such that the map ϕ_{M} arises from this structure via the preceding construction.

The author was supported in part by NSF grant DMS-0701315.

If 0 → M′ → M → M′′ → 0 is an exact sequence of A[F]-modules, then we may form the diagram

\xymatrix{0 \ar[r] & F^{∗}M′ \ar[r] \ar[d]^{ϕ_{M′}} & F^{∗}M \ar[r] \ar[d]^{ϕ_{M}} & F^{∗}M′′ \ar[r] \ar[d]^{ϕ_{M′′}} & 0 \\ 0 \ar[r] & M′ \ar[r] & M \ar[r] & M′′ \ar[r] & 0,}

whose top row is again exact (since F is flat). The snake lemma gives rise to an exact sequence

(1) 0 → ker ϕ_{M′} → ker ϕ_{M} → ker ϕ_{M′′} → coker ϕ_{M′} → coker ϕ_{M} → coker ϕ_{M′′} → 0.

We remark that the ring A[F] is typically non-Noetherian. Our main goal in this section is to prove that it is nevertheless left coherent (i.e. that finitely generated left submodules of A[F] are finitely presented).

1.1. Lemma. If M is a left A[F]-submodule (i.e. left ideal) of A[F], then M is finitely generated over A[F] if and only if coker ϕ_{M} is finitely generated over A.

Proof. If A[F]^{n} → M is surjective, then so is the induced map

A^{n} ⟶^{∼} coker ϕ_{A[F]^{n}} → coker ϕ_{M}.

This proves the “only if” direction.

Suppose, conversely, that coker ϕ_{M} is finitely generated over A. For any d ≥ 0, write M^{≤d} := M ∩ (⨁_{i=0}^{d} AF^{i}) ⊂ M. Since A is Noetherian, each M^{≤d} is finitely generated over A, and clearly M = ⋃_{d≥0} M^{≤d}. If we choose d ≥ 0 such that the natural map M^{≤d} → coker ϕ_{M} is surjective, then for any d′ > d, we find that M^{≤d′} ⊂ M^{≤d} + F M. On the other hand, clearly F M ⋂ M^{≤d′} = F M^{≤d′−1}. Thus

M^{≤d′} ⊂ M^{≤d} + F M^{≤d′−1}.

Proceeding by recursion on d′, we find that M^{≤d′} ⊂ A[F]M^{≤d}, and thus (since d′ > d was arbitrary) that M = A[F]M^{≤d}. This proves the “if” direction. □

1.2. Lemma. If M is a finitely generated left A[F]-module, then M is finitely presented if and only if ker ϕ_{M} is finitely generated over A.

Proof. Since M is finitely generated over A[F], we may choose a presentation of M of the form 0 → M′ → A[F]^{n} → M → 0. Applied to this short exact sequence, the exact sequence (1) reduces to

0 → ker ϕ_{M} → coker ϕ_{M′} → A^{n} → coker ϕ_{M} → 0.

In particular, we see that ker ϕ_{M} is finitely generated over A if and only if the same is true of coker ϕ_{M′}. On the other hand, lemma 1.1 shows that coker ϕ_{M′} is finitely generated over A if and only if M′ is finitely generated over A[F]. The lemma follows. □

COHERENT RINGS AND REPRESENTATIONS OF GL_{2} OF A p-ADIC FIELD 3

1.3. Proposition. The ring A[F] is left coherent.

Proof. If M is a finitely generated left submodule of A[F], then the inclusion M ⊂ A[F] induces an inclusion ker ϕ_{M} ⊂ ker ϕ_{A[F]} = 0. Thus ker ϕ_{M} vanishes, and so by the preceding lemma, M is finitely presented over A[F]. □

As a corollary, the kernel and image (as well as the cokernel – although this doesn’t require coherence) of any map between finitely presented left A[F]-modules is again finitely presented.

1.4. Example. If A is a smooth algebra over a field k of characteristic p > 0, then both the absolute Frobenius endomorphism F_{A} of A and the relative Frobenius endomorphism F_{A/k} of A are flat maps. Thus the rings A[F_{A}] and A[F_{A/k}] are left coherent. More generally, if X is a smooth k-scheme, then we may form sheaves of rings O_{X}[F] and O_{X}[F_{X/k}], both of which are then seen to be left coherent. (The first of these coincides with the sheaf denoted O_{F,X} in [7].)

1.5. Example. Continuing to suppose that X is a smooth scheme over a field k of characteristic p > 0, a quasi-coherent sheaf of left O_{X}[F_{X}]-modules M is said to be unit if the map ϕ_{M} : F_{X}^{∗}M → M is an isomorphism. In particular, the kernel of ϕ_{M} then vanishes, and so it follows from Lemma 1.2 that a locally finitely generated unit O_{X}[F_{X}]-module is in fact locally finitely presented as a left O_{X}[F_{X}]-module, a result which was proved by a different method in [7]. (See Remark 6.1.2 of that reference.)

