Draft: May 19, 2008
ON A CLASS OF COHERENT RINGS, WITH APPLICATIONSTO THE SMOOTH REPRESENTATION THEORY OF GL2 (Qp ) IN
CHARACTERISTIC p
MATTHEW EMERTONIn 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 OF,X on a smooth finite
type k-scheme X, introduced in [7], is coherent. (See Example 1.4 below.) Thissheds 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 GL2 (Qp ), then any smooth P -representation over kis naturally an A[F ]-module. (See Section 4 below.) In particular, this is true ofa smooth GL2 (Qp )-representation over k. Suppose that V is a finitely generated
admissible smooth GL2 (Qp )-representation over k, and let V0 be finite-dimensional
k-subspace invariant under KZ (where, as usual, K := GL2 (Zp ) and Z denotes
the centre of GL2 (Qp )) which generates V over GL2 (Qp ). We show that the A[F ]submodule of V generated by V0 is finitely presented over A[F ], and is admissible
as an A-module – i.e. is Pontrjagin dual to a finitely generated A-module. (SeeTheorem 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 theclassification of irreducible admissible smooth GL2 (Qp )-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 takenwith respect to the map F : A → A. If M is an A[F ]-module, then the action
The author was supported in part by NSF grant DMS-0701315.
1

2

MATTHEW EMERTON

of the element F on M induces an A-linear map φM : F ∗ M → M, defined via
φM : a ⊗ m 7→ aF · m. Conversely, if M is an A-module, and φM : F ∗ M → M isan 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 φM arises from this structure via
the preceding construction.
If 0 → M 0 → M → M 00 → 0 is an exact sequence of A[F ]-modules, then we may
form the diagram/ F ∗M/0/ F ∗M 0/ F ∗ M 00
0
φM 0

0

φM


/ M0


/M

φM 00


/ M 00/ 0,whose top row is again exact (since F is flat). The snake lemma gives rise to an
exact sequence(1)

0 → ker φM 0 → ker φM → ker φM 00→ coker φM 0 → coker φM → coker φM 00 → 0.

We remark that the ring A[F ] is typically non-Noetherian. Our main goal in thissection 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 φM is finitely generated over A.
Proof. If A[F ]n → M is surjective, then so is the induced map∼

An −→ coker φA[F ]n → coker φM .
This proves the “only if” direction.
Suppose, conversely, that coker φM is finitely generated over A. For any d ≥ 0,
Ld
write M ≤d := M ∩ ( i=0 AF i ) ⊂ M.
is Noetherian, each M ≤d is finitely
S Since A≤dgenerated over A, and clearly M = d≥0 M . If we choose d ≥ 0 such that thenatural map M ≤d → coker φM is surjective, then ford0 > d, we find that
T any
0
≤d0
≤d
≤d0
M
⊂ M + F M. On the other hand, clearly F M M= F M ≤d −1 . Thus00

M ≤d ⊂ M ≤d + F M ≤d −1 .
0Proceeding by recursion on d0 , we find that M ≤d ⊂ A[F ]M ≤d , and thus (since
d0 > 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 φM 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 0 → A[F ]n → M → 0. Applied to this short exact sequence, theexact sequence (1) reduces to
0 → ker φM → coker φM 0 → An → coker φM → 0.
In particular, we see that ker φM is finitely generated over A if and only if the sameis true of coker φM 0 . On the other hand, lemma 1.1 shows that coker φM 0 is finitely
generated over A if and only if M 0 is finitely generated over A[F ]. The lemma
follows.
COHERENT RINGS AND REPRESENTATIONS OF GL2 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 φM ⊂ ker φA[F ] = 0. Thus ker φM 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 thisdoesn’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, thenboth the absolute Frobenius endomorphism FA of A and the relative Frobenius
endomorphism FA/k of A are flat maps. Thus the rings A[FA ] and A[FA/k ] are left
coherent. More generally, if X is a smooth k-scheme, then we may form sheaves ofrings OX [F ] and OX [FX/k ], both of which are then seen to be left coherent. (The
first of these coincides with the sheaf denoted OF,X in [7].)1.5. Example. Continuing to suppose that X is a smooth scheme over a field k ofcharacteristic p > 0, a quasi-coherent sheaf of left OX [FX ]-modules M is said to be∗unit if the map φM : FX
M → M is an isomorphism. In particular, the kernel ofφM then vanishes, and so it follows from Lemma 1.2 that a locally finitely generated
unit OX [FX ]-module is in fact locally finitely presented as a left OX [FX ]-module,a result which was proved by a different method in [7]. (See Remark 6.1.2 of thatreference.)
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 willagain denote by F ) compatible with the endomorphism F of A, in the sense thatthe diagram
F /
A
A

