HOMOLOGICAL STABILITY FOR COMPLETED HOMOLOGYFRANK CALEGARI AND MATTHEW EMERTON

Contents
1. Introduction
1.1. Acknowledgements
2. Arithmetic Manifolds
2.1. Cohomology of arithmetic quotients
3. Stability for ` 6= p
4. Hecke Operators
4.1. Proof of Theorem 1.2
5. Stability for ` = p
5.1. Consequences for classical cohomology groups
5.2. Relation to K-theory
References

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1. Introduction
Let GLN (Z, M ) denote the principal congruence subgroup of GLN (Z) of level M , explicitly,
the group of invertible matrices with integer coefficients which are congruent to the identity
modulo M . A theorem of Charney [Cha80] and (independently) Maazen [Maa79] shows that,for fixed d, the homology groups
Hd (GLN (Z, 1), Z) = Hd (GLN (Z), Z)
are stable, that is, they are independent of N for N sufficiently large. Moreover, the naturaltransition maps induced from the inclusions GLN ⊂ GLN +1 in this range are isomorphisms.
In [Cha84], Charney extends this result to prove a stability theorem for congruence subgroups.
In particular, she proves that the homology groups 1
Hd GLN (Z, M ), ZM
are also stable as N → ∞. Note that the analogous result with coefficients in R is (in bothcases) a consequence of a theorem of Borel [Bor74]. The result is no longer true if one does
not invert primes dividing the level. For example, if p > 2 is prime, it is an elementary
consequence of the congruence subgroup property [BLS64, Men65] that the homomorphism
2

H1 (GLN (Z, p), Z) → GLN (Z, p)/GLN (Z, p2 ) ' (Z/pZ)N −1
is an isomorphism for N ≥ 3, and the latter group is certainly not stable. The starting point
for the Church–Farb theory of representation stability [CF11] is the observation that, as arepresentation of SLN (Fp ), the right hand side has a description which is independent of N (it2010 Mathematics Subject Classification. 11F80, 11F75, 19F99.
The first author was supported in part by NSF Career Grant DMS-0846285. The second author was
supported in part by NSF Grant DMS-1003339.
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FRANK CALEGARI AND MATTHEW EMERTON

is the adjoint representation). Alternatively, one might view this calculation as saying that all
the cohomology of GLN (Z, p) in degree 1 (for N ≥ 3) arises via pullback from the homology ofcongruence subgroups of the local p-adic lie group GLN (Zp ). This latter homology is evidently
insensitive to the finer arithmetic properties of GLN (Z), and so it is natural to try to excise
the cohomology arising for “local” reasons and consider what remains. This is exactly whatis achieved by replacing the homology groups by completed homology groups, as first definedand studied in [Eme06, CE09, CE12].
Fix a tame level M , and let ΓN (pk ) = GLN (Z, M pk ) denote the principal congruencesubgroup of level M pk . Recall that the completed homology and cohomology groups aredefined as follows [CE12]:e ∗ (Fp ) := lim H∗ (ΓN (pr ), Fp )H←−e ∗ (Zp ) := lim H∗ (ΓN (pr ), Zp )H←−
Our main result is as follows:e ∗ (Fp ) := lim H ∗ (ΓN (pr ), Fp ),H−→e ∗ (Qp /Zp ) := lim H ∗ (ΓN (pr ), Qp /Zp ).
H−→

e d (Fp ) stabilize as N → ∞ and are finite Zp e d (Zp ) and H
Theorem 1.1. The modules H
modules and Fp -vector spaces respectively. Moreover, the action of SLN (Qp ) on either module
is trivial.
The question of computing these groups and understanding their relationship to arithmetic
and K-theory will be taken up in the companion paper [Cal13], although see § 5.1 for some
immediate consequences of stability of completed homology for classical homology groups.
(We confine ourselves here to the non-obvious remark that all the classical homology groups
at level ΓN (pr ) can be recovered from the completed homology groups.)
Our second result concerns the action of Hecke operators on cohomology classes in low
degree. Let Γ = ΓN (M ) for some level M . If p is prime and F is a field of characteristicp, then each of the cohomology groups H d (Γ, F) admits a natural action of a commutative
ring T of Hecke operators T`,k for 1 ≤ k ≤ N and for primes ` not dividing M p. Let[c] ∈ H d (Γ, F) denote an eigenclass for the action of T with eigenvalues a(`, k) ∈ F. A
theorem of Scholze [Sch13] (Conjecture B of [Ash92]) says that, associated to [c], there exists
a continuous semisimple Galois representation ρ : GQ → GLN (F) unramified outside N p such
that
X
(?)
(−1)k `k(k−1)/2 a(`, k)X k = det(I − ρ(Frob` )−1 X).
We determine exactly which Galois representations arise in low degree relative to N . Letω : GQ → F× denote the mod-p cyclotomic character.
Theorem 1.2. Fix an integer d, and suppose that N is sufficiently large compared to d. Thenfor any eigenclass [c] ∈ H d (Γ, F), there exists a character χ of conductor M and a Galois
representation:ρ = χ ⊗ 1 ⊕ ω ⊕ ω 2 ⊕ . . . ⊕ ω N −1 ,such that ρ(Frob` ) satisfies ?.
Remark 1.3. It follows from the argument that N ≥ 2d + 6 will suffice.
This theorem should be interpreted as saying that the action of Hecke on stable cohomology
is trivial, which is indeed how we shall prove this theorem. Note that if [c] comes fromcharacteristic zero, then the result follows from a theorem of Borel [Bor74], which shows inHOMOLOGICAL STABILITY FOR COMPLETED HOMOLOGY