2. A homological application

Let A and F be as in the preceding section. Suppose given a map A → B, with B again taken to be Noetherian, and a flat endomorphism of B (which we will again denote by F) compatible with the endomorphism F of A, in the sense that the diagram

\xymatrix{A \ar[r]^{F} \ar[d] & A \ar[d] \\ B \ar[r]^{F} & B}

commutes. The map A → B then extends naturally to a map of rings A[F] → B[F].

2.1. Lemma. There is a natural B-linear isomorphism of δ-functors

Tor_{•}^{A[F]}(B[F], –) ⟶^{∼} Tor_{•}^{A}(B, –).

Proof. Clearly A[F] is free, and so flat, as a left A-module. Thus if M is a left A[F]- module, and P_{•} is a left-resolution of M by free A[F]-modules, then P_{•} is also a resolution of M by free A-modules. Furthermore, there is an evident isomorphism of (B, A[F])-bimodules B⊗_{A}A[F] → B[F], which gives rise to a B-linear isomorphism

B[F] ⊗_{A[F]} P_{•} ⟶^{∼} B ⊗_{A} P_{•}.

Passing to homology on the source and target of this isomorphism gives the required isomorphism of Tor-functors. □

In particular, if M is a left A[F]-module, then the B-module structure on each Tor_{•}^{A}(B, M) extends in a natural way to a left B[F]-module structure.

2.2. Proposition. If M is a finitely presented left A[F]-module, then each of the Tor-modules Tor_{•}^{A}(B, M) is finitely presented as a left B[F]-module.

Proof. Since A[F] is left coherent, we may choose a resolution P^{•} of M by free A[F]-modules, each member of which is finitely generated over A[F]. The preceding lemma shows that we may compute the modules Tor_{•}^{A}(B, M), with their natural B[F]-module structure, as the homology of the complex B[F] ⊗_{A[F]} P^{•}. This is a complex of finite rank free left B[F]-modules, and so, by the left coherence of B[F], has finitely presented homology modules. □

2.3. Example. Let f : Y → X be a morphism of smooth k-schemes, where k is a field of characteristic p > 0. In [7, §2] a functor f^{!} from complexes of left O_{X}[F_{X}]-modules to complexes of left O_{Y} [F_{Y}]-modules is defined. Up to a shift, it coincides with the sheaf-theoretic pull-back f^{−1}, followed by the left-derived tensor product with O_{Y} [F_{Y}] over f^{−1}(O_{X}[F_{X}]). As Lemma 2.1 shows, on the underlying O_{X} and O_{Y} -modules, this functor coincides (up to a shift) with the left-derived functor of the usual pull-back f^{∗}. Proposition 2.2 shows that f^{!} takes complexes with finitely presented cohomology sheaves to complexes with finitely presented cohomology sheaves. (In the case of complexes with unit cohomology sheaves, the fact that f^{!} preserves the property of having finitely generated, or equivalently, by Examples 1.5, finitely presented, cohomology sheaves, is proved in [7, Prop. 6.7].)

3. The case when A is a discrete valuation ring

As indicated in the section title, we now suppose that A is a discrete valuation ring. We fix a uniformizer t of A, and we write k := A/tA to denote the residue field of A. We also suppose that the flat map F : A → A is local, and that the induced endomorphism of k is the identity. Thus k[F] is the usual commutative polynomial ring over k.

We may apply the discussion of the preceding section to the surjection A → k, and conclude that if M is any A[F]-module, then the modules Tor_{•}^{A}(k, M) have a natural k[F]-module structure. Of course, since A is a discrete valuation ring, it has projective dimension 1, and the modules Tor_{•}^{A}(k, M) admit the following explicit descriptions, for any A-module M:

Tor_{0}(k, M) ⟶^{∼} M/tM, Tor_{1}(k, M) ⟶^{∼} M[t],

where M[t] denotes the submodule of M consisting of elements annihilated by t.

If M is a left A[F]-module, then the induced k[F]-module structures on these Tor-modules are easy to describe. Namely: the action of F on M/tM is given by

m mod tM ↦ F m mod tM,

while the action of F on M[t] is given by

M[t] ∋ m ↦ \frac{F (t)}{t}F m

(where one notes that since F is local, the image F (t) of t does indeed lie in tA, so that F (t)/t is a well-defined element of A).

We make the following definition.

3.1. Definition. We say that an A-module M is admissible if M is A-torsion, and if M[t] is finite-dimensional over k.

If we let K denote the field of fractions of A, and Â denote the completion of A, then the functor M ↦ Hom_{A}(M, K/A) induces an anti-equivalence of categories between the category of admissible A-modules and the category of finitely generated Â-modules. If Hom_{A}(M, K/A) has free rank r over Â, then we say that the admissible A-module M has corank r. Since F is a local endomorphism, we see that if M is an admissible A-module, of corank r, then the same is true of F^{∗}M.

Our goal in this section is to prove some basic results concerning finitely generated left A[F]-modules that are admissible as A-modules.

3.2. Proposition. If M is a finitely generated left A[F]-module that is admissible as an A-module, then M is finitely presented as a left A[F]-module.

