1
Completed cohomology – a survey
Frank Calegari and Matthew Emerton

This note summarizes the theory of p-adically completed cohomology.
This construction was first introduced in paper [8] (although insufficientattention was given there to the integral aspects of the theory), andthen further developed in the papers [4] and [12]. The papers [8] and [4]may give the impression that p-adically completed cohomology is some
sort of auxiliary construction that can be used to prove theorems (ofeither a p-adic or classical nature) about automorphic forms. However,
we believe that p-adically completed cohomology is in fact an object offundamental importance, and that it provides the best approximationthat we know of to spaces of “p-adic automorphic forms”. (In particular,
unlike the spaces that go by this name that are sometimes constructed byarithmetico-geometric means in the theory of modular curves, or moregenerally Shimura varieties, p-adically completed cohomology admits arepresentation of the p-adic group, and thus allows the introduction ofrepresentation-theoretic methods into the study of p-adic properties ofautomorphic forms.)
A systematic exposition of the theory, and of its (largely conjectural,at this point) applications to the p-adic aspects of the Langlands correspondence between automorphic eigenforms and Galois representations,will be given in the paper [6]. These notes provide a summary of someof the basic points of the theory, as well as one of the main conjecturesof [6] (Conjecture 1.5 below). The anticipated connection of the theory
with the p-adic aspects of the Langlands program are discussed in the
final Section 1.8. We don’t attempt to formulate any formal conjecturesin this section, but simply try to indicate, in very general and somewhat
idealized terms, what form we expect this connection to take. Our main
goal for doing so here is to explain some the motivations behind making

2Calegari and Emerton

Conjecture 1.5. It remains to be seen how closely our expectations, andour conjecture, conform to the reality of the situation.

1.1 Definitions
Let G0 be a pro-finite group, assumed to admit a countable basis ofneighbourhoods of the identity, consisting of normal open subgroups,say· · · ⊂ Gr ⊂ · · · ⊂ G1 ⊂ G0 .
Suppose given a tower of topological spaces· · · → Xr → · · · → X1 → X0 ,each equipped with an action of G0 , such that:
1. The maps Xr+1 → Xr are G0 -equivariant.
2. Gr acts trivially on Xr , and realizes Xr as a G0 /Gr -torsor over X0 .
(In particular, all the maps in the tower are finite coverings.)
In this context we may define the p-adically completed homology andcohomology modules attached to the tower X• (having fixed the primep), namely:H̃• := lim
H• (Xr , Zp )
←−
r

and
H̃ • := lim
lim H • (Xr , Z/ps Z).
←− −→
s

r

We can also consider Borel–Moore and compactly supported variants:
H̃•BM := lim
H•BM (Xr , Zp )
←−
r

and
H̃c• := lim
lim Hc• (Xr , Z/ps Z).
←− −→
s

r

There are natural maps H̃• → H̃•BM and H̃c• → H̃ • .
From now on we suppose that each Xr is a manifold which is homotopic to a finite simplicial complex. This ensures us that all the homology spaces (usual or Borel–Moore) and cohomology spaces (usual orCompleted cohomology – a survey