3

particular that the only rational cohomology in low degrees arises from the trivial automorphic
representation.
Example 1.4. A special case of a construction due to Soulé [Sou79] implies that K22 (Z)contains an element of order 691.1 Associated to [c] via the Hurewicz map is a stable class[c] in H22 (GLN (Z), Z) for all sufficiently large N . Our theorem implies that the class [c] isassociated to the representation ρ := 1⊕ω⊕. . .⊕ω N −1 via ?. On the other hand, the existenceof [c] corresponds — implicitly — to the existence of a non-semisimple Galois representation% with %ss = 1 ⊕ ω 11 such that the extension class in Ext1 (ω 11 , 1) is unramified everywhere.
Is there a generalization of Ash’s conjectures that predicts the existence of a non-semisimple
Galois representations associated to Eisenstein Hecke eigenclasses?
Remark 1.5. That Theorem 1.2 (or something similar) might be true was suggested by
Akshay Venkatesh in discussions with the first author. (See [Ash12] for a discussion of thisconjecture and some partial results.)
1.1. Acknowledgements. The first author would like to thank Akshay Venkatesh, and both
authors are grateful to Andy Putman for answering questions about his paper [Put12]. We
would also like to thank Konstantin Ardakov, Kevin Buzzard, Thomas Church, Jean-François
Dat, Benson Farb, Toby Gee, Florian Herzig, Akshay Venkatesh, and Simon Wadsley for interesting conversations and helpful remarks. An earlier version of this preprint from 2012
established Theorem 1.1 conditionally on Conjecture 5.6; the unconditional proof of Theorem 1.1 was found in October of 2013. The first author would especially like to thankFlorian Herzig for some remarks concerning admissible representations that led to the proofof Theorem 1.1.
2. Arithmetic Manifolds
Torsion free arithmetic groups are well known to act freely and properly discontinuouslyon their associated symmetric spaces. This allows one to translate questions concerning thecohomology of arithmetic groups into questions about cohomology of the associated arithmetic
quotients. While one can study such spaces independently of any adelic framework, it is morenatural from the perspective of Hecke operators to work in this generality, and this is theapproach we adopt in this paper.
2.1. Cohomology of arithmetic quotients. Let K∞ denote a fixed maximal compact
0 denote the connected component containing the identity.
subgroup of GLn (R), and let K∞
0 ' SO(n). Let A denote the adeles of Q. For
One has isomorphisms K∞ ' O(n) and K∞
any finite set of places S, let AS denote the adeles with the components at the places v|Smissing, so (for example) A∞ denotes the finite adeles. Fix a compact open subgroup K ` ofGLn (A`,∞ ), let K` denote a compact open subgroup of GLn (Z` ), and let K = K ` K` . Let
0
Y (K) = GLn (Q)\GLn (A)/K∞
K ` K`

denote the corresponding arithmetic quotient.
Assume that F is a finite field of characteristic p 6= `. We now let Πn,` denote the directlimitΠn,` = lim H d (Y (K), F).
−→K`

1Indeed, K (Z) ' Z/691Z.
22

4

FRANK CALEGARI AND MATTHEW EMERTONFor K` sufficiently small, the quotient Y (K) is a manifold consisting of a finite number ofconnected components, all of which are K(π, 1) spaces. In particular, if K` is the level`k congruence subgroup, and K ` is the open subgroup of tame level N , then writing theassociated space as Y (K, `k ) there is an isomorphismM
H d (Y (K, `k ), F) 'H d (Γ(`k ), F),A
×∞,×
where A := Q \A
/ det(K) is a ray class group of conductor N times a power of `, and Γ

