Draft: January 13, 2010
ON THE EFFACEABILITY OF CERTAIN δ-FUNCTORS
MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

1. Introduction
Let F be a finite extension of Qp and let o be its ring of integers. Let G := GL2 (F ),let K := GL2 (o), and let Z be the centre of G. Let A be a finite local Artinian
Zp -algebra with maximal ideal m and residue field k containing the residue
field of F . Recall that a representation V of G on an A-module is said to besmooth if for all v ∈ V the stabilizer of v is an open subgroup of G. Let Modsm
G (A)denote the category of smooth A-representations. Further recall that a smooth
A-representation V is admissible if for every open subgroup J of G the space V Jof J-invariants is finite dimensional. Let Modadm
G (A) denote the full subcategory of
Modsm
(A)
consistingofadmissible
representations.
The categories Modsm
G
G (A) andsm
ModG (A) are abelian. In practice, one is interested in admissible representations,sm
but Modsm
G (A) does not have enough injectives. The category ModG (A) has enoughinjectives, but it is too big. To remedy this the first author, in [2], [3], has introduced
an intermediate category of locally admissible representations Modl.adm(A). WeG
recall the definition: If V is a smooth A-representation of G, a vector v ∈ V iscalled locally admissible if the A[G]-submodule of V generated by v is admissible; asmooth representation V of G over A is then called locally admissible if every v ∈ V
is locally admissible. We let Modl.adm(A) denote the full subcategory of Modsm
G
G (A)
l.adm
consisting of locally admissible representation. The category ModG (A) is abelian
and has enough injectives [2, Prop. 2.2.15], [3, Prop. 2.1.1].
We introduce some variants of the preceding categories:l.adm
If ζ : Z → A× is a smooth character, then we denote by Modadm
G,ζ (A), ModG,ζ (A),
sm
adm
l.adm
and ModG,ζ (A) the full subcategories of ModG (A), ModG (A), and Modsm
G (A)respectively, consisting of representations admitting ζ as a central character. We
smalso let Modsm
K,ζ (A) denote the full subcategory of ModK (A) consisting of Krepresentations admitting ζ|Z∩K as a central character. The categories Modadm
G,ζ (A),
l.adm
sm
sm
ModG,ζ (A), ModG,ζ (A), and ModK,ζ (A) are abelian, and the last two have enoughinjectives. (See Lemma 2.5 (1) below.)

Let $ be a uniformizer of F , and view $ as an element of Z via the isomorphism
l.adm
sm
Z∼
= F × . If u ∈ A× , then we let Modadm
G,$=u (A), ModG,$=u (A), and ModG,$=u (A)
adm
l.adm
sm
denote the full subcategories of ModG (A), ModG (A), and ModG (A) respectively, consisting of representations on which $ acts via u. Again these categories
are abelian and the last two have enough injectives. (See Lemma 2.5 (2) below.)
In this note we show that the restriction to K of an injective object in Modl.adm
G,ζ (A)l.adm
sm(resp. Modl.adm
(A)
or
Mod
(A))
is
an
injective
objectinMod(A)(resp.
G,$=u
G
K,ζ
The first author was supported in part by the NSF grant DMS-0701315.
1

2

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

sm
Modsm
K (A)). (Here ModK,ζ (A) has the obvious meaning, namely it is the full subsm
category of ModK (A) consisting of K-representations admitting ζ|Z∩K as a central
character.) This implies that certain δ-functors defined in [3] are effaceable, andsmremain effaceable when restricted to Modsm
G,ζ (A) or ModG,$=u (A). In particular, it
proves Conjecture 3.7.2 of [3] for GL2 (F ).