Proof. Consider the map ϕ_{M} : F^{∗}M → M. This is an A-linear map between two admissible A-modules of equal corank. Since coker ϕ_{M} is finitely generated over A (as M is finitely generated over A[F]), we conclude that the same is true of ker ϕ_{M}. Lemma 1.2 implies that M is finitely presented over A[F]. □

3.3. Proposition. If M is a finitely generated left A[F]-module that is admissible as an A-module, then any subquotient of M is again finitely generated over A[F] and admissible as an A-module.

Proof. Any subquotient of an admissible A-module is again admissible as an Amodule, while any quotient of a finitely generated A[F]-module is again finitely generated over A[F]. It follows that any subquotient of M is admissible as an A-module, and that any quotient M′′ of M is finitely generated over A[F]. Proposition 3.2 shows that M and M′′ are furthermore both finitely presented over A[F]. If M′ is an A[F]-submodule of M, then applying this to M′′ = M/M′, and recalling that A[F] is left coherent, we conclude that M′ is also finitely presented, and so in particular finitely generated, over A[F]. Having proved the proposition for quotients and subobjects, it obviously also holds for arbitrary subquotients. □

3.4. Corollary. If M is a finitely generated left A[F]-module that is admissible as an A-module, then M is of finite length.

Proof. Since M is admissible as an A-module, it is Artinian as an A-module, and so also as an A[F]-module. Propositions 3.2 and 3.3 show that any A[F]-submodule of M is finitely generated, and thus that M is also Noetherian as an A[F]-module. The corollary follows. □

The following proposition gives a criterion for recognizing when a finitely generated left A[F]-module is admissible over A.

3.5. Proposition. If M is a finitely generated left A[F]-module that is torsion over A, then the following conditions are equivalent:

(1) M is admissible as an A-module.

(2) The quotient M/tM is finite-dimensional over k.

(3) The quotient M/tM is torsion over k[F].

Proof. That 1 implies 2 is valid for any admissible A-module, while any k[F]-module that is finite-dimensional over k is certainly k[F]-torsion, so that 2 implies 3. It remains to show that 3 implies 1. Thus for the remainder of the proof we assume that M/tM is torsion over k[F]. Since M is A-torsion by assumption, we must prove that M[t] is finite-dimensional over k.

We argue by induction on the number of generators of M as an A[F]-module. Suppose first that M is cyclic. Let m ∈ M be a generator, and let M_{0} ⊂ M denote the A-submodule of M that m generates. By assumption, M_{0} ⟶^{∼} A/t^{r}A for some r ≥ 0. The inclusion M_{0} ⊂ M induces a surjection A[F] ⊗_{A} M_{0} → M of A[F]- modules, with kernel N, say. Note that there are isomorphisms (of k[F]-modules)

(2)  (A[F] ⊗_{A} M_{0})/t(A[F] ⊗_{A} M_{0}) ⟶^{∼} k[F]

and

(3)  (A[F] ⊗_{A} M_{0})[t] ⟶^{∼} k[F].

The short exact sequence 0 → N → A[F] ⊗_{A} M_{0} → M → 0 induces an exact sequence

N/tN → (A[F] ⊗_{A} M_{0})/t(A[F] ⊗_{A} M_{0}) → M/tM.

Taking into account the isomorphism of (2), and the fact that M/tM is torsion over k[F] by assumption, we see that may choose an element n ∈ N whose image in (A[F] ⊗_{A} M_{0})/t(A[F] ⊗_{A} M_{0}) is non-zero. Let M′ denote the left A[F]-submodule of A[F]⊗_{A} M_{0} generated by n; since n is non-zero, so is M′. Write M′′ := (A[F]⊗_{A} M_{0})/M′.

Now consider the long exact sequence of Tor-modules associated to the short exact sequence 0 → M′ → A[F] ⊗_{A} M_{0} → M′′ → 0. Taking into account the isomorphisms (2) and (3), we may write it in the following form:

(4)  0 → M′[t] → k[F] → M′′[t] → M′/tM′ → k[F] → M′′/tM′′ → 0.

Since M′ is cyclic over A[F] by construction, the quotient M′/tM′ is cyclic over k[F], while, again by construction, it has non-zero image in k[F]. Thus the fifth arrow in (4) must be injective, and so (4) gives rise to a short exact sequence

0 → M′[t] → k[F] → M′′[t] → 0.

Since M′ is a non-zero submodule of the torsion A-module A[F]⊗_{A} M_{0}, we see that M′[t] is non-zero. Thus M′′[t] is isomorphic to the quotient of k[F] by a non-zero submodule, and hence is finite-dimensional over k. It follows that M′′ is admissible over A, and thus so is its quotient M.