F / 
B
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 ], –) −→ TorA
• (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 aresolution 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
TorA
• (B, M ) extends in a natural way to a left B[F ]-module structure.

4

MATTHEW EMERTON

2.2. Proposition. If M is a finitely presented left A[F ]-module, then each of theTor-modules TorA
• (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 precedinglemma shows that we may compute the modules TorA
• (B, M ), with their natural
B[F ]-module structure, as the homology of the complex B[F ] ⊗A[F ] P • . This is acomplex 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
OX [FX ]-modules to complexes of left OY [FY ]-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 OY [FY ] over f −1 (OX [FX ]). As Lemma 2.1 shows, on the underlying
OX and OY -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, thefact 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 ringAs 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 theinduced 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 TorA
• (k, M ) have anatural k[F ]-module structure. Of course, since A is a discrete valuation ring, it has
projective dimension 1, and the modules TorA
• (k, M ) admit the following explicit
descriptions, for any A-module M :∼Tor0 (k, M ) −→ M/tM,∼

Tor1 (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 bym mod tM 7→ F m mod tM,
while the action of F on M [t] is given by
F (t)
Fm
t
(where one notes that since F is local, the image F (t) of t does indeed lie in tA, sothat F (t)/t is a well-defined element of A).
We make the following definition.
M [t] 3 m 7→

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.

COHERENT RINGS AND REPRESENTATIONS OF GL2 OF A p-ADIC FIELD

5b denote the completion of A,
If we let K denote the field of fractions of A, and A
then the functor M 7→ HomA (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 HomA (M, K/A) has free rank r over Â, then we say that theadmissible A-module M has corank r. Since F is a local endomorphism, we seethat 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 φM : F ∗ M → M . This is an A-linear map between two
admissible A-modules of equal corank. Since coker φM is finitely generated over A(as M is finitely generated over A[F ]), we conclude that the same is true of ker φM .
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 00 of M is finitely generated over A[F ]. Proposition 3.2 shows that M and M 00 are furthermore both finitely presented over A[F ].
If M 0 is an A[F ]-submodule of M , then applying this to M 00 = M/M 0 , and recalling that A[F ] is left coherent, we conclude that M 0 is also finitely presented, andso 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 asan 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 ]-submoduleof 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 ]-modulethat 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.

6

MATTHEW EMERTON

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 M0 ⊂ M denote∼
the A-submodule of M that m generates. By assumption, M0 −→ A/tr A for somer ≥ 0. The inclusion M0 ⊂ M induces a surjection A[F ] ⊗A M0 → M of A[F ]modules, with kernel N , say. Note that there are isomorphisms (of k[F ]-modules)(2)∼(A[F ] ⊗A M0 )/t(A[F ] ⊗A M0 ) −→ k[F ]

and
(3)

∼

(A[F ] ⊗A M0 )[t] −→ k[F ].

The short exact sequence 0 → N → A[F ] ⊗A M0 → M → 0 induces an exact
sequence
N/tN → (A[F ] ⊗A M0 )/t(A[F ] ⊗A M0 ) → 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 M0 )/t(A[F ] ⊗A M0 ) is non-zero. Let M 0 denote the left A[F ]-submoduleof A[F ]⊗A M0 generated by n; since n is non-zero, so is M 0 . Write M 00 := (A[F ]⊗A
M0 )/M 0 .
Now consider the long exact sequence of Tor-modules associated to the short
exact sequence 0 → M 0 → A[F ] ⊗A M0 → M 00 → 0. Taking into account theisomorphisms (2) and (3), we may write it in the following form:(4)

0 → M 0 [t] → k[F ] → M 00 [t] → M 0 /tM 0 → k[F ] → M 00 /tM 00 → 0.