3compactly supported) of Xr that we have written down are finitely generated over the indicated ring of coefficients, and that they are related
by appropriate forms of duality.1
We also suppose from now on that G0 is a p-adic analytic group.
Without this hypothesis, it seems impossible to control the inverse limits
involved in our constructions. On the other hand, with this hypothesis,
one can control these limits in a very satisfactory way. Indeed, Lazard[14] has proved that completed group ring Zp [[G0 ]] is Noetherian, andSchneider and Teitelbaum [16] have explained how this result can beapplied to control various analytic difficulties that arise in the study of
p-adic Banach-space representations of G0 .
In particular, we have the following result (which strengthens the results stated in [8], in so far as it deals explicitly with homology as well as
cohomology, and pays attention to the integral aspects of the constructions, and not just to the objects obtained after tensoring up with Qp ),whose proof will appear in [6].
Theorem 1.1 1. The natural action of Zp [[G0 ]] on the modules H̃•and H̃•BM makes each of them a finitely generated left module overZp [[G0 ]]. Furthermore, the canonical topology on each of these modules (obtained by writing it as a quotient of a finite direct sum of
copies of Zp [[G0 ]]) coincides with projective limit topology.
Note that this implies in particular that each of the torsion submodules H̃• [p∞ ] and H̃•BM [p∞ ] is finitely generated over Zp [[G0 ]], and so
in particular, is of bounded exponent.
2. The spaces H̃ • and H̃c• are p-adically complete and their p-power
torsion submodules have bounded exponent.
3. There are short exact sequences
0 → Homcont (H̃•−1 , Qp /Zp ) → H̃ • → Homcont (H̃• , Zp ) → 0(here cont means continuous with respect to the canonical topology on
the source, and the evident topology — either discrete or p-adic —on the target) and0 → Hom(H̃ •+1 [p∞ ], Qp /Zp ) → H̃• → Hom(H̃ • , Zp ) → 0(since the natural topology on H̃ • is the p-adic topology, and since
all Zp -linear maps are automatically p-adically continuous, there is1

The paper [12] provides a sheaf-theoretic description of H̃ • and H̃c• , which allows
for more flexibility in the basic set-up of the theory than we permit here.
However, we will have no need for this extra generality in the applications that
we have in mind.

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Calegari and Emerton

no need to explicitly specify any continuity conditions on the maps
appearing in this exact sequence), as well as similar short exact sequences relating H̃•BM and H̃c• . These exact sequences are compatible
with the maps H̃• → H̃•BM , and the maps H̃c• → H̃ • .
4. For each r ≥ 0, there are Hochschild–Serre type spectral sequences
E2i,j := H i (Gr , H̃ j ) =⇒ H i+j (Xr , Zp )andE2i,j := H i (Gr , H̃cj ) =⇒ Hci+j (Xr , Zp ),compatible with the maps H̃c• → H̃ • .
Remark 1.2 If M is any continuous G0 -module over Zp , then we may
associate a family of local systems M• on the tower X• to M , and there
is an analogue of part (4) of the theorem for the cohomology of the local
system Mr .
Remark 1.3 The theorem shows the advantages of working with both
homology and cohomology. It is on the homology side that the algebraic
aspects of the theory are most transparent, while the cohomology side
is better adapted to comparison with the situation at finite levels.
It is useful to make some additional definitions on the cohomology
side, as follows:Ĥ • := the p-adic completion of lim
H • (Xr , Zp )
−→
r


= lim
lim H • (Xr , Zp )/ps H • (Xr , Zp )
←− −→
s

r

and
Tp H • := the p-adic Tate module of lim
H • (Xr , Zp )
−→
r

= lim
lim H • (Xr , Zp )[ps ]
←− −→
s

r

(the transition maps being given by multiplication by p). Note thatthe p-adic completion kills the p-divisible part of lim
H • (Xr , Zp ), while−→r

the p-adic Tate module knows only about the torsion subgroup of thisdivisible part. There is an exact sequence
0 → Ĥ • → H̃ • → Tp H •+1 → 0.

(1.1)

Completed cohomology – a survey

5

Since Tp H •+1 is p-torsion-free and p-adically complete, we may take theZp dual of this exact sequence to obtain a short exact sequence
0 → Hom(Tp H •+1 , Zp ) → Hom(H̃ • , Zp ) → Hom(Ĥ • , Zp ) → 0.
Recall from Theorem 1.1 (3) that Hom(H̃ • , Zp ) is the p-torsion freequotient of H̃• .
There are similar constructions for compactly supported cohomology,
and their formation is compatible with the maps H̃c• → H̃ • .