Acknowledgments. The authors would like to thank Florian Herzig for useful comments, which have improved the exposition.The second author’s work on this notewas undertaken while he was visiting Université Paris-Sud, supported by Deutsche
Forschungsgemeinschaft, and he would like to thank these institutions.
2. Injectives
We establish some simple results about injective objects in various contexts.
2.1. Lemma. If H is a compact p-adic analytic group, if V is an injective object ofsm
Modsm
H (k), and if W is an injective envelope of V in ModH (A), then the inclusion∼
V ,→ W induces an isomorphism V −→ W [m].
Proof. Certainly the inclusion V ,→ W factors through an inclusion V ,→ W [m].
Since the source is injective, this inclusion splits. If C denotes a complement to theinclusion, then V ∩ C = 0, and thus C = 0 (as W is an essential extension of V ).
This proves the lemma.
2.2. Lemma. Let H be a finite index open subgroup of G
(1) An object of Modsm
G (A) is admissible (resp. locally admissible) as a Grepresentation if and only if it is so as an H-representation.
∼G(2) If V is an object of Modsm
H (A), so that IndH V (−→ A[G] ⊗A[H] V ) is ansmG
object of ModG (A), then IndH V is admissible (resp. locally admissible) asa G-representation if and only if V is admissible (resp. locally admissible)
as an H-representation.
Proof. The admissibility claim of part (1) is clear, since H contains a cofinal collection of open subgroups of G. Since H has finite index in G, the group ring A[G] isfinitely generated as an A[H]-module, and thus an A[G]-module is finitely generated
if and only if it is finitely generated as an A[H]-module. The local admissibility
claim of part (1) follows from this, together with the admissibility claim, since an
A[G]-module (resp. A[H]-module) is locally admissible if and only if every finitely
generated submodule is admissible.
To prove the if direction of claim (2), suppose first that V is an admissible Hrepresentation.
If we write G as a union of finitely many left H-cosets,S
Tn say G−1=
0
00
0
,
if
H
is
an
open
subgroup
of
H,
and
if
we
write
H
:=
H
∩
i=1 gi Hgi ,i=1n gi H
then
0

00

∼

H
H
(IndG
⊂ (IndG−→ (A[G] ⊗A[H] V )H
HV)
HV)
∼

00

00

−1

00

−→ ⊕ni=1 (gi V )H = ⊕ni=1 gi V gi H gi .
Since gi−1 H 00 gi is an open subgroup of H, each of the summands appearing on theright-hand side is finite dimensional, and thus so is their direct sum. Thus IndG
HV

ON THE EFFACEABILITY OF CERTAIN δ-FUNCTORS

3

is admissible as claimed. If we suppose that V instead is locally admissible, or
equivalently, is the inductive limit of its admissible subrepresentations, we see thatG
the same is true of IndG
H V , since IndH commutes with the formation of induction
limits (being naturally isomorphic to A[G] ⊗A[H] –).
To prove the other direction of (2), note first that the inclusion A[H] ⊂ A[G] gives∼Grise to an H-equivariant embedding V ,→ A[G]⊗A[H] V −→ IndG
H V. Thus if IndH V
is (locally) admissible as a G-representation, and hence also (locally) admissible asan H-representation, by part (1), the same is true of its H-subrepresentation V . 
2.3. Proposition. If H is an open subgroup of G of finite index, then an object V
adm
l.adm
of the category Modsm(A)) is injective if andG (A) (resp. ModG (A), resp. ModG
G
only if IndH V is injective as an object of the same category.
Proof. Consider the sequence of adjunction isomorphisms∼∼

G
HomA[G] (U, IndG
H V ) −→ HomA[H] (U, V ) −→ HomA[G] (IndH U, V ).

Since the composite of IndG
H (which is naturally equivalent to A[G] ⊗A[H] –) and theadmforgetful functor induces an exact functor from Modsm
G (A) (resp. ModG (A), resp.
l.adm
ModG (A)) to itself (here we are taking into account Lemma 2.2), the proposition
follows.