Suppose now that M is generated over A[F] by n elements, m_{1}, …, m_{n}. Let M′ denote the A[F]-submodule of M generated by m_{1}, and write M′′ := M/M′. Clearly M′′ is A-torsion, while M′′/tM′′, being a quotient of M/tM, is k[F]-torsion. Since M′′ is generated by n − 1 elements, we conclude by induction on n that M′′ is admissible. Now consider the long exact sequence of Tor-modules associated to the short exact sequence 0 → M′ → M → M′′ → 0:

0 → M′[t] → M[t] → M′′[t] → M′/tM′ → M/tM → M′′/tM′′ → 0.

Since M′′ is admissible, we see that M′′[t] is finite-dimensional over k, while by assumption M/tM is k[F]-torsion. Thus M′/tM′ is also k[F]-torsion. Since M′ is also cyclic over A[F], and A-torsion, we conclude from the case n = 1 of the proposition that we have already proved that M′ is admissible over A. As M is an extension of admissible A-modules, it is itself admissible over A. □

4. Applications to the representation theory of GL_{2}

Let E be an unramified finite extension of ℚ_{p}, of degree d, with ring of integers O.

Write G := GL_{2}(E), K = GL_{2}(O), P = (\begin{matrix}E^{×} & E \\ 0 & E^{×} \end{matrix}) (the Borel subgroup of

upper triangular matrices in G), N = (\begin{matrix}1 & E \\ 0 & 1 \end{matrix}) (the unipotent radical of P), N_{0} =

(\begin{matrix}1 & O \\ 0 & 1 \end{matrix}) (a compact open subgroup of N), Γ = (\begin{matrix}O^{×} & 0 \\ 0 & 1 \end{matrix}), F = (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) ∈ P, and

let Z denote the centre of G.

Let k be a finite field of characteristic p, and write A = k[[N_{0}]], the completed group ring of N_{0} over k. Recall that A is a complete regular local ring of dimension d. We denote its maximal ideal by 𝔪.

Note that N_{0} is closed under conjugation by F; indeed, F nF^{−1} = n^{p} for all n ∈ N_{0}. Thus the endomorphism of A induced by conjugation by F is equal to the relative Frobenius F_{A/k}, and so in particular is a flat endomorphism of A, to which the theory of the preceding sections applies. We write A[F] to denote the non-commutative polynomial ring over A in which F acts via F_{A/k}. We also write A[F, Γ] to denote the twisted group ring of Γ over A[F]: the elements of Γ commute with F and with the field k of scalars, and conjugate the elements of N_{0} just as they do in the group G, namely, if (\begin{matrix}a & 0 \\ 0 & 1 \end{matrix}) ∈ Γ and (\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) ∈ N_{0}, then

(\begin{matrix}a & 0 \\ 0 & 1 \end{matrix})(\begin{matrix}1 & x \\ 0 & 1 \end{matrix})(\begin{matrix}a & 0 \\ 0 & 1 \end{matrix})^{−1} = (\begin{matrix}1 & ax \\ 0 & 1 \end{matrix}).

The endomorphism F_{A/k} is local, and induces the identity endomorphism on k. Thus if M is any A[F]-module, then Lemma 2.1 shows that the Tor-modules Tor_{•}^{A}(k, M) are naturally k[F]-modules, where k[F] is the usual commutative polynomial ring over k.

If V is a smooth P -representation defined over k, then it is in particular smooth as an N_{0}-representation, and so is naturally a torsion A-module. Combining this A-module structure with the action of F and Γ on V, we see that V is naturally an A[F, Γ]-module.

4.1. Definition. If V is a smooth P -representation over k, and if V_{0} is a Γsubrepresentation of V, then we let M(V, V_{0}) denote the left A[F]-submodule of V generated by V_{0} (which we observe is naturally an A[F, Γ]-module).

Let C denote the abelian category of left A[F, Γ]-modules that are torsion as Amodules, and let C′ denote the full subcategory consisting of modules each element of which is annihilated by a power of F. Evidently C′ is a Serre subcategory of C, and we let C′′ denote the quotient category C′′ := C/C′.

4.2. Definition. If V is smooth P -representation, then we define M(V) := V, regarded as an object of C′′.

4.3. Proposition. The formation of M(V) yields an exact and faithful functor from the category of smooth P -representations over k to the category C′′.

Proof. The exactness of M(V) is immediate from the definition. To see that it is faithful, it suffices to note that if V is non-zero, then M(V) is not the zero object in C′′. Indeed, since the action of F on V is injective, if V is non-zero then M(V) does not belong to the category C′. □

The following proposition provides a means of studying the functor M(V).

4.4. Proposition. If V is a smooth representation of P over k, and if V_{0} is a ΓZ-invariant k-vector subspace of V that generates V over P, then the inclusion M(V, V_{0}) ⊂ V induces an isomorphism in the quotient category C′′.