1.2 Non-commutative Iwasawa theory
If M is any finitely generated left Zp [[G0 ]]-module, then we defineE • (M ) := Ext• (M, Zp [[G0 ]]).
These are again naturally Zp [[G0 ]]-modules. (We compute the Exts via
the left Zp [[G0 ]]-module structure on Zp [[G0 ]], leaving a right Zp [[G0 ]]module structure on E • (M ). We convert this back to a left-module structure via the canonical anti-involution of Zp [[G0 ]] induced by g 7→ g −1 .)We note that these modules are unchanged (up to a natural isomorphism,
and applying the forgetful functor from Zp [[G0 ]]-modules to Zp [[G00 ]]modules) if we replace G0 by any open subgroup G00 .
We define the codimension of M (or codim M for short) to be theminimal value of i such that E i (M ) 6= 0. If G0 is commutative, then thisagrees with the usual notion of codimension of support of the coherent
sheaf associated to M on Spec Zp [[G0 ]]. In general, even if G0 is noncommutative (which is typically the case), this notion behaves entirely
analogously to the usual notion of codimension of support of a sheaf
in algebraic geometry [18]. In addition to the codimension of M , it issometimes convenient to speak of the dimension of M (or the Iwasawa
dimension of M , for emphasis), which we define to be the quantity 1 +dim G0 − codim M . (If we think of codim M as being the codimension of
the support of the — in general non-existent — coherent sheaf associatedto M on the — in general non-existent — Spec of Zp [[G0 ]], then the
Iwasawa dimension of M would be the dimension of its support.)
We also remark that if we shrink G0 sufficiently (i.e. replace G0 by
Gr for a large enough value of r), then Zp [[G0 ]] admits a skew-field of
fractions, say L, and hence we can speak of a finitely generated Zp [[G0 ]]module M being torsion-free (the natural map M → L ⊗ M is injective)

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Calegari and Emerton

or torsion (L ⊗ M = 0), and we can speak of the rank of the (torsionfree part of) M (i.e. the rank of the L-vector space L ⊗ M ). Having
positive rank is equivalent to being of codimension 0, while being torsion (or equivalently, being of rank 0) is equivalent to being of positive
codimension.

1.3 Poincaré duality
We suppose that X0 (and hence every Xr ) is equidimensional of somedimension d. We then have two spectral sequences that express Poincaré
duality in the p-adically completed situation (the detailed consructionsof which will appear in [6]). Note that both work with homology, rather
than cohomology. (The reason for this is that the functors E i intervene.)
Here are the spectral sequences:BM
E2i,j := E i (H̃j ) =⇒ H̃d−i−jandE2i,j := E i (H̃jBM ) =⇒ H̃d−i−j .
They are compatible with the maps H̃• → H̃•BM .
We point out the amplitude of the δ-functor E • is given by the dimension of the group G0 . In applications, this can often be quite a bit larger
than the dimension d of the spaces Xr , which can lead to interestingtension in these spectral sequences.

1.4 A simple example of everything so far
An illustrative example is given by taking G0 = Zp , Gr := pr Zp for∼r > 0, and Xr := R/pr Z for each r ≥ 0. We let G0 /Gr := Zp /pr Zp −→Z/pr Z act on Xr in the obvious manner by translations. Thus each Xr
is a circle, and each map Xr+1 → Xr is a degree p covering map. Notethat Zp [[G0 ]] = Zp [[T ]].
Clearly H̃0 = Zp , with trivial G0 -action, while H̃1 = 0 (the inverse
limit of a sequence of copies of Zp under the multiplication by p map
obviously vanishes). Similarly H̃ 0 = Zp , while H̃ 1 = 0. (Since the spaces
Xr are compact, Borel–Moore homology agrees with usual homology,
and compactly supported cohomology agrees with usual cohomology.)
Since Gr is procyclic, we see that H 0 (Gr , Zp ) = H 1 (Gr , Zp ) = Zp ,Completed cohomology – a survey

7

and thus we do indeed recover the cohomology of the circle Xr from thecompleted cohomology via the Hochschild–Serre type spectral sequence.
Similarly, using the resolution
T·

0 → Zp [[T ]]→Zp [[T ]] → Zp → 0,
we compute that E 0 (Zp ) = 0, while E 1 (Zp ) = Zp . This is immediately
seen to be consistent with the Poincaré duality spectral sequence.