Since M 0 is cyclic over A[F ] by construction, the quotient M 0 /tM 0 is cyclic overk[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 0 [t] → k[F ] → M 00 [t] → 0.
Since M 0 is a non-zero submodule of the torsion A-module A[F ] ⊗A M0 , we see thatM 0 [t] is non-zero. Thus M 00 [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 00 is admissible
over A, and thus so is its quotient M .
Suppose now that M is generated over A[F ] by n elements, m1 , . . . , mn . Let
M 0 denote the A[F ]-submodule of M generated by m1 , and write M 00 := M/M 0 .
Clearly M 00 is A-torsion, while M 00 /tM 00 , being a quotient of M/tM , is k[F ]-torsion.
Since M 00 is generated by n − 1 elements, we conclude by induction on n that M 00
is admissible. Now consider the long exact sequence of Tor-modules associated tothe short exact sequence 0 → M 0 → M → M 00 → 0:
0 → M 0 [t] → M [t] → M 00 [t] → M 0 /tM 0 → M/tM → M 00 /tM 00 → 0.
Since M 00 is admissible, we see that M 00 [t] is finite-dimensional over k, while byassumption M/tM is k[F ]-torsion. Thus M 0 /tM 0 is also k[F ]-torsion. Since M 0
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 0 is admissible over A. As M is anextension of admissible A-modules, it is itself admissible over A.
COHERENT RINGS AND REPRESENTATIONS OF GL2 OF A p-ADIC FIELD

7

4. Applications to the representation theory of GL2
Let E be an unramified finite extension of Qp , of degreed, with ring of integers O.
E× E
Write G := GL2 (E), K = GL2 (O), P =(the Borel subgroup of
×

 0 E
1 E
upper triangular matrices in G), N =(the unipotent radical of P ), N0 =
0 1 





1 O
O× 0
p 0
(a compact open subgroup of N ), Γ =,F =∈ P , and
0 1
0 1
0 1
let Z denote the centre of G.
Let k be a finite field of characteristic p, and write A = k[[N0 ]], the completedgroup ring of N0 over k. Recall that A is a complete regular local ring of dimension d. We denote its maximal ideal by m.
Note that N0 is closed under conjugation by F ; indeed, F nF −1 = np for alln ∈ N0 . Thus the endomorphism of A induced by conjugation by F is equal to
the relative Frobenius FA/k , and so in particular is a flat endomorphism of A, towhich the theory of the preceding sections applies. We write A[F ] to denote thenon-commutative polynomial ring over A in which F acts via FA/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 N0 just as
they do in the group G, namely, if a0 10 ∈ Γ and 10 x1 ∈ N0 , then



−1 

a 0 1 x a 0
1 ax=.
0 1 0 10 1
0 1
The endomorphism FA/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
TorA• (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 N0 -representation, and so is naturally a torsion A-module. Combining thisA-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 V0 is a Γsubrepresentation of V , then we let M (V, V0 ) denote the left A[F ]-submodule of
V generated by V0 (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 0 denote the full subcategory consisting of modules each elementof which is annihilated by a power of F . Evidently C 0 is a Serre subcategory of C,
and we let C 00 denote the quotient category C 00 := C/C 0 .
4.2. Definition. If V is smooth P -representation, then we define M (V ) := V ,regarded as an object of C 00 .
4.3. Proposition. The formation of M (V ) yields an exact and faithful functorfrom the category of smooth P -representations over k to the category C 00 .
Proof. The exactness of M (V ) is immediate from the definition. To see that it isfaithful, it suffices to note that if V is non-zero, then M (V ) is not the zero object
in C 00 . 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 0 .
8

MATTHEW EMERTON

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 V0 is a
ΓZ-invariant k-vector subspace of V that generates V over P , then the inclusion
M (V, V0 ) ⊂ V induces an isomorphism in the quotient category C 00 .
Proof. By assumption V = k[P ]V0 . Since P = F −N N0 F N ΓZ, and since V0 is ΓZinvariant, we may rewrite this equality as V = k[F −1 ]M (V, V0 ). Thus any elementof the quotient V /M (V, V0 ) is annihilated by some power of F , i.e. V /M (V, V0 ) isan object of C 0 . This proves the proposition.