Proof. By assumption V = k[P]V_{0}. Since P = F^{−ℕ}N_{0}F^{ℕ}ΓZ, and since V_{0} is ΓZinvariant, we may rewrite this equality as V = k[F^{−1}]M(V, V_{0}). Thus any element of the quotient V/M(V, V_{0}) is annihilated by some power of F, i.e. V/M(V, V_{0}) is an object of C′. This proves the proposition. □

The preceding proposition applies in particular if V is a smooth G-representation, and if V_{0} is a KZ-invariant subspace of V that generates V over G. Indeed, G = P K, and so V = k[G]V_{0} = k[P]V_{0}. Unfortunately, this proposition is not so useful in general, since we can’t say much about the A[F, Γ]-modules M(V, V_{0}). Indeed, the only general result on their structure that we can prove at the moment is the following:

4.5. Proposition. If V is an admissible smooth representation of G over k, and if V_{0} is a finite-dimensional KZ-invariant k-vector subspace of V, then each of the Tor-modules Tor_{•}^{A}(k, M(V, V_{0})) is a torsion k[F]-module.

Proof. If W denotes the G-subrepresentation of V generated by V_{0}, then since M(W, V_{0}) = M(V, V_{0}), it is no loss of generality to suppose that V is generated by V_{0}. Proposition 4.4 then shows that V/M(V, V_{0}) is an object of C′. This is easily seen to imply that each element of any of the Tor-modules Tor_{•}^{A}(k, V/M(V, V_{0})) is annihilated by some power of F. In particular, each of these Tor-modules is k[F]-torsion. A consideration of the long exact sequence of Tor-modules associated to the short exact sequence

0 → M(V, V_{0}) → V → V/M(V, V_{0}) → 0

then shows that to prove the proposition, it suffices (in fact, is equivalent) to prove that each of the Tor-modules Tor_{•}^{A}(k, V ) is a torsion k[F]-module. Since Tor_{•}^{A}(k, V ) ⟶^{∼} H^{d−•}(N_{0}, V), this follows from [6, Thm. 3.2.3(1)]. (Note that under the preceding isomorphism, the action of F on the source corresponds to the action of F, thought of as an element of Z_{M}^{+}, in the notation of [6], on the target.) □

The preceding result suggests the following question.

4.6. Question. If V is a finitely generated admissible smooth G-representation over k, can we choose a finite-dimensional KZ-invariant k-vector subspace V_{0} of V which generates V over G, with the additional property that M(V, V_{0}) is finitely presented over A[F]?

An affirmative answer to the preceding question, together with Propositions 2.2 (applied with B = k) and 4.5, would imply that M(V, V_{0})[𝔪] (which coincides with Tor_{d}^{A}(k, M(V, V_{0}))) is a finitely generated torsion k[F]-module, and hence is finitedimensional over k, or equivalently, that M(V, V_{0}) is an admissible A-module (in an obvious sense, generalizing that given by Definition 3.1 in the case of a discrete valuation ring).

At the moment we can answer the preceding question in the case when E = ℚ_{p}, in which case it does indeed have an affirmative answer, as we now show. (Note, though, that we will reverse the logic in the argument of the preceding paragraph: by applying the results of Section 3, we will first show that M(V, V_{0}) is admissible, and then as a consequence deduce that it is finitely presented over A[F].)

4.7. Theorem. If E = ℚ_{p}, if V is an admissible smooth representation of G over k, and if V_{0} is a finite-dimensional k-vector subspace of V, then M(V, V_{0}) is finitely presented and of finite length as a left A[F]-module, and is admissible as an A-module.

Proof. Since d = 1, the complete local ring A is a discrete valuation ring; let t denote a uniformizer. We begin by noting that since M(V, V_{0}) is finitely generated over A[F] by construction, it follows from Proposition 3.2 and Corollary 3.4 that if M(V, V_{0}) is admissible over A, then it is both finitely presented and of finite length over A[F]. Thus it suffices to prove that M(V, V_{0}) is admissible.

Write W := k[G]V_{0} and V_{1} := k[KZ]V_{0}. Since V is admissible, we see that V_{1} is finite-dimensional over k, while since G = P K, we see that V_{1} generates W over P. Also, we have that M(W, V_{1}) ⊃ M(W, V_{0}) = M(V, V_{0}), and hence M(V, V_{0}) is admissible if M(W, V_{1}) is. Since Proposition 4.5 shows that M(W, V_{1})/tM(W, V_{1}) is k[F]-torsion, it follows from Proposition 3.5 that M(W, V_{1}) is admissible, as required. □

4.8. Remark. The preceding result is essentially due to Colmez [4], although he phrases it differently, as we now explain.

In the context of Theorem 4.7, Colmez writes Π rather than V, and W rather than V_{0}, and writes I^{+}(W, Π) to denote the A[F]-module that we have called M(V, V_{0}). In the following discussion we will use Colmez’s notation, to facilitate the comparison with [4].