1.5 Congruence quotients of symmetric spaces
We now suppose that G is a connected reductive linear algebraic groupover Q. We let A denote the adèle ring over Q, let A∞ denote the ringof finite adèles, and let A∞,p denote the ring of prime-to-p finite adèles.
We fix a compact open subgroup K ∞,p of G(A∞,p ) (the tame level),take G0 to be a sufficiently small compact open subgroup of G(Qp ), andlet (Gr )r≥1 be a sequence of normal open subgroups of G0 that form aneighbourhood basis of the identity in G0 .
o
We let K∞
denote the connected component of the identity of a maximal compact subgroup K∞ of G(R), and let Ao∞ denote the connectedcomponent of the identity of the group of real points A∞ of a maximal
Q-split torus in the centre of G.
We defineo
Xr := G(Q)\G(A)/Ao∞ K∞
Gr K ∞,p .

The Xr form a tower with an action of G0 satisfying the axioms of Section 1, and so the preceding theory applies. Furthermore, the usual argument about limits of (co)homology of arithmetic quotients in the adèlicsetting shows that H̃• , H̃ • , etc. all inherit not just an action of G0 , but of
the entire p-adic group G(Qp ). They also inherit an action of a suitablycompleted Hecke algebra T (built up out of spherical Hecke operators
at primes ` 6= p over which G and the tame level K ∞,p are unramified),oand an action of the component group π0 := (A∞ K∞ )/(Ao∞ K∞).
The following theorem is the main result of [4]. (The result about
Tp H •+1 being torsion, even in the middle dimension in the discrete seriescase, is not stated there, but follows from the proof.)
Theorem 1.4 If, in the preceding setting, the group G is semi-simple,
then the modules H̃n are torsion Zp [[G0 ]]-modules, unless the group G(R)admits discrete series and n is the “middle dimension” (i.e. one half

8Calegari and Emertonof the dimension of the Xr ). In this latter case, Hom(Ĥ n , Zp ) (andhence H̃n ) has positive rank, while Hom(Tp H n+1 , Zp ) is torsion overZp [[G0 ]].

We believe that this is actually only the beginning of the story withregard to the codimensions of the various H̃• . In the following section,
we will go on to describe how we expect the rest of the story to playout. But first, we will close this section by discussing the boundary longexact sequence.
Each Xr embeds as an open subset of its Borel-Serre compactification
X r . We let ∂Xr denote X r \Xr ; it is the boundary of X r . As r varies thespaces X r form a tower to which the p-adically completed cohomology
machinery applies, and hence we may define completed (co)homology
spaces H̃• (∂) and H̃ • (∂). There are long exact sequences· · · → H̃• (∂) → H̃• → H̃•BM → H̃•−1 (∂) → · · ·(1.2)· · · → H̃ •−1 (∂) → H̃c• → H̃ • → H̃ • (∂) → · · · .

(1.3)

and

The complement ∂Xr is a union of strata ∂XP,r indexed by the (equivalence classes of) proper parabolic subgroups P of G. For fixed P andvarying r the strata ∂XP,r again form a tower to which the p-adicallycompleted cohomology machine applies, allowing us to define H̃• (∂P )and H̃ • (∂P ). There is a Mayer–Vietoris spectral sequence relating thevarious H̃• (∂P ) (resp. H̃ • (∂P )) to H̃• (∂) (resp. H̃ • (∂)).
If we let P = MN be a Levi decomposition of the proper parabolic P,
then each ∂P,r is a bundle over a union of congruence quotients associatedto M, whose fibre is a compact nilmanifold (of dimension equal to dim N).
A generalization to a nilmanifolds of the example of Section 1.4 showsthat these nilmanifolds contribute to completed (co)homology only indegree 0, and so H̃• (∂P ) and H̃ • (∂P ) are supported in the same degrees
as the p-adically completed (co)homology associated to M. To describe
the precise relationship between the p-adically completed cohomology
for ∂P and for M, it is convenient to take a direct limit over all tame

Completed cohomology – a survey

9levels. We then find (at least when p is odd) that:lim−→tame levelsH̃ • (∂P )

∼

G(A )

−→ IndP(Aff)invariants under ker(π0M → π0 )tame levels
of the H̃ • associated to M , (1.4)
lim−→where the induction is p-adically completed at p, and smooth at primes
away from p. (We have written π0M to denote the analogue for the Levi
subgroup M of the component group π0 which was defined above for
G. Since π0 and π0M are finite 2-groups, there is the possibility of extra
complications when p = 2, which we won’t attempt to address here.)
We note that much of this general set-up for congruence quotients has
been described by Richard Hill in [12, §4].