is the corresponding classical congruence subgroup of level N .
The module Πn,` is endowed tautologically with an action of GLn (Q` ) that is admissible;that is, letting G(`k ) denote the full congruence subgroup of GLn (Z` ) of level `k , we havethatG(`k )
dimF Πn,` < ∞for any k. This property records nothing more than the finite dimensionality of the cohomology groups H d (Y (K, `k ), F).
3. Stability for ` 6= p
We use the following two key inputs to prove our result. The first is as follows:
Proposition 3.1. There is an isomorphism Πn,` → Πn+1,` for all n ≥ 2d + 6.
Proof. For any fixed `-power congruence subgroup, this follows immediately from the main
result of Charney [Cha84] (in particular, §5.4 Example (iv), p.2118). The theorem then follow
by taking direct limits.
Lemma 3.2. For n ≥ 2d + 6, the action of GLn (Q` ) on Πn,` is via the determinant.
Proof. By [Cha84], the action of SLn (Z` ) on H ∗ (Γ(`m ), F) is trivial in the stable range (seealso Corollary 7 of [Ash12]), and hence the action of SLn (Z` ) on Πn,` is also trivial. Let V
be an irreducible sub-quotient of Πn,` . The only admissible representations of GLn (Q` ) thatare trivial on SLn (Z` ) are given by characters (the normal closure of SLn (Z` ) inside SLn (Q` )is the entire group), and it follows that every irreducible constituent of Πn,` is a character.
Since the action of GLn (Q` ) on the extension of any two characters still acts through thedeterminant, the result follows for Πn,` .
Remark 3.3. One easy way to see that SLn (Z` ) must act trivially on H ∗ (Γ(`m ), F) in thestable range is that the latter module has fixed dimension for all n, whereas the smallest
non-trivial representation of SLn (Z` ) over F has unbounded dimension as n increases.
We now sketch two further proofs of Lemma 3.2 (one incomplete). The first proof “explains”
the triviality of the action of SLn (Q` ) by showing that Πn,` is very small in a precise way.
Proposition 3.4. For sufficiently large n, there is, for all m, an isomorphism
H d (Γ(`), F) ' H d (Γ(`m ), F).
Proof. The main result of [Cha84] identifies the stable cohomology groups H ∗ (Γ(`), F) as
well as the groups H ∗ (Γ(`m ), F) with the cohomology of the homotopy fibre of the map
BSL(Z)+ → BSL(Z/N `)+ and BSL(Z)+ → BSL(Z/N `m )+ respectively (under the assumption that F has characteristic prime to N `). The natural map between these fibre sequencesinduces an isomorphism on F-homology, trivially for BSL(Z)+ and for BSL(Z/N `m )+ →HOMOLOGICAL STABILITY FOR COMPLETED HOMOLOGY

5

BSL(Z/N `)+ by Gabber’s rigidity theorem [Gab92]. By the Zeeman comparison theorem,
this implies the result for the homotopy fibre.
More details of this argument can be found in the companion paper [Cal13] (see Remark 1.13). Alternatively, one can prove the result directly in this case, since the transfer map from Γ(`m ) to Γ(`) is an isomorphism over any ring such that ` is inverted (thegroup Γ(`)/Γ(`m ) is an `-group), and hence it suffices to show that the SLn (Z` ) action on
H d (Γ(`m ), F) is trivial. Lemma 3.2 is an easy consequence of Proposition 3.4. For our next
argument, we begin with the following:
Proposition 3.5. Let Vb be an irreducible admissible infinite dimensional representation ofGLn (Q` ) in characteristic zero. Then the Gelfand–Kirillov dimension of Vb is at least n − 1.
Proof. The Gelfand–Kirillov dimension of Vb can be interpreted in two ways. On the onekhand, it is that value of d ≥ 0 for which dim V G(` )  `dk ; on the other hand, if we pull backthe Harish–Chandra character to a neighbourhood of 0 in g, the Lie algebra of GLn (Q` ), via
the exponential map, then a result of Howe and Harish–Chandra shows that the resulting
function χ admits an expansion
X
χ=
cO µ
bO ,
O

where the sum ranges over the nilpotent GLn (Q` )-orbits in g∨ , and µbO denotes the Fouriertransform of a suitably normalized GLn (Q` )-invariant measure on O (see e.g. the introductionof [GS05], whose notation we are following here, for a discussion of these ideas); the Gelfand–
1
Kirillov dimension of Vb is then also equal maxO | cO 6=0 dim O (as one sees by pairing the2characteristic function of G(`k ) — pulled back to g — against χ). Since Vb has Gelfand–
Kirillov dimension 0 if and only if it is finite dimensional, and since the minimal non-zeronilpotent orbit in g has dimension 2(n − 1) (see §1.3 p. 459 and Table 1 on p. 460 of [Smi83]),
the proposition follows.

Let V be an irreducible sub-quotient of Πn,` . Since the cohomology of pro-` groups vanishes
in characteristic p 6= `, we deduce that, for all k,kG(`k )

dim V G(` ) ≤ dim Πn,`G(`k )= dim Πm,` ,where m = 2d + 6 is fixed, and where the equality follows from Propostion 3.1. Yet, for a
fixed m, there is the trivial inequality relating the growth of cohomology to the growth of theindex (up to a constant) and thusG(`k )

dim Πm,`2

 [GLm (Z` ) : G(`k )]  `km .

The bound on invariant growth then implies that the Gelfand–Kirillov dimension of Vb is at
most m2 . Suppose that V lifted to a (necessarily irreducible) representation Vb in characteristic 0. We deduce a corresponding bound for the invariants of V . By Proposition 3.5,
this is a contradiction for sufficiently large n unless Vb and thus V is one-dimensional. In
general, it follows from the main theorem of Vignéras ([Vig01], p.182) that any irreducible
admissible irreducible representation V of GLn (Q` ) over F lifts to a virtual representation[Vb ] in characteristic 0. We expect that [Vb ] may be chosen to contain a unique irreducible
representation of largest Gelfand–Kirillov dimension. If this is so, then the bound of Proposition 3.5 applies also to mod-p representations of GLn (Q` ), and it would follow that every