2.4. Definition.
(1) If ζ : Z → A× is a smooth character and V is a representation of G (resp. K) over A, then we let
V Z=ζ := {v ∈ V | z · v = ζ(z)v for all z ∈ Z}
(resp. V Z∩K=ζ := {v ∈ V | z · v = ζ(z)v for all z ∈ Z ∩ K}.)(2) If u ∈ A× and V is a representation of G over A, then we let
V $=u := {v ∈ V | $ · v = uv}.
Since the subrepresentation of a smooth admissible (resp. smooth locally admissible, resp. smooth) representation is again smooth admissible (resp. smooth locallyadmissible, resp. smooth), we see, in the context of the preceding definition, thatadm
the construction V 7→ V Z=ζ induces a functor Modadm
G (A) → ModG,ζ (A) (resp.
l.adm
l.adm
sm
sm
ModG (A) → ModG,ζ (A), resp. ModG (A) → ModG,ζ (A)) that is right adjoint
to the forgetful functor, that the construction V 7→ V Z∩K=ζ induces a functorsm
Modsm
K (A) → ModK,ζ (A) that is right adjoint to the forgetful functor, and thatadm
the construction V 7→ V $=u induces a functor Modadm
G (A) → ModG,$=u (A) (resp.
l.adm
l.adm
sm
sm
ModG (A) → ModG,$=u (A), resp. ModG (A) → ModG,$=u (A)) that is right adjoint to the forgetful functor. In particular, the functors V 7→ V Z=ζ , V 7→ V Z∩K=ζ ,and V 7→ V $=u all preserve injectives.
2.5. Lemma.
(1) If ζ : Z → A× is a smooth character, then each of the catel.adm
sm
sm
gories Modadm
G,ζ (A), ModG,ζ (A), ModG,ζ (A), ModK,ζ (A) are abelian, andthe last three have enough injectives.
l.adm(2) If u ∈ A× , then each of the categories Modadm
G,$=u (A), ModG,$=u (A), andsm
ModG,$=u (A) is abelian, and the last two have enough injectives.

4

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

Proof. The abelianess claims are evident. To establish the claim of (1) regardingsm
sm
injectives, let V be an object of Modl.adm
G,ζ (A) (resp. ModG,ζ (A), resp. ModK,ζ (A)),
l.adm
and let V ,→ W be an A[G]-linear embedding into an injective object in ModG
(A)
sm
sm
(resp. ModG (A), resp. ModK (A)). This embedding then factors through an embedding V ,→ W Z=ζ (resp. V ,→ W Z∩K=ζ in the Modsm
K,ζ case), and the lattersmobject is injective in Modl.adm(A)(resp.
Mod
(A),
resp.
Modsm
G,ζ
G,ζ
K,ζ (A)), as wasnoted above. The claim of (2) regarding injective is proved analogously, utilizing
the functor V 7→ V $=u .

3. Main result
We introduce notation for some subgroups of G that we will need to 
consider,+o× onamely: we write G := {g ∈ G : valF (det g) ≡ 0 (mod 2)}, write I := $o×o(an Iwahori subgroup of K), let NG (I) denote the normalizer in G of I, set G :=NG (I)/$Z , and write N0 := ( 10 o1 ) .
We recall the following result, proved by Christophe Breuil and the second authorin [1], which is the main input into the theorems of this note.
3.1. Theorem. Let V be an object in Modadm
G,$=1 (k), and if p = 2 then assume00that V ∼V
forsome
representation
Vof G+ . Then there exists a smooth+= IndGG
adm
admissible representation Ω of G in ModG,$=1 (k) and a G-equivariant injection
V ,→ Ω such that V |K ,→ Ω|K is an injective envelope of V |K in Modsm
K (k).
Proof. [1] Corollary 9.11. If p = 2 then [1] Proposition 9.2 implies that V satisfies
the conditions of [1] Corollary 9.11.
The next two propositions will let us bootstrap the preceding theorem from thecase of coefficients in a field to the more general case of coefficients in an Artinianring.
3.2. Proposition. Suppose that p is odd. If V is an object of Modadm
G,$=1 (A) withthe property that V [m] is injective as an object of Modsm
K (k), and if(3.3)

V ,→ Wis an injective envelope of V in the category Modsm
K (A), then the K-action on
W may be extended to a G-action in such a way that W becomes an object ofModadm
G,$=1 (A), and the embedding (3.3) is G-equivariant.
Proof. We adapt the method of proof of [4, Thm. 6.1]. First, since V is an essential
extension of V [m] (even as an A-module), we see that W is equally well the Kinjective envelope of V [m]. It then follows from Lemma 2.1 that the embedding
V [m] ,→ W [m] is an isomorphism. Thus W is an essential extension (even as
an A-module) of V [m], and since by [3, Prop. 2.1.2] it is injective as a smooth
I-representation, it is an I-injective envelope of V [m], and hence also of V .
Since I is open in K, it follows from [3, Prop. 2.1.2] that V [m] is injective as asmooth I-representation over k. Since p is odd and G/I has order two, in fact V [m]is injective as a smooth G-representation over k. Thus an analogous argument toON THE EFFACEABILITY OF CERTAIN δ-FUNCTORS