1.6 Conjectures on codimensions
We maintain the set-up of the preceding section. We write G∞ := G(R),and we begin by defining two quantities associated to G:
l0 := rank of G∞ − rank of A∞ K∞ ,and
q0 = ( dimension of G∞ − dimension of A∞ K∞ − l0 )/2 = (d − l0 )/2,
where d denotes the dimension of the quotients Xr . Note that if G issemi-simple, then these quantities coincide with the quantities denotedby the same symbols in Borel–Wallach [2]. Namely, l0 denotes the “defect” of G∞ with regard to possessing discrete series, while q0 denotes
the first “interesting” dimension for the (co)homology of Xr .
We now state our basic conjecture on codimensions.
Conjecture 1.5 1. If n < q0 , then the codimension of H̃n is greaterthan l0 + q0 − n.
2. The codimension of H̃q0 equals l0 .
3. If n < q0 , then the codimension of H̃nBM is greater than l0 + q0 − n.
4. The codimension of H̃qBM
equals l0 .
0
5. H̃q0 is p-torsion-free.
6. H̃qBM
is p-torsion-free.
0
7. H̃n = 0 if n > q0 .

10Calegari and Emerton

8. H̃nBM = 0 if n > q0 .
In fact, these conjectures are not independent, as the following theorem makes clear.
Theorem 1.6 1. If parts (1) and (2) (resp. parts (3) and (4)) of Conjecture 1.5 hold for G, and for all the proper Levi subgroups of G, thenparts (3) and (4) (resp. parts (1) and (2)) of the conjecture also holds
for G (and also for all its Levi subgroups), as do parts (7) and (8).
2. If part (7) (resp. part (8)) of Conjecture 1.5 holds for G × G, thenpart (5) (resp. part (6)) of the conjecture holds for G.
Proof Suppose that parts (1) and (2), or parts (3) and (4), of the conjecture hold for all the proper Levi subgroups of G. Applying the theorem
inductively, we conclude that parts (1), (2), (3), (4), (5), and (6) of theconjecture hold for these Levi subgroups. Formula (1.4) then allows us
to bound from below the codimensions of the p-adically completed homology spaces associated to the various boundary strata. A comparisonof the invariants l0 and q0 for G and for its Levi subgroups, along witha consideration of the Mayer–Vietoris spectral sequence that computes
the p-adically completed homology of the boundary in terms of the padically completed cohomology of its various strata, as well as of thelong exact sequence (1.2), then shows that parts (1) and (2) of the conjecture for G are equivalent to parts (3) and (4) for G. Looking at the
Poincaré duality spectral sequence then shows that parts (7) and (8) of
the conjecture also hold.
Part (2) of the theorem follows by applying a Kunnëth-type theorem
to compare the completed cohomology for G to that for G × G.
Remark 1.7 In particular, the preceding theorem shows that if either
parts (1) and (2), or part (3) and (4), of Conjecture 1.5 hold for every
group G, then all parts of the conjecture hold for all groups G.
Remark 1.8 Richard Hill has also made a conjecture about the vanishing of completed cohomology of arithmetic quotients, namely [12,Conj. 3]. Conjecture 1.5 implies Hill’s conjecture, but is quite a bitstronger in general. The various vanishing results proved in [12, §5],
being consistent with [12, Conj. 3], are thus also consistent with ourgeneral conjecture.
Remark 1.9 Eric Urban has made a conjecture somewhat analogous
to Conjecture 1.5 concerning the dimensions of irreducible componentsof “eigenvarieties” [17, Conj. 5.7.3]. In fact, since “eigenvarieties” canCompleted cohomology – a survey