The dual Hom_{k}(I^{+}(W, Π), k) is a finitely generated A-module (since K^{+}(W, Π) is admissible as an A-module, by Theorem 4.7), which Colmez denotes D_{W}^{♮} (Π). The cokernel of the map ϕ_{I^{+}(W,Π)} : F^{∗}I^{+}(W, Π) → I^{+}(W, Π) is finitely generated and torsion over A (since I^{+}(W, Π) is finitely generated over A[F] and torsion over A), and thus the the kernel of the induced map D_{W}^{♮} (Π) → F^{∗}D_{W}^{♮} (Π) is finitely generated and torsion over A. If we write K := k((t)) to denote the fraction field of A, then the tensor product K ⊗_{A} D_{W}^{♮} (Π) is well-defined (up to a canonical isomorphism) independently of the choice of W (as follows from Proposition 4.4, or more accurately, its proof). Colmez denotes this tensor product by D(Π). The induced map D(Π) → F^{∗}D(Π) is then an isomorphism of finite-dimensional Kvector spaces. Together with the Γ-action on D(Π) induced by the Γ-action on I^{+}(W, Π), the inverse of this isomorphism equips D(Π) with the structure of a finite-dimensional (ϕ, Γ)-module over K. It is the finite-dimensionality of D(Π), rather than Theorem 4.7 itself, which is proved by Colmez [4, Thm. 4.13].

Let us also remark that Colmez in fact assumes that the representation Π is of finite length, and that his proof relies strongly on the classification of irreducible representations (due to Barthel-Livné [1] and Breuil [3]). In [5] we proved, again relying on the classification of irreducibles, that any finitely generated admissible smooth GL_{2}(ℚ_{p})-representation is in fact of finite length. As we observe in the following corollary, this follows directly from Theorem 4.7. We will also show how the theory of A[F]-modules can be used to give a simple proof of the classification of irreducible admissible smooth representations of GL_{2}(ℚ_{p}).

It follows from Proposition 4.4 and Theorem 4.7, that on the category of finitely generated admissible smooth GL_{2}(ℚ_{p}) representations, the functor M(V) restricts to a faithful exact functor with image lying in the full subcategory of C′′ consisting of finite-length objects.

4.9. Corollary. If E = ℚ_{p}, then any finitely generated admissible smooth Grepresentation over k is of finite length as a P -representation (and so in particular, as a G-representation).

Proof. If V is a finitely generated admissible smooth G-representation, then Proposition 4.4 and Theorem 4.7 shows that M(V) is a finite length object of C′′. Since the functor M is faithful and exact, we conclude that V is of finite length as a P -representation, as claimed. □

We conclude this section by noting the following result, a kind of converse to Theorem 4.7 (which however holds for arbitrary unramified E):

4.10. Proposition. If V is a smooth G-representation over k, and if V_{0} is a KZinvariant subspace of V that generates V_{0} over G which has the additional property that M(V, V_{0}) is admissible over A, then V is admissible as a G-representation.

Proof. If K_{1} denotes the first congruence subgroup of K, then (using, e.g., the Iwahori decomposition of K_{1}), one sees that M(V, V_{0}) is K_{1}-invariant. Since M(V, V_{0}) is admissible over A by assumption, it is certainly admissible as a K_{1}- representation. If g ∈ K, then gM(V, V_{0}) is also K_{1}-invariant (since K_{1} is normal in K), and is again admissible over K_{1}. The Cartan decomposition of G shows that G = ∐_{g∈K/K_{1},n≥0} gK_{1}F^{n}KZ, and thus that V = ∑_{g∈K/K_{1}} gM(V, V_{0}). Hence V is the sum of a finite number of admissible K_{1}-subrepresentations, and thus is itself admissible as a K_{1}-representation, and hence also as a G-representation. □

5. Irreducible supersingular representations of GL_{2}(ℚ_{p})

We maintain the notation of the preceding section, and in addition set E = ℚ_{p}. We also let I and I_{1} have their usual meanings (so I is the upper triangular Iwahori subgroup of K, and I_{1} its pro-p Sylow subgroup).

Up to a twist, any irreducible representation of K over k is of the form Sym^{r}k^{2}, where 0 ≤ r ≤ p − 1. We regard Sym^{r}k^{2} as a KZ-representation by requiring the element p ∈ ℚ_{p}^{×} ⟶^{∼} Z to act trivially. Recall that the Hecke algebra H := End_{G}(c − Ind_{KZ}^{G} Sym^{r}k^{2}) is isomorphic to a polynomial ring in one generator T over k. The Hecke operator T can be defined in various ways; we recall one useful point of view here. (This description of T was inspired by an argument of Kevin Buzzard.) Recall that for any smooth G-representation V, there is a functorial isomorphism

Hom_{G}(c − Ind_{KZ}^{G} Sym^{r}k^{2}, V) ⟶^{∼} Hom_{KZ}(Sym^{r}k^{2}, V).

Thus rather than describing T directly as an endomorphism of c − Ind_{KZ}^{G} Sym^{r}k^{2}, we instead describe the induced endomorphism of the functor Hom_{KZ}(Sym^{r}k^{2}, –).

The space of invariants (Sym^{r}k^{2})^{I_{1}} is one-dimensional, and I acts on this space through the character z^{r} ⊗ 1. Thus there is a surjection

(5)  Ind_{IZ}^{KZ} z^{r} ⊗ 1 → Sym^{r}k^{2},

which fits into a short exact sequence

0 → (Sym^{p−1−r}k^{2}) ⊗ det^{r} → Ind_{IZ}^{KZ} z^{r} ⊗ 1 → Sym^{r}k^{2} → 0.