5

that of the preceding paragraph shows that if V ,→ W1 denotes an injective envelopeof V as an G-representation over A, then W1 is also an I-injective envelope.
∼Thus we may find an I-equivariant isomorphism W −→ W1 , respecting the given
embeddings of V into source and target. We use this isomorphism to transport
the G-action on W1 to a corresponding action on W . Since G is the amalgam
of KZ and NG (I) along IZ, we may then glue the K-action and G-action on W0
to obtain a G-action on W with respect to which ( $0 $ ) acts trivially, which byconstruction is compatible with the G-action on V . Finally, since V is admissible
as a G-representation (and so, equivalently, as a K-representation), [3, Prop. 2.1.9]
shows that W is admissible as a K-representation (and so, equivalently, as a Grepresentation). This completes the proof of the proposition.

3.4. Proposition. Suppose that p = 2. If V is an object of Modadm
G,$=1 (A) with theproperty that V [m] is injective as an object of Modsm
(k),
and
if
K
IndG
G+ V ,→ W

(3.5)

sm
is an injective envelope of IndG
G+ V in the category ModK (A), then the K-action
on W may be extended to a G-action in such a way that W becomes an object ofModadm
G,$=1 (A), and the embedding (3.5) is G-equivariant.

Proof. Let V ,→ W1 denote an injective envelope of V in the category Modsm
K (A).
∼
G
G
Note that IndG+ V −→ V ⊗A IndG+ 1 (where here 1 denotes the free rank one trivial
representation of G+ over A). Since IndG
G+ 1 is trivial when restricted to K, we find∼that there is a corresponding A[K]-linear isomorphism W −→ W1 ⊗A IndG
G+ 1.
∼

There is a natural isomorphism G/I −→ G/G+ , and hence natural G-equivariant
∼
∼
G
G
G
isomorphisms IndG
G+ 1 −→ IndI 1 and IndG+ V −→ IndI V. Thus there is a commutative diagram of isomorphismW/ IndG V
I _

∼

IndG
G+_ V∼/ W1 ⊗A IndG+ 1
G∼/ W1 ⊗A IndG 1I∼/ IndG W1 ,I

in which the top horizontal arrow is G-equivariant, the left-hand vertical arrow is Kequivariant, the right hand vertical arrow is G-equivariant, and the bottom three arrows are, reading from left to right, K-equivariant, I-equivariant, and I-equivariant
respectively. We may use the bottom row of this diagram (whose composite is Iequivariant) to transport the G-action on IndGI W1 to a corresponding action on W .
Since G is the amalgam of KZ and NG (I) along IZ, we see this action glues with0
the K-action on W to induced a G-action on W with respect to which ( $0 $ ) acts
trivially. A consideration of the diagram shows that the embedding (3.5) is bothK-equivariant and G-equivariant, and hence is G-equivariant. Finally, since V isadmissible as a G-representation (and so, equivalently, as a K-representation), thesame is true of IndG
G+ V , by Lemma 2.2. Thus [3, Prop. 2.1.9] shows that W isadmissible as a K-representation (and so, equivalently, as a G-representation), andthe proof of the proposition is complete.

We can now prove our first main result.

6

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

3.6. Theorem. If u ∈ A× , then any injective object of Modl.adm
G,$=u (A) is injective
as a smooth K-representation over A.
Proof. We begin with a straightforward reduction to the case u = 1. Namely, let
B = A[x]/(x2 − u); note that B ∼= A ⊕ A as an A-module. Thus if V is an injective
l.adm
object of ModG,$=u(A), then V is a direct summand of B ⊗A V when the latter isl.adm
regarded as an object of ModG,$=u(A), and so it suffices to verify that the latteris injective as a smooth K-representation over A. A consideration of the natural∼
isomorphism HomA[K] (U, B ⊗A V ) −→ HomB[K] (B ⊗A U, B ⊗A V ), for objects U
smof ModK (A), shows in turn that it suffices to prove that B ⊗A V is injective as an
object of Modsm
K (B).
Now B is Gorenstein over A, i.e. there is an isomorphism of B-modules∼

B −→ HomA (B, A),and hence there is an isomorphism of B[G]-modules∼

B ⊗A V −→ HomA (B, V ).
A consideration of the resulting isomorphism ∼∼HomB[G] (W, B ⊗A V ) −→ HomB[G] W, HomA (B, V ) −→ HomA[G] (W, V ),l.adm
for objects W of ModG,$=u(B), shows that B ⊗A V is injective as an object ofl.adm
ModG (B). Thus the claim of the theorem for Modl.adm
G,$=u (A) follows from thel.adm
corresponding claim for ModG,$=u (B).