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FRANK CALEGARI AND MATTHEW EMERTON

irreducible constituent of Πn,` is a character. Since the action of GLn (Q` ) on the extensionof any two characters still acts through the determinant, this would yield a third proof of
Proposition 3.2.
Remark 3.6. In a preliminary version of this paper, we made the error of assuming that
any irreducible GLn (Q` )-representation V in characteristic p lifted to a characteristic zero
representation Vb . We thank Jean-François Dat for pointing out to us that this is not always
the case, and also for suggesting a possible approach to proving our modified expectation via
induction on the ordering of partitions associated to V . Because of the availability of otherarguments, however, we have not made a serious attempt to carry out this one. Although
this alternate argument is therefore incomplete, it is the one that is most amenable to generalization to the case of ` = p (see §5), and so for psychological reasons we have presented it
here.
4. Hecke Operators
Let g ∈ GLn (A∞ ) be invertible. Associated to g one has the Hecke operator T (g), definedby considering the composition:H • (Y (K), F) → H • (Y (gKg −1 ∩ K), F) → H • (Y (K ∩ g −1 Kg), F) → H • (Y (K), F),
the first map coming from the obvious inclusion, the final coming from corestriction map. The
Hecke operators preserve H • (Y (K), F), but not necessarily the cohomology of the connectedcomponents. Indeed, the action on the component group is via the determinant map on
G(A∞ ) and the natural action of A∞,× on A. The Hecke operator T`,k is defined by taking g
to be the diagonal matrix consisting of k copies of ` and n − k copies of 1. The algebra T of
endomorphisms generated by T`,k on H ∗ (Y (K), F) for ` prime to the level of K generates acommutative algebra. For any such T = T (g), let hT i denote the isomorphism of H ∗ (Y (K), F)
that acts by permuting the components according to the image of det(g) in A.
There are many definitions of the notion of “Eisenstein” in many different contexts. For
our purposes, the following very restrictive definition is appropriate:
Definition 4.1. A cohomology class [c] ∈ H d (Y (K), F) is Eisenstein if T [c] = hT i deg(T )[c]for any Hecke operator T . A maximal ideal m of T is Eisenstein if and only if m contains
T − deg(T ) for all T with hT i = 1 in A.
If [c] ∈ H d (Γ, F) is a Hecke eigenclass such that T [c] = deg(T )[c] for all hT i = 1, then [c] isnecessarily Eisenstein, and moreover T [c] = χ(hT i) deg(T )[c] for some character χ : A → F×of A. By class field theory, χ corresponds to a finite order character of GQ of conductor
dividing M . A easy computation of deg(T`,k ) then implies that ? will be satisfied with ρequals χ ⊗ (1 ⊕ ω ⊕ . . . ⊕ ω n−1 ) if and only if [c] is Eisenstein.
Lemma 4.2. If [c] is a Hecke eigenclass such that T [c] = det(T )[c] for almost all T withhT i = 1 in A, then [c] is Eisenstein.
Proof. It suffices to show that T [c] = det(T )[c] for all T of level prime to N p with hT i = 1
in A. This is an immediate consequence of the Cebotarev density theorem and the fact that
there exists a Galois representation associated to [c] by [Sch13].
HOMOLOGICAL STABILITY FOR COMPLETED HOMOLOGY

74.1. Proof of Theorem 1.2. Given any eigenclass [c] ∈ H d (Y (K), F), consider its image[ι(c)] in Πn,` . If this image is non-zero, we can determine the eigenvalues of [c] by determining
those of [ι(c)]. Since the GLn (Q` )-action on Πn,` factors through det, the action of any
g ∈ GLn (Q` ) on [ι(c)], as well as the action of the deg(T (g)) representatives in the double
coset decomposition of g, is via hT i; hence T [ι(c)] − hT i deg(T )[ι(c)] is zero in Πn,` . It follows
that either [c] is Eisenstein or it lies in the kernel of the map H d (Y (K), F) → Πn,` . Hence
Theorem 1.2 follows by induction from the following Lemma.
Lemma 4.3. Suppose that for every non-Eisenstein maximal ideal m of T the cohomology
groups H i (Y (L), F)m vanish for all i < d, where L = K ` L` is any compact normal open
subgroup L` ⊂ GLn (Z` ). If [c] lies in the kernel of the map H d (Y (K), F) → Πn,` , then [c] isEisenstein.
Proof. By assumption, the class [c] lies in the kernel of the map
H d (Y (K), F) → H d (Y (L), F)
for some L as above. There is a Hochschild–Serre spectral sequence:
E2ij = H i (K/L, H j (L, F)) → H i+j (K, F).
This spectral sequence is compatible with Hecke operators T away from N p and `. By
Lemma 4.2,we may detect whether a class is Eisenstein by considering Hecke operators awayfrom `. Localizing the spectral sequence at a non-Eisenstein maximal ideal m, we deduce thatE2ij vanishes for j < d. Hence the first d rows vanish, and we are left with an isomorphism:
L/K

H d (Y (K), F)m = H 0 (K/L, H d (Y (L), F)m ) = H 0 (Y (L), F)m
It follows that if [c] is not Eisenstein, then it is trivial. The result follows.