11

also be constructed from completed cohomology via applying the locallyanalytic Jacquet module functor of [7] (see [8]), it seems likely that theeventual relationship of Conjecture 1.5 to Urban’s conjecture will be
more than one of mere analogy. On the other hand, neither conjecture
seems to be an immediate logical consequence of the other, and a precise
understanding of their mutual relationship remains to be found.
There are several different heuristics and motivations behind Conjecture 1.5 (as well as a small amount of actual evidence); some of theseheuristics are explained in Section 1.8. For now, let us remark that it isconsistent with Künneth, and it is consistent with the Poincaré duality
spectral sequence. Of course, in the case when G is semi-simple, it is also
consistent with Theorem 1.4, which shows at least that the codimensionof H̃n is zero (respectively, positive) exactly when it is predicted to be
so by the conjecture.
We now discuss some illustrative examples.
Example 1.10 If G is a torus, then q0 = 0, and a generalizationof the analysis of the example in Section 1.4 shows that in this case,
Conjecture 1.5 is equivalent to Leopoldt’s conjecture. (See [12, Cor. 5].)
Example 1.11 If G = SL2 , then l0 = 0 and q0 = 1, and one finds thatH̃0 = Zp has codimension 3, while H̃1 has codimension zero. (Apply
Theorem 1.4.)
Example 1.12 If K is a quadratic imaginary field, and G = SL2/K(regarded as a group over Q by restriction of scalars, as usual), thenl0 = 1 and q0 = 1, and one finds that H̃0 = Zp , and so has codimension
6, that H̃1 has codimension 1, and that H̃2 = 0. (Apply Theorem 1.4
together with the Poincaré duality spectral sequence. Note that the term
E 6 (H̃0 ) = E 6 (Zp ) = Zp plays a crucial role in showing that H̃1 6= 0.)
Example 1.13 If G is semi-simple and simply connected and satisfies
the congruence subgroup property, then H̃1 is finite.
Example 1.14 If G = Sp4 , then l0 = 0 and q0 = 3. One finds thatH̃0 = Zp , H̃1 is finite (since G satisfies the congruence subgroup property), H̃2 has codimension at least 1 (by Theorem 1.4), and H̃3 has
codimension 0 (again by Theorem 1.4). The Poincaré duality spectral
sequence then shows that H̃n = 0 for n ≥ 4.
Remark 1.15 It is not clear in general how one might go about computing the codimension of support of (or any other information about)

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Calegari and Emerton

the completed cohomology. However, suppose that one knew that the Zp cohomology of each Xr was p-torsion-free (or, more generally, of bounded
exponent). The Tate module term in the exact sequence (1.1) would then∼vanish, and hence there would be an isomorphism Ĥ • −→ H̃ • . Now Ĥ •is the p-adic completion of the classical cohomology, and this classicalcohomology may be described in terms of classical automorphic forms(at least after tensoring with Qp over Zp ). Thus it seems not totallyinconceivable that one might be able to prove results about (e.g. thecodimension of support of) Ĥ • . (This is done for modular curves in [9],although admittedly the arguments there rely on the full strength of thep-adic Langlands program for GL2 (Qp ).) Consequently, it seems worth
investigating and attempting to establish torsion-freeness results for thecohomology of congruence quotients, say in the Shimura variety context.

1.7 Mod p analogues
Returning to the notation of Section 1.1, we observe that we may define
mod p analogues of completed homology and cohomology. Namely, wewriteH̃•,Fp := lim
H• (Xr , Fp )
←−
r

and
HF•p := lim
H • (Xr , Fp ).
−→r

We can also consider Borel–Moore and compactly supported variants:BMH̃•,F:= limH•BM (Xr , Fp )
p
←−
r

and
•
Hc,F
:= limHc• (Xr , Fp ).
p
−→
r

BM
•
There are natural maps H̃•,Fp → H̃•,F
and Hc,F→ HF•p . Note that inpp
the case of cohomology, there is no completion at all. We then have thefollowing result (whose proof will appear in [6]), which provides the analogue of Theorem 1.1 for Fp -coefficients, as well as universal coefficient
type results relating the completed (co)homology to its mod p variant.