Replacing r by p−1−r and twisting, we obtain a corresponding short exact sequence

(6)  0 → Sym^{r}k^{2} → Ind_{IZ}^{KZ} 1 ⊗ z^{r} → (Sym^{p−1−r}k^{2}) ⊗ det^{r} → 0.

If W is any smooth G-representation, we can then construct the following natural map:

(7) Hom_{KZ}(Sym^{r}k^{2}, W) ↪ Hom_{KZ}(Ind_{IZ}^{KZ} z^{r} ⊗ 1, W)

⟶^{∼} Hom_{IZ}(z^{r} ⊗ 1, W) ⟶^{{wF \atop ∼}} Hom_{IZ}(1 ⊗ z^{r}, W)

⟶^{∼} Hom_{KZ}(Ind_{IZ}^{KZ} 1 ⊗ z^{r}, W) → Hom_{KZ}(Sym^{r}k^{2}, W),

in which the first arrow is induced by the surjection (5) (and hence is injective), the first and third isomorphisms are provided by Frobenius reciprocity, w denotes

the matrix (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ∈ K (so that wF = (\begin{matrix}0 & 1 \\ p & 0 \end{matrix})), and the final arrow is induced

by the injection in (6). The composite (7) induces an endomorphism of the functor Hom_{KZ}(Sym^{r}k^{2}, –), which as noted above corresponds in turn to an endomorphism of c − Ind_{KZ}^{G} Sym^{r}. This endomorphism is the Hecke operator T.

Recall that any absolutely irreducible admissible smooth G-representation over k is, up to a twist, a quotient of (c − Ind_{KZ}^{G} Sym^{r}k^{2})/(T − λ)(c − Ind_{KZ}^{G} Sym^{r}k^{2}), for some r ∈ {0, …, p − 1} and some λ ∈ k. In the case when λ ≠ 0, these quotients are easily analyzed [1], and give rise to one-dimensional, principal series, or special representations as their irreducible subquotients. In the case when λ = 0, Breuil [3] showed that the corresponding quotients are irreducible and admissible. We will give another proof of Breuil’s result here, via an analysis of the associated A[F]-modules.

5.1. Theorem. If r ∈ {0, …, p−1}, then (c−Ind_{KZ}^{G} Sym^{r}k^{2})/T (c−Ind_{KZ}^{G} Sym^{r}k^{2}) is an irreducible admissible smooth representation of G.

Proof. Write V := (c − Ind_{KZ}^{G} Sym^{r}k^{2})/T (c − Ind_{KZ}^{G} Sym^{r}k^{2}), let V_{0} denote the KZ-subrepresentation Sym^{r}k^{2} of V, and write v to denote a basis for the onedimensional subspace V_{0}^{N_{0}} of v. Since v is fixed by N_{0}, we have that

(8)  tv = 0.

(Recall that t denotes the uniformizer of A := k[[N_{0}]].) Also, Γ acts on v via the character \overline{ε}^{r}, where \overline{ε} denotes the 𝔽_{p}^{×}-valued character of Γ defined by (\begin{matrix}a & 0 \\ 0 & 1 \end{matrix}) ↦ a mod p.

Since, by assumption, T annihilates the embedding V_{0} ↪ V, a consideration of the description of T afforded by (7) shows that under the action of KZ, wF v generates a copy of V_{1} := (Sym^{p−1−r}k^{2}) ⊗ det^{r} in V, with wF v spanning the onedimensional subspace V_{1}^{N_{0}}. (This is a well-known result, due originally to Breuil.) In particular

(9)  twF v = 0.

Since w ∈ K, we see that F v also lies in V_{1}, and (recalling the manner in which w acts on V_{1}) that

(10)  t^{p−1−r}F v = cwF v,

for some non-zero scalar c ∈ k^{×}. Since (wF)^{2} = 1, we find that wF wF v = v. Recalling the manner in which w acts on V_{0}, we deduce that F wF v generates V_{0} over N_{0}, or equivalently, over A, and, in particular, that

(11)  t^{r}F wF v = dv,

for some non-zero scalar d ∈ k^{×}. Combining the relations (8), (9), (10), and (11), we find that M(V, V_{0}) is a cyclic A[F, Γ]-module, with v as a generator, and that v satisfies (at least) the relations

(12)  tv = t^{p−r}F v = (t^{(p−1)(p−r)}F^{2} − cd)v = 0, γv = \overline{ε}(γ)^{r}v for γ ∈ Γ.

These relations imply that M(V, V_{0}) is A-torsion and that M(V, V_{0})/tM(V, V_{0}) = 0, and thus Propositions 3.5 and 4.10 show that V is admissible.