.
This completes the proof of Theorem 1.2.
Remark 4.4. By the universal coefficient theorem, one hasΠ∨lim Hd (Y (K), F).
n,` := Hom(Πn,` , F) = ←−Suppose that one definesΠ∨H (Y (K), Z` ).
n,` (Z` ) = lim←− dThen it follows from Lemma 3.2 for d and d + 1 that the action of GLn (Q` ) on Π∨n,` (Z` ) is
via the determinant for sufficiently large n.
Remark 4.5. One can ask whether a Hecke operator T with hT i trivial in A acts via thedegree on the entire cohomology group H d (Γ, Z). Our argument shows that, for such T , theimage of T − deg(T ) on H d (Γ, Z) is — in the notation of [CV12] — congruence; i.e., it lies in
the kernel of the map H d (Γ, Z) → H d (Γ(M ), Z) for some M .
Remark 4.6. The main theorem and its proof remain valid, and essentially unchanged, if
one replaces GLn (Z) by GLn (OF ) for any number field F .

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FRANK CALEGARI AND MATTHEW EMERTON

5. Stability for ` = p
It is natural to wonder whether the methods of the previous sections can be extended to` = p. One obvious obstruction is that the naı̈ve notion of stability fails, even for d = 1(as noted in the introduction). We proceed instead by using completed cohomology, whose
definition for a finite field F we recall below. It is completed cohomology that is the correct
analogue of the modules Πn,` defined above.
We fix, once and for all, a tame level M , and let Γ(pk ) = Γn (pk ) denote the principal
congruence subgroup of SLn (Z) of level M pk .
e nd are defined as follows [CE12]:
Definition 5.1. The completed cohomology groups H
e nd := lim H d (Γ(pk ), F)H−→
Although this definition is formally the same as Πn,` , the theory when ` = p is quite
different to the ` 6= p case, due to the non-semisimple nature of representations of pro-pgroups on mod-p vector spaces. However, we still prove the following result:e ndTheorem 5.2 (Stability of completed cohomology). For n sufficiently large, the modules H
are finite dimensional over F and are independent of n.
e d , F) ' lim Hd (Γ(pk ), F). If we define
e d,n := Hom(HRemark 5.3. We have an isomorphism Hn←−ke
Hd,n (Zp ) := lim Hd (Γ(p ), Zp ),←−e d,n (Zp ) is a finite Zp -module
then Theorem 5.2 also implies, by Nakayama’s lemma, that H
for sufficiently large n. Hence Theorem 5.2 implies Theorem 1.1.
Remark 5.4. Theorem 5.2 is immediate when d = 0 and d = 1. Indeed, for d = 0 one has
e 0 (F) = F for all n, while for d = 1 one has H
e 1 (F) = 0 for all n ≥ 3 by the congruenceHnnsubgroup property.
The proof of Theorem 5.2 will occupy most of the remainder of the paper. We begin byrecalling some facts concerning non-commutative Iwasawa theory. Let K = SLn (Zp ) and
let K(pk ) denote the principal congruence subgroups of K. By construction, the module
e nd is naturally a module over the completed group ring Λ = ΛF := Fp [[K(p)]]. If M is a
H
p
e d,n
Λ-module, then let M ∨ := Hom(M, Fp ) be the dual Λ-module. By Nakyama’s Lemma, Hde
is finitely generated, which implies (by definition) that Hn is co-finitely generated. The ring
Λ is Auslander regular [Ven02], which implies that there is a nice notion of dimension andco-dimension of finitely generated Λ-modules. One characterization of dimension for modules
is given by the following result of Ardakov and Brown [AB06]:
Proposition 5.5. If M is finitely generated and M ∨ is the co-finitely generated dual, thenM has dimension at most m if and only if, as k increases without bound,kdim(M ∨ )K(p )  pmk .
The following is the natural analogue of Proposition 3.5 in this context.
Conjecture 5.6. Let M be a finitely generated Zp [[K(p)]]-module. If M is infinite dimensional over F, then the dimension of M is at least n − 1.

HOMOLOGICAL STABILITY FOR COMPLETED HOMOLOGY

9Remark 5.7. The analogue of this conjecture for ΛQp is true by Theorem A of [AW12]. In
particular, if any finitely generated pure ΛFp -module M is the reduction of of a p-torsion
free ΛZp = Zp [[K(p)]]-module, then the conjecture is true. Such a lifting always exists forcommutative regular local rings, but it is unclear whether one should expect it to hold for Λ.
Our arguments proceed in a manner quite similar to the ` 6= p case. One missing ingredientis that Proposition 3.1 is no longer valid. We use the central stability results of Putman [Put12]
as a replacement.
Let (Vn ) be a collection of representations of Sn . Recall from [Put12] that a sequence...