Theorem 1.16 1. There is a natural action of the completed groupBMring Fp [[G0 ]] on each of H̃•,Fp and H̃•,F, which makes each of thesepCompleted cohomology – a survey

13

spaces a finitely generated left module over Fp [[G0 ]]. Furthermore, thecanonical topology on each of these modules (obtained by writing it asa quotient of a finite direct sum of copies of Fp [[G0 ]]) coincides withprojective limit topology.
•
2. The spaces HF•p and Hc,F
are admissible smooth representations ofp
G0 over Fp .
3. There are natural isomorphisms∼

HF•p −→ Homcont (H̃•,Fp , Fp )(here cont means continuous with respect to the canonical topology on
the source, and the evident topology — either discrete or p-adic —on the target) and∼H̃•,Fp −→ Hom(HF•p , Fp )(since the natural topology on H̃ • is the p-adic topology, and since
all Zp -linear maps are automatically p-adically continuous, there isno need to explicitly specify any continuity conditions on the maps
appearing in this exact sequence), as well as similar isomorphisms
BM•relating H̃•,F
and Hc,F. These isomorphisms are compatible withp
p•
BM
→ HF•p .
the maps H̃•,Fp → H̃•,Fp , and the maps Hc,Fp
4. For each r ≥ 0, there are Hochschild–Serre type spectral sequences
E2i,j := H i (Gr , H j ) =⇒ H i+j (Xr , Fp )andE2i,j := H i (Gr , Hcj ) =⇒ Hci+j (Xr , Fp ),•→ HF•p .
compatible with the maps Hc,Fp
5. There are short exact sequences

0 → H̃• /pH̃• → H̃•,Fp → H̃•−1 [p] → 0
and
0 → H̃ • /pH̃ • → HF•p → H̃ •+1 [p] → 0,BM•
as well as similar isomorphisms relating H̃•,F
to H̃•BM and Hc,F
to
p
p
•
H̃c .

If M is any finite generated left Fp [[G0 ]]-module, we define EF•p (M ) :=Ext• (M, Fp [[G0 ]]). We can then define the Fp -codimension of M (orcodimFp M , for short) to be the minimal i such that EFi p (M ) 6= 0. Of

14Calegari and Emertoncourse, we may also regard M as a left Zp [[G0 ]]-module, and there areshort exact sequences
0 → E • (M ) → EF•p (M ) → E •+1 (M ) → 0(as follows by applying Ext• (M, –) to the short exact sequence 0 →p·Zp [[G0 ]] → Zp [[G0 ]] → Fp [[G0 ]] → 0), from which one deduces that
the Fp -codimension of M is one less than the codimension of M . In
particular, we find that the Iwasawa dimension of M is equal to dim G0 −codimFp M.
If M is a finitely generated Zp [[G0 ]]-module, then M/pM is a finitely
generated Fp [[G0 ]]-module, and we have the following simple lemma.
Lemma 1.17 If M is a finitely generated Zp [[G0 ]]-module which isfurthermore p-torsion free, then the codimension of M and the Fp codimension of M/pM coincide.
Proof Computing E i (M ) using a projective resolution of M , one immediately finds (using the p-torsion-freeness of M ) that∼E • (M )/pE • (M ) −→ EF•p (M/pM ).
The lemma follows from this.
Returning to the case when M is a finitely generated Fp [[G0 ]]-module,
we note that V := Homcont (M, Fp ), when equipped with the contragredient action of G0 , is then an admissible smooth representation of G0
over Fp . (We also recall that, conversely, M can be recovered from V
via the double duality isomorphism M = Hom(V, Fp ).) One nice aspectof the mod p situation is that the Iwasawa dimension of M can be recovered from the behaviour of the corresponding smooth representation
V “at finite level”, as the following theorem shows. (For the first statement of the following result, see [1]; the proof of the second statement
will appear in [6].)Theorem 1.18 Let M be a finitely generated Fp [[G0 ]]-module, andwrite V := Homcont (M, Fp ), so that V is an admissible smooth G0 representation over Fp . If