If χu : F × → B × denotes the character defined by χu (a) = uvalF (a) , then twistl.adm
ing by χu ◦ det induces an equivalence of categories between ModG,$=1(B) andl.adm−1ModG,$=u (B) (with an essential inverse given via twisting by χu ). Thus thel.adm
claim of the theorem for ModG,$=u(B) follows from the corresponding claim forl.adm
ModG,$=1 (B). Altogether, we have reduced to a consideration of the case whenu = 1, and we assume that indeed u = 1 from now on.
In the remainder of the proof we treat the cases when p is odd and p = 2 separately.
Thus suppose first that p is odd, and let V be an injective object of Modl.adm
G,$=1 (A).
If U is an admissible G-subrepresentation of V , then by Theorem 3.1 there is aG-equivariant map U [m] ,→ Ω, where Ω is an object of Modadm
G (k) which is also aninjective envelope of U [m] as a smooth K-representation over k. By the injectivityof V , we may extend the inclusion U [m] ⊂ V to a G-equivariant map Ω → V .
Since Ω is an essential extension of U [m] (even as a K-representation) this latter
map must again be an embedding; we thus regard Ω as a G-subrepresentation of V
containing U [m].
If we write X = U + Ω ⊂ V , then X is an admissible G-subrepresentation of V
containing U , and X[m] = Ω is injective as a smooth K-representation over k.
Thus by Proposition 3.2, we may find a G-equivariant embedding X ,→ W, where
W is an object of Modadm
G,$=1 (A) which is an injective envelope of X as a smooth
K-representation over A. By the injectivity of V we may extend the inclusion
X ⊂ V to a G-equivariant map W → V, which must again be an inclusion, since
W is an essential extension of X (even as a K-representation). Thus we see that U

ON THE EFFACEABILITY OF CERTAIN δ-FUNCTORS

7

is contained in an admissible G-subrepresentation W of V which is injective whenregarded as a smooth K-representation over A.
Since V is locally admissible, it is isomorphic to the inductive limit of its admissible
G-subrepresentations. The result of the preceding paragraph shows that it is in fact
the inductive limit of those of its admissible G-subrepresentations that are injective
as smooth K-representations over A. It follows from [3, Prop. 2.1.3] that V itself
is injective as an object of Modsm
K (A).
Suppose now that p = 2. In this case, the preceding argument breaks down, and wemust use the technique of passing from V to IndG
G+ V . (In fact, we apply it twice.)
To this end, we first note that since V is an injective object of Modl.adm(A), the sameGG
is true of IndG
G+ V , by Proposition 2.3. Now the natural embedding V ,→ IndG+ V
l.adm
of injective objects in ModG (A) must split, and thus V is a direct summandGof IndG
G+ V . Hence it suffices to verify that IndG+ V is injective as an object ofsm
G
ModK (A). Iterating this argument, it suffices in fact to show that IndG
G+ IndG+ V
sm
is injective as an object of ModK (A), and this is what we will do.
Let U be an admissible subrepresentation of V . By Theorem 3.1 there is a Gadm
equivariant map IndG
G+ U [m] ,→ Ω, where Ω is an object of ModG (k) which is alsoG
an injective envelope of IndG+ U [m] as a smooth K-representation over k. By theGGinjectivity of IndG
G+ V , we may extend the inclusion IndG+ U [m] ⊂ IndG+ V to aG
G-equivariant map Ω → IndG+ V . Since Ω is an essential extension of IndG
G+ U [m](even as a K-representation) this latter map must again be an embedding; we thusG
regard Ω as a G-subrepresentation of IndG
G+ V containing IndG+ U [m].
Write X = IndG
G+ U + Ω ⊂ V , so that X is an admissible G-subrepresentation of
IndG
Vcontaining
IndG+
G
G+ U , with X[m] = Ω an injective smooth K-representation
over k. Since V is the inductive limit of its admissible subrepresentations U , the inG
duction IndG
G+ V is the inductive limit of its admissible subrepresentations IndG+ U,
and hence also is the inductive limit of its admissible subrepresentations X for whichGGX[m] is injective as an object of Modsm
K (k). The representation IndG+ IndG+ V isG
thus the inductive limit of its corresponding subrepresentations IndG+ X.
Given such an X, by Proposition 3.4 we may find a G-equivariant embedding
l.adm
IndG
G+ X ,→ W, where W is an object of ModG,$=1 (A) which is an injective envelopesm
G
Gof IndG
G+ X in the category ModK (A). Since IndG+ IndG+ V is injective as an object
l.adm
of ModG,$=1 (A), we may extend the embedding
G
G
IndG
G+ X ,→ IndG+ IndG+ V

to a G-equivariant map
G
W ,→ IndG
G+ IndG+ V,

which must again be an embedding, since W is an essential extension of IndG
G+ X
G
V
is
in
fact
the
inductive
limit(even as a K-representation). Thus IndG
Ind
+
+
G
G
of those of its admissible G-subrepresentations that are injective as objects of
G
G
Modsm
K (A). It follows from [3, Prop. 2.1.3] that IndG+ V IndG+ V is an injectivesm
object of ModK (A). As already observed, this suffices to complete the proof of thetheorem.
8

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

Our next result removes the hypothesis of having fixed action of $ from the preceding theorem.
3.7. Theorem. If V is an injective object of Modl.adm(A), then V is also injectiveG
as an object of Modsm
(A).
K
∼

Proof. LetL
B = A[$±1 ] −→ A[t±1 ]. If U is any locally admissible G-representation,then U = n Un , where n runs over the maximal ideals of B and Un denotes the
localization of U at n. Furthermore,[
Un = U [n∞ ] :=U [ni ],i≥1iwhere U [n ] denotes the subspace of U consisting of elements annihilated by ni .
Each maximal ideal n is of the form (m, f ), where m is the maximal ideal of A,
and f ∈ A[t] is a monic polynomial. Since A is Artinian, so that m is a nilpotent
ideal, we see that the powers ni are cofinal with the sequence of principal ideals(f i ). Thus we may equally well write[Un =U [f i ],i≥1iwhere of course U [f ] denotes the subspace of U consisting of elements annihilated
by f i .
Suppose now that V is an injective object of Modl.adm(A). Since, by the discusG
sion of the preceding paragraph, V is the inductive limit of the V [f i ] (where f iruns over the various powers of the various monic polynomials associated to thevarious maximal ideals n of B), in order to show that V is injective as an object
iof Modsm
K (A), it suffices, by [3, Prop. 2.1.3], to show that each V [f ] is an injectivesm
object of ModK (A).
If we write C := B/(f i ), and if we let u denote the image of t in C, then V [f i ]l.adm
is an object of Modl.adm(A),G,$=u (C). Since V is injective as an object of ModG
l.admi
one immediately sees that V [f ] is injective as an object of ModG,$=u (C). It thenfollows from Theorem 3.6 that V [f i ] is injective as an object of Modsm
K (C). Since
C is finite flat over A, we see that V [f i ] is equally well injective as an object of
sm
sm
Modsm
K (A), as required. (The forgetful functor from ModK (C) to ModK (A) isright adjoint to the exact functor C ⊗A –, and so preserves injectives.)Finally, we prove an analogue of the preceding results in the context of a fixed
central character.
3.8. Theorem. If ζ : Z → A× is a smooth character, and V is an injective object
smin Modl.adm
G,ζ (A), then the restriction of V to K is an injective object in ModK,ζ (A).
Proof. Write u = ζ($). If V ,→ W is an A[G]-linear embedding of V into anZ∩K=ζinjective object W of Modl.adm. The
G,$=u (A), then it factors through V ,→ W
l.adm
target of this embedding lies in ModG,ζ (A), and so the injectivity of V implies
that this embedding is split; i.e. V is a direct summand of W Z∩K=ζ as an A[G]module. It thus suffices to note that W Z∩K=ζ is an injective object of Modsm
K,ζ (A),Z∩K=ζ
since W is injective as an object of Modsm(A),
by
Theorem
3.6,
and
U
→
7
U
K
sm
sm
is right adjoint to the embedding of ModK,ζ (A) into ModK (A).