- Vn−1

φn−1

- Vn

φn

- Vn+1- ...

is centrally stable if:(1) The φ∗ are equivariant with respect to the natural inclusions on symmetric groups,
S(2) For each n, the morphism IndSn+1
Vn → Vn+1 induced by φn identifies Vn+1 with thenS(unique) maximal quotient of IndSn+1
Vn such that the 2-cycle (n, n + 1) acts triviallynon φn−1 (Vn−1 ).
e d,n are centrally stable forLemma 5.8. If Theorem 5.2 holds for j < d, then the modules H
sufficiently large n.
Proof. This argument is essentially taken directly from Putman [Put12], and we follow hisargument closely. Putman considers a spectral sequence at a fixed level, which we may
take to be Γ(pk ). Taking the inverse limit, one obtains a corresponding spectral sequence
for completed homology. In order for this sequence to degenerate at the relevant terms on
2 are actually zero. By assumption, thepage 2, it suffices to show that the appropriate Ei,je j,n for j < d are finite dimensional vector spaces that are independent of n andmodules H
have a trivial Sn -action. Thus it suffices (following Putman) to show that the (n+1)st central
stability complex for the trivial sequence:
F → F → F → ...
of modules for the symmetric group (starting at the trivial group) is exact. This is a very
special case of Proposition 6.1 of [Put12], but can be verified directly in this case. Hence one
e d,n are centrally stable.
deduces — as in Putman — that the HRemark 5.9. It is expected that forthcoming work of Church and Ellenberg will provide
a direct proof that the groups Hd (Γ(pk ), F) are centrally stable. After taking inverse limits
e d are centrally stable without any induction hypothesis andin d, this would imply that the H
provide another (although not entirely unrelated) proof of Lemma 5.8.
e d,n as a ΛF -module
Corollary 5.10. If Theorem 5.2 holds for j < d, then the dimension of Hpis bounded independently of n.
Proof. This is obvious for any fixed collection of n. Yet, for n sufficiently large, central
stability implies that the natural map
S
e d,n → He d,n+1
IndSn+1Hninduced by the n + 1 embeddings of SL(n) into SL(n + 1) is surjective. Passing to cohomology,
ed → He nd induce homomorphisms from the Γn+1 (pk )-invariants of Hed
the natural maps H
n+1
n+1

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FRANK CALEGARI AND MATTHEW EMERTON

e nd . In particular, the dimension of the Γn+1 (pk )-invariants ofto the Γn (pk )-invariants of H
d
k
e
ed
H
n+1 is at most n times the dimension of the Γn (p )-invariants of Hn , and so it follows
e d,n+1 has Iwasawa dimension bounded by the Iwasawa dimension
by Proposition 5.5 that H
e d,n .
of H
Remark 5.11. Using Putman’s spectral sequence, one may prove unconditionally thatdimF H d (Γ(pk ), F)  pmkfor some constant m that does not depend on n (although the implied constant does depend
on n). This leads to an alternate proof of Corollary 5.10 via the Hochschild–Serre spectral
sequence and Proposition 5.5.
e d,nWe prove Theorem 5.2 by induction. By the proceeding corollary, the dimension of H
must be bounded for sufficiently large n. If we assume Conjecture 5.6, then we immediately
e d,n is finite
deduce that this dimension is actually zero for sufficiently large n, and hence H
e d,n is trivial. In particular, the action of Sn must also be
and the action of SLn (Qp ) on H
either trivial, or via the sign character, for sufficiently large n. By Lemma 5.8, we also deduce
e d,n is centrally stable. This rules out the possibility that Sn acts via thethat the sequence H
sign character (for large enough n), since these do not fit into a centrally stable sequence.
It follows that we must have a centrally stable sequence of trivial Sn -representations. Yet ae d,n stabilizes.
centrally stable sequence of trivial modules is stable, and hence H
How may we upgrade this to an unconditional proof? One thing to observe is that thee d,n have an action of SLn (Qp ) that has not been exploited. For technical reasons, it
modules H
is actually more convenient to have an action of the group GLn (Qp ). In order to upgrade theaction of SLn (Qp ) to GLn (Qp ), one should take the direct limit (in cohomology) as follows:
lim H d (Y (K), F).
−→
Kp

e nd to denote this limit (rather than the limit over the cohomology ofFrom now on, we use H
congruence subgroups that it denoted up till now).
e d is cofinitely generated over Fp [[GLn (Zp )]] rather than ΛF =With this new definition, HnpFp [[SLn (Zp )]]. It then suffices to show that these completions are eventually stable and finite
over the ring Fp [[det(GLn (Zp ))]] = Fp [[Z×p ]]. Instead of appealing to a general conjecture
concerning finitely generated ΛFp -modules, we only need consider such modules whose dual
admits a smooth admissible action of GLn (Qp ). In particular, it suffices to show that the
only irreducible GLn (Qp )-representations that occur as sub-quotients of these completions
are either trivial or have large dimension. So, instead of proving Conjecture 5.6, it suffices toprove the following:
Conjecture 5.12. Let π be an irreducible infinite-dimensional smooth admissible representation of GLn (Qp ), and let π ∨ = Hom(π, Fp ). Then the dimension of π ∨ as a ΛFp -moduleis ≥ n − 1.
Although this conjecture is presumably easier than Conjecture 5.6, we still do not know
how to prove it. Instead, we prove the following two lemmas:
Lemma 5.13. Let π be an irreducible infinite-dimensional smooth admissible representationof GLn (Qp ), and let π ∨ = Hom(π, Fp ). If π is not a supercuspidal representation, then thedimension of π ∨ as a ΛFp -module is ≥ n − 1.