The preceding proposition applies in particular if V is a smooth G-representation,and if V0 is a KZ-invariant subspace of V that generates V over G. Indeed, G =P K, and so V = k[G]V0 = k[P ]V0 . Unfortunately, this proposition is not so usefulin general, since we can’t say much about the A[F, Γ]-modules M (V, V0 ). Indeed,
the only general result on their structure that we can prove at the moment is thefollowing:
4.5. Proposition. If V is an admissible smooth representation of G over k, andif V0 is a finite-dimensional KZ-invariant
k-vector subspace of V , then each of theTor-modules TorA
• k, M (V, V0 ) is a torsion k[F ]-module.
Proof. If W denotes the G-subrepresentation of V generated by V0 , then sinceM (W, V0 ) = M (V, V0 ), it is no loss of generality to suppose that V is generated
by V0 . Proposition 4.4 then shows that V /M (V, V0 ) is an object of C 0 . This is easily
seen to imply that each element of any of the Tor-modules TorA
• k, V /M (V, V0 )is annihilated by some power of F . In particular, each of these Tor-modules isk[F ]-torsion. A consideration of the long exact sequence of Tor-modules associatedto the short exact sequence
0 → M (V, V0 ) → V → V /M (V, V0 ) → 0
then shows that to prove the proposition, it suffices (in fact, is equivalent) tois a torsion k[F ]-module. Since
prove that each of the Tor-modules TorA
• k, V ∼Ad−•Tor• k, V −→ H
(N0 , 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 ZM, in the notation of [6], on thetarget.)
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 V0 of V
which generates V over G, with the additional property that M (V, V0 ) 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, V0 )[m] (which coincides with
TorA
d k, M (V, V0 ) ) is a finitely generated torsion k[F ]-module, and hence is finitedimensional over k, or equivalently, that M (V, V0 ) is an admissible A-module (inan 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 = Qp ,in which case it does indeed have an affirmative answer, as we now show. (Note,COHERENT RINGS AND REPRESENTATIONS OF GL2 OF A p-ADIC FIELD

9though, 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, V0 ) is admissible,
and then as a consequence deduce that it is finitely presented over A[F ].)4.7. Theorem. If E = Qp , if V is an admissible smooth representation of G
over k, and if V0 is a finite-dimensional k-vector subspace of V , then M (V, V0 ) 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 tdenote a uniformizer. We begin by noting that since M (V, V0 ) is finitely generated
over A[F ] by construction, it follows from Proposition 3.2 and Corollary 3.4 that if
M (V, V0 ) 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, V0 ) is admissible.
Write W := k[G]V0 and V1 := k[KZ]V0 . Since V is admissible, we see that V1 isfinite-dimensional over k, while since G = P K, we see that V1 generates W over P .
Also, we have that M (W, V1 ) ⊃ M (W, V0 ) = M (V, V0 ), and hence M (V, V0 ) is
admissible if M (W, V1 ) is. Since Proposition 4.5 shows that M (W, V1 )/tM (W, V1 )is k[F ]-torsion, it follows from Proposition 3.5 that M (W, V1 ) is admissible, as
required.

4.8. Remark. The preceding result is essentially due to Colmez [4], although hephrases it differently, as we now explain.
In the context of Theorem 4.7, Colmez writes Π rather than V , and W rather
than V0 , and writes I + (W, Π) to denote the A[F ]-module that we have called
M (V, V0 ). In the following discussion we will use Colmez’s notation, to facilitate
the comparison with [4].

The dual Homk 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 φI + (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 fieldof A, then the tensor product K ⊗A D\W (Π) is well-defined (up to a canonicalisomorphism) 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 afinite-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 offinite 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 GL2 (Qp )-representation is in fact of finite length. As we observe in thefollowing 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 classificationof irreducible admissible smooth representations of GL2 (Qp ).