Let M denote the A[F, Γ]-module cyclically generated by the element v satisfying the relations (12). We will show that M is irreducible. This will imply that M(V, V_{0}) (which we have shown is a non-zero quotient of M) is isomorphic to M, and hence is irreducible. Since the functor V ↦ M(V) is exact and faithful, we will conclude that V is irreducible as a G-representation, as claimed, and so will complete the proof of the theorem.

We first observe that M[t] = k⟨v, t^{p−1−r}F v⟩, as the reader may easily verify. Also, by assumption the group Γ acts on v via the character \overline{ε}^{r}. The reader may then easily deduce that Γ acts trivially on t^{p−1−r}v. Suppose that N is a non-zero A[F, Γ]-submodule of M. We will show that N contains either v or t^{p−1−r}F v, and so is equal to M. (Indeed, v is a cyclic generator of M, and we have the relation t^{r}F t^{p−1−r}F v = cdv, with cd ≠ 0.)

As N is non-zero, its submodule N[t] is a non-zero subspace of M[t], and so contains some non-zero linear combination αv + βt^{p−1−r}F v. If β = 0, then α ≠ 0, and so M contains v. If α = 0, then β ≠ 0, and so M contains t^{p−1−r}F v. Suppose that neither α nor β vanishes. If r ≥ p − r, then applying t^{r}F to the element αv+βt^{p−1−r}F v, we find that M contains t^{(p−1)(p−r)}F^{2}v = cdv, and thus contains v. If p − 2 − r ≥ r (so that p − 1 − r ≥ r + 1), then applying t^{p−1−r}F to the element αv + βt^{p−1−r}F v, we find that M contains t^{p−1−r}F v. Thus, except in the case r = (p − 1)/2, we conclude that M is irreducible even as an A[F]-module.

Finally, suppose that r = (p − 1)/2. Note that in this case necessarily p > 2 and 0 < r < p − 1. We consider the action of the group Γ on N[t]. Since Γ acts on the element v (resp. t^{(p−1)/2}F v) via the character \overline{ε}^{(p−1)/2} (resp. the trivial character), and since these characters are distinct, we find that since N[t] contains some non-zero linear combination αv + βt^{(p−1)/2}F v, it necessarily contains one of v or t^{(p−1)/2}F v. This completes the proof of the theorem. □

5.2. Remark. In fact, the proof of the theorem shows that V is even irreducible as a P -representation – a result originally due to Berger [2, Thm. 2.3.1 (1)], and which also follows from the G-irreducibility together with [8, Thm. 4.3].

5.3. Remark. If we let D denote the (ϕ, Γ)-module attached to

V := (c − Ind_{KZ}^{G} Sym^{r}k^{2})/T (c − Ind_{KZ}^{G} Sym^{r}k^{2})

as in Remark 4.8 (so D = k((t)) ⊗_{k[[t]]} Hom_{k}(M(V, V_{0}), k)), then the proof of the preceding result shows that D is irreducible just as a ϕ-module over k((t)), except when r = (p − 1)/2 (in which case one may check that it is in fact reducible as a ϕ-module, though as the preceding theorem shows, it is irreducible as a (ϕ, Γ)- module). The (ϕ, Γ)-module D corresponds to the representation Ind_{ℚ_{p^{2}}}^{ℚ_{p}} ω_{2}^{r+1} of the Galois group G_{ℚ_{p}} (where ω_{2} denotes a fundamental character of level 2, which is a character of G_{ℚ_{p^{2}}} ), and the irreducibility of D as a ϕ-module for r ≠ (p − 1)/2

corresponds to the fact that Ind_{ℚ_{p^{2}}}^{ℚ_{p}} ω_{2}^{r+1} remains irreducible when restricted to Galois group G_{ℚ_{p,∞}} of the p-adic cyclotomic extension ℚ_{p,∞} of ℚ_{p}, except in the case when r = (p − 1)/2.

References

[1] Barthel L., Livné R., Irreducible modular representations of GL_{2} of a local field, Duke Math. J. 75 (1994), 261–292.

[2] Berger L., Représentations modulaires de GL_{2}(ℚ_{p}) et représentations galoisiennes de dimension 2, to appear in Asterisque.

[3] Breuil C., Sur quelques représentations modulaires et p-adiques de GL_{2}(ℚ) I, Compositio Math. 138 (2003), 165–188.

[4] Colmez P., Représentations de GL_{2}(ℚ_{p}) et (ϕ, Γ)-modules (version provisoire et partielle),

preprint (2007), available at http://math.jussieu.fr/~colmez/publications.html

[5] Emerton M. Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties, preprint (2008), available at

http://math.northwestern.edu/~emerton/preprints.html

[6] Emerton M. Ordinary parts of admissible representations of p-adic reductive groups II. Derived functors, preprint (2007), available at

http://math.northwestern.edu/~emerton/preprints.html

[7] Emerton M., Kisin M., The Riemann–Hilbert correspondence for unit F –crystals, Asterisque 293, 2004.

[8] Paskunas V., On the restriction of representations of GL_{2}(F) to a Borel subgroup, to appear in Compositio Math.

Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston, IL 60208

E-mail address: emerton@math.northwestern.edu