HOMOLOGICAL STABILITY FOR COMPLETED HOMOLOGY

11

Lemma 5.14. Any infinite-dimensional irreducible smooth admissible representation π thate d is not supercuspidal.
occurs as a sub-quotient of H
n+1
Taken together, these lemmas serve as a replacement for either Conjecture 5.12 or Conjecture 5.6, and imply the main theorem. To recapitulate the argument, from Lemma 5.14, we
e d is not supercuspidal. From Lemma 5.13,
deduce that every irreducible constitutent of H
n+1de
we deduce that either H
containsarepresentationof dimension at least n, which contran+1de
dicts the uniformly bounded dimension of H
n+1 for all n, or the only irreducible constituentsdeof Hn+1 are finite, in which case the action of SLn (Qp ) is also trivial.
Proof of Lemma 5.13. Let G denote GLn (Qp ) and let B denote a Borel of G. The Lemma iseasy to verify directly for the Steinberg representation, for which the Iwasawa dimension of
the dual is equal to dim(G/B). By the main theorem of [Her11], we may therefore assume
that there exists a proper parabolic P with Levi M such that:G
π = IndG
P (σ1 ⊗ . . . ⊗ σr ) = IndP (σ).

The dual of such an induced representation has dimension equal to the sum of the dimensionof the dual of σ and the dimension of G/P , and hence has dimension at least equal to thatof G/P . It is easy enough to prove this directly, but we instead give a proof here using
the chacacterization of dimension coming from Proposition 5.5 (which applies just as well toGLn (Zp ) as it does to SLn (Zp )).
Using the Iwasawa decomposition G = KP (where now K denotes GLn (Zp )), we see that
we may rewrite π (thought of as a K-representation) as π = IndKP ∩K (σ). A short calculation
then shows, since the level pk congruence subgroup K(pk ) is normal in K, thatk

G(Z/pk )k

π K(p ) = IndP (Z/pk ) (σ M ∩K(p ) ).
Now for k > 0, the representation inside the induction is nonzero. Hencekdim π K(p ) ≥ [G(Z/pk ) : P (Z/pk )] ∼ pmk ,where m = dim(G/P ). As claimed, this gives a lower bound of dim(G/P ) on the Iwasawa
dimension of π.
The minimum value of dim(G/P ) amongst all proper parabolics is n − 1, and so the lemma
follows.
Proof of Lemma 5.14. Let M ⊂ GLn+1 (Qp ) denote the Levi subgroup Q×
p × GLn (Qp ) of
GLn+1 (Qp ), and let P be the corresponding parabolic. Associated to P is a natural arithmetic
group ΓP ⊂ GLn+1 (Z). Let ΓP (pk ) = ΓP ∩ Γn+1 (pk ). The group ΓP (pk ) is a semi-direct
product:ΓP (pk ) ' (pk Z)n o Γn (pk ).
Let us compute the completed cohomology groups of ΓP . There is a Hochschild–Serre spectral
sequence:
H i (Γn (pk ), H j ((pk Z)n , Fp )) ⇒ H i+j (ΓP (pk ), Fp ).
There is an isomorphism H j (pn Z, Fp ) = ∧j (Fp )n , but the composition maps from ΓP (pk ) to
ΓP (pk+1 ) induce the zero map on these cohomology groups unless j = 0. Hence, taking direct
limits, we get an isomorphism:
e Pd := lim H d (ΓP (pn ), Fp ) ' H
e d.
H
n
−→

12

FRANK CALEGARI AND MATTHEW EMERTON

Omit the degree d from the notation. We therefore have maps as follows:
e n+1 → H
eP → Hen.
H
The first map is P -equivariant, the second is an isomorphism, and the composition is thenatural map. It follows that there exists a G-equivariant map
e n+1 → IndGeH
P Hn .
e n from
Note that Sn+1 ∩ P = Sn , and hence the right hand side includes the induction from H
Sn to Sn+1 . Hence, by central stability, we actually have an injection:
e n+1 → IndGeH
P Hn .
To complete the proof of the lemma, it suffices to show that no constituent of the righthand side is supercuspidal. But this also follows immediately from Herzig’s classificatione n is a smooth admissible M of irreducible admissible representations of G [Her11], since H
representation, where M is the Levi of P .

5.1. Consequences for classical cohomology groups. The homology (or cohomology)
of arithmetic groups with F coefficients can be recovered from completed homology via the
Hochschild–Serre spectral sequence [CE12]. For example, if we take our arithmetic group
to be the principal congruence subgroup Γ(p) of SLn (Z), then, recalling that G(p) is theprincipal subgroup of SLn (Zp ), this spectral sequence has the form
e j =⇒ H i+j Γ(p), F .
E2i,j := H i G(p), H
e 0 = F, and Theorem 5.2 implies that each H
e j is finite with trivial G(p)As noted above, H
action. Hence Theorem 5.2 implies that the cohomology group H d (Γ(p), F) has a filtrationconsisting of three types of classes: those arising via pullback from H d (G(p), F), those arisinge d (which dovia (higher) transgressions from H j of G(p) for j < d, and those arising from H
not depend on n). In this optic, the notion of representation stability developed by Church
and Farb [CF11] is then seen (in this context) to have its origin in the mod-p cohomology of
p-adic Lie groups, rather than in the properties of arithmetic groups.
One interpretation of Theorem 5.2 is that the ` = p and ` 6= p theories after completion arequite similar. The difference in phenomenology between these two cases is then a consequenceof the vanishing of H i (G(`k ), F) when k ≥ 1 and the chararacteristic p of F is not `. Another way to state these results is to say that the only irreducible GLn (A∞ )-representations
occurring inside cohomology in sufficiently small degree are one-dimensional.
e d is finite, it is not at all apparent how
5.2. Relation to K-theory. Even knowing that H
e d in terms of
to compute it. One might wonder whether there is some interpretation of H
K-theory (some related speculation along these lines occurs in §8.3 of [CV12]). This question
is taken up in the companion paper [Cal13].
References[AB06]