10

MATTHEW EMERTON

It follows from Proposition 4.4 and Theorem 4.7, that on the category of finitely
generated admissible smooth GL2 (Qp ) representations, the functor M (V ) restricts
to a faithful exact functor with image lying in the full subcategory of C 00 consistingof finite-length objects.
4.9. Corollary. If E = Qp , 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 00 . 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 V0 is a KZinvariant subspace of V that generates V0 over G which has the additional property
that M (V, V0 ) is admissible over A, then V is admissible as a G-representation.
Proof. If K1 denotes the first congruence subgroup of K, then (using, e.g., theIwahori decomposition of K1 ), one sees that M (V, V0 ) is K1 -invariant. Since
M (V, V0 ) is admissible over A by assumption, it is certainly admissible as a K1 representation. If g ∈ K, then gM (V, V0 ) is also K1 -invariant (since K1 is normalin K),`and is again admissible over K1 . The Cartan
Pdecomposition of G shows thatG = g∈K/K1 ,n≥0 gK1 F n KZ, and thus that V = g∈K/K1 gM (V, V0 ). Hence V isthe sum of a finite number of admissible K1 -subrepresentations, and thus is itself
admissible as a K1 -representation, and hence also as a G-representation.

5. Irreducible supersingular representations of GL2 (Qp )We maintain the notation of the preceding section, and in addition set E = Qp .
We also let I and I1 have their usual meanings (so I is the upper triangular Iwahori
subgroup of K, and I1 its pro-p Sylow subgroup).
Up to a twist, any irreducible representation of K over k is of the form Symr k 2 ,where 0 ≤ r ≤ p − 1. We regard Symr k 2 as a KZ-representation by requiring∼
the element p ∈ Q×p −→ Z to act trivially. Recall that the Hecke algebra H :=r 2
EndG (c − IndG
KZ Sym k ) 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 usefulpoint 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
∼

r 2
r 2
HomG (c − IndG
KZ Sym k , V ) −→ HomKZ (Sym k , V ).
r 2Thus rather than describing T directly as an endomorphism of c − IndG
KZ Sym k ,r 2
we instead describe the induced endomorphism of the functor HomKZ (Sym k , –).
The space of invariants (Symr k 2 )I1 is one-dimensional, and I acts on this space
through the character z r ⊗ 1. Thus there is a surjection

(5)

r 2
r
IndKZ
IZ z ⊗ 1 → Sym k ,COHERENT RINGS AND REPRESENTATIONS OF GL2 OF A p-ADIC FIELD

11

which fits into a short exact sequencer 2
r
0 → (Symp−1−r k 2 ) ⊗ detr → IndKZIZ z ⊗ 1 → Sym k → 0.

Replacing r by p−1−r and twisting, we obtain a corresponding short exact sequence(6)p−1−r 2r
0 → Symr k 2 → IndKZk ) ⊗ detr → 0.
IZ 1 ⊗ z → (Sym

If W is any smooth G-representation, we can then construct the following natural
map:
(7)

r
HomKZ (Symr k 2 , W ) ,→ HomKZ (IndKZIZ z ⊗ 1, W )wF∼∼−→ HomIZ (z r ⊗ 1, W ) −→HomIZ (1 ⊗ z r , W )

∼

r 2
r
−→ HomKZ (IndKZIZ 1 ⊗ z , W ) → HomKZ (Sym k , W ),in which the first arrow is induced by the surjection (5) (and hence is injective),
the first andthirdisomorphisms are providedby Frobenius reciprocity, w denotes
0 1
0 1
the matrix∈ K (so that wF =), and the final arrow is induced
1 0
p 0
by the injection in (6). The composite (7) induces an endomorphism of the functor
HomKZ (Symr k 2 , –), which as noted above corresponds in turn to an endomorphismrof c − IndG
KZ Sym . This endomorphism is the Hecke operator T .
Recall that any absolutely irreducible admissible smooth G-representation over

G
r 2
r 2
k is, up to a twist, a quotient of c − IndG
KZ Sym k /(T − λ) c − IndKZ Sym k ,for some r ∈ {0, . . . , p − 1} and some λ ∈ k. In the case when λ 6= 0, these quotientsare 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 associatedA[F ]-modules.
G
r 2
r 2
5.1. Theorem. If r ∈ {0, . . . , p−1}, then (c−IndG
KZ Sym k )/T (c−IndKZ Sym k )is an irreducible admissible smooth representation of G.