ON THE EFFACEABILITY OF CERTAIN δ-FUNCTORS

9

l.adm
3.9. Corollary. If V is injective any of the categories ModG,ζ(A) (for some
l.adm
×
×
smooth character ζ : Z → A ), ModG,$=u (A) (for some u ∈ A ), of Modl.adm(A),G
then V |N0 is an injective object in Modsm
N0 (k).

Proof. In the latter two cases this follows from Theorems 3.6 and 3.7, together
with [3, Prop. 2.1.5]. In the case of Modl.adm
G,ζ (A), it suffices, by Theorem 3.8, toshow that an injective object of Modsm
K,ζ (A) is injective as an N0 -representation.
To see this, note first that any injective object of Modsm
K,ζ (A) is the inductive limitof injective modules that are direct summands of C S (K, A)r , for some r ≥ 0. By
a consideration of Lemma 2.1.10 and Proposition 2.1.11 of [3] we then see that if
suffices to show that C S (K, A)Z=ζ is injective as an N0 -representation. If we writeT∼
K1 := K SL2 (F ), then clearly C S (K, A)Z=ζ −→ C S (K1 , A). The latter is injective
as an N0 -representation, by [3, Prop. 2.2.11], and the corollary is proved.

Let G be the group of Qp -valued points of a connected reductive linear algebraic
group over Qp . Let P be a parabolic subgroup of G with a Levy subgroup M and
let P be the parabolic subgroup of G opposite to P with respect to M . In [2], thefirst author defined a left exact functor OrdP : Modl.adm(A) → Modl.adm(k) such
G
M
l.adm
l.adm
that for all U in ModM ) and V in ModG (A) one has∼ HomM (U, OrdP (V )).
HomG (IndG U, V ) =P

Further, for i ≥ 0 in [3] there are defined functors H i OrdP : Modl.adm
(A) →
G
0
i
Modl.adm
(A)
such
that
H
Ord
=
Ord
and
{H
Ord
:
i
≥
0}
is
a
δ-functor.
P
PP
M
It is conjectured there that for i ≥ 1 the functors H i OrdP are effaceable, which
would imply that they are universal, and hence coincide with the derived functorsof OrdP .
3.10. Corollary. If G = GL2 (F ) and V is an injective object in Modl.adm(A) (resp.
G
l.adm
i
Modl.admG,ζ (A), resp. ModG,$=u (A)), then H OrdP (I) = 0 for all i ≥ 1.
Proof. Since by Corollary 3.9, I|N0 is an injective object in Modsm
N0 (k) we have that
H i (N0 , I) = 0 for all i ≥ 1. The claim follows from the definition of H i OrdP , see[3, Def.3.3.1].

l.adm
Since Modl.adm(A), Modl.adm
GG,ζ (A), and ModG,$=u (A) each have enough injectives,i
we conclude that the H OrdP are effaceable for i ≥ 1 on any of these categories.
In particular, we have verified [3, Conj. 3.7.2] in the case G = GL2 (F ).

3.11. Remark. The authors of this note strongly believe that an analogue Theorem 3.1 holds for other groups than GL2 (F ). If this is the case than our proof
should go through to establish [3, Conj. 3.7.2] for these groups.
References[1] C. Breuil and V. Paškūnas ‘Towards a modulo p Langlands correspondence for GL2 ’, toappear in Memoirs of the Amer. Math. Soc.
[2] M. Emerton ‘Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties’, to appear in Astérisque.
[3] M. Emerton ‘Ordinary parts of admissible representations of p-adic reductive groups II. Derived functors’, to appear in Astérisque.

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MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS[4] V. Paškūnas, ‘Admissible unitary completions of locally Qp -rational representations of
GL2 (F )’, preprint (2008), available at http://arxiv.org/abs/0805.1006.
(Matthew Emerton) Mathematics Department, Northwestern University, 2033 Sheridan
Rd., Evanston, IL 60208
E-mail address, Matthew Emerton: emerton@math.northwestern.edu