K. Ardakov and K. A. Brown, Ring-theoretic properties of Iwasawa algebras: a survey, Doc. Math.
(2006), no. Extra Vol., 7–33. MR 2290583 (2007k:11185)[Ash92] Avner Ash, Galois representations attached to mod p cohomology of GL(n, Z), Duke Math. J. 65(1992), no. 2, 235–255. MR 1150586 (93c:11036), Hecke operators and the stable homology of GL(n), Comm. Algebra 40 (2012), no. 4, 1369–[Ash12]1387. MR 2912991

HOMOLOGICAL STABILITY FOR COMPLETED HOMOLOGY

13[AW12] Konstantin Ardakov and S. J. Wadsley, On irreducible representations of compact p-adic analytic
groups, preprint, 2012.
[BLS64] H. Bass, M. Lazard, and J.-P. Serre, Sous-groupes d’indice fini dans SL(n, Z), Bull. Amer. Math.
Soc. 70 (1964), 385–392. MR 0161913 (28 #5117)[Bor74] Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974),
235–272 (1975). MR 0387496 (52 #8338)[Cal13] Frank Calegari, The stable homology of congruence subgroups, preprint, 2013.
[CE09] Frank Calegari and Matthew Emerton, Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms, Ann. of Math. (2) 170 (2009), no. 3, 1437–1446. MR 2600878(2011c:22032)[CE12], Completed cohomology—a survey, Non-abelian fundamental groups and Iwasawa theory,
London Math. Soc. Lecture Note Ser., vol. 393, Cambridge Univ. Press, Cambridge, 2012, pp. 239–
257. MR 2905536[CF11] Thomas Church and Benson Farb, Representation theory and homological stability, preprint, 2011.
[Cha80] Ruth M. Charney, Homology stability for GLn of a Dedekind domain, Invent. Math. 56 (1980), no. 1,
1–17. MR 557579 (81h:18010)[Cha84] Ruth Charney, On the problem of homology stability for congruence subgroups, Comm. Algebra 12(1984), no. 17-18, 2081–2123. MR 747219 (86b:20053)[CV12] Frank Calegari and Akshay Venkatesh, Towards a Torsion Langlands correspondence, draft, 2012.
[Eme06] Matthew Emerton, On the interpolation of systems of eigenvalues attached to automorphic Hecke
eigenforms, Invent. Math. 164 (2006), no. 1, 1–84. MR 2207783 (2007k:22018)[Gab92] Ofer Gabber, K-theory of Henselian local rings and Henselian pairs, Algebraic K-theory, commutative
algebra, and algebraic geometry (Santa Margherita Ligure, 1989), Contemp. Math., vol. 126, Amer.
Math. Soc., Providence, RI, 1992, pp. 59–70. MR 1156502 (93c:19005)[GS05] Wee Teck Gan and Gordan Savin, On minimal representations definitions and properties, Represent.
Theory 9 (2005), 46–93 (electronic). MR 2123125 (2006a:22015)[Her11] Florian Herzig, The classification of irreducible admissible mod p representations of a p-adic GLn ,Invent. Math. 186 (2011), no. 2, 373–434. MR 2845621[Maa79] Hendrik Maazen, Stabilité de l’homologie de GLn , C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 15,
707–708. MR 532394 (80f:20056)[Men65] Jens L. Mennicke, Finite factor groups of the unimodular group, Ann. of Math. (2) 81 (1965), 31–37.
MR 0171856 (30 #2083)[Put12] Andy Putman, Stability in the homology of congruence subgroups, preprint, 2012.
[Sch13] Peter Scholze, On torsion in the cohomology of locally symmetric varieties, preprint, 2013.
[Smi83] S. P. Smith, Krull dimension of factor rings of the enveloping algebra of a semisimple Lie algebra,
Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 3, 459–466. MR 698349 (85a:17007)[Sou79] C. Soulé, K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math.
55 (1979), no. 3, 251–295. MR 553999 (81i:12016)[Ven02] Otmar Venjakob, On the structure theory of the Iwasawa algebra of a p-adic Lie group, J. Eur. Math.
Soc. (JEMS) 4 (2002), no. 3, 271–311. MR 1924402 (2004h:16029)[Vig01] Marie-France Vignéras, Correspondance de Langlands semi-simple pour GL(n, F ) modulo ` 6= p,
Invent. Math. 144 (2001), no. 1, 177–223. MR 1821157 (2003f:11182)