G
r 2
r 2
Proof. Write V := c − IndG
KZ Sym k /T c − IndKZ Sym k , let V0 denote theKZ-subrepresentation Symr k 2 of V , and write v to denote a basis for the onedimensional subspace V0N0 of v. Since v is fixed by N0 , we have that(8)

tv = 0.

(Recall that t denotes the uniformizer of A := k[[N0 ]].) Also, Γ acts onv viathea0character εr , where ε denotes the F×
p -valued character of Γ defined by 0 1 7→
a mod p.
Since, by assumption, T annihilates the embedding V0 ,→ V, a considerationof the description of T afforded by (7) shows that under the action of KZ, wF v
generates a copy of V1 := (Symp−1−r k 2 ) ⊗ detr in V , with wF v spanning the onedimensional subspace V1N0 . (This is a well-known result, due originally to Breuil.)
In particular(9)

twF v = 0.

12

MATTHEW EMERTON

Since w ∈ K, we see that F v also lies in V1 , and (recalling the manner in which
w acts on V1 ) that(10)tp−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 V0 , we deduce that F wF v generates V0
over N0 , or equivalently, over A, and, in particular, that(11)

tr F wF v = dv,

for some non-zero scalar d ∈ k × . Combining the relations (8), (9), (10), and (11),
we find that M (V, V0 ) is a cyclic A[F, Γ]-module, with v as a generator, and that v
satisfies (at least) the relations(12)

tv = tp−r F v = (t(p−1)(p−r) F 2 − cd)v = 0,γv = ε(γ)r v for γ ∈ Γ.

These relations imply that M (V, V0 ) is A-torsion and that M (V, V0 )/tM (V, V0 ) = 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, V0 ) (which we have shown is a non-zero quotient of M ) is isomorphic to M ,and hence is irreducible. Since the functor V 7→ M (V ) is exact and faithful, we
will conclude that V is irreducible as a G-representation, as claimed, and so willcomplete the proof of the theorem.
We first observe that M [t] = khv, tp−1−r F vi, as the reader may easily verify.
Also, by assumption the group Γ acts on v via the character εr . The reader may
then easily deduce that Γ acts trivially on tp−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 tp−1−r F v, andso is equal to M . (Indeed, v is a cyclic generator of M , and we have the relation
tr F tp−1−r F v = cdv, with cd 6= 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 + βtp−1−r F v. If β = 0, then α 6= 0,
and so M contains v. If α = 0, then β 6= 0, and so M contains tp−1−r F v. Supposethat neither α nor β vanishes. If r ≥ p − r, then applying tr F to the element
αv+βtp−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 tp−1−r F to the element
αv + βtp−1−r F v, we find that M contains tp−1−r F v. Thus, except in the caser = (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 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 ofv 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)], andwhich 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
G
r 2
r 2
V := (c − IndG
KZ Sym k )/T (c − IndKZ Sym k )COHERENT RINGS AND REPRESENTATIONS OF GL2 OF A p-ADIC FIELD

13

as in Remark 4.8 (so D = k((t)) ⊗k[[t]] Homk (M (V, V0 ), k)), then the proof of thepreceding 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 asa φ-module, though as the preceding theorem shows, it is irreducible as a (φ, Γ)Q
module). The (φ, Γ)-module D corresponds to the representation IndQp2 ω2r+1 ofp
the Galois group GQp (where ω2 denotes a fundamental character of level 2, which
is a character of GQp2 ), and the irreducibility of D as a φ-module for r 6= (p − 1)/2
corresponds to the fact that IndQp2 ω2r+1 remains irreducible when restricted top
Galois group GQp,∞ of the p-adic cyclotomic extension Qp,∞ of Qp , except in thecase when r = (p − 1)/2.
Q

References[1] Barthel L., Livné R., Irreducible modular representations of GL2 of a local field, Duke
Math. J. 75 (1994), 261–292.
[2] Berger L., Représentations modulaires de GL2 (Qp ) et représentations galoisiennes de dimension 2, to appear in Asterisque.
[3] Breuil C., Sur quelques représentations modulaires et p-adiques de GL2 (Q) I, Compositio
Math. 138 (2003), 165–188.
[4] Colmez P., Représentations de GL2 (Qp ) 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 GL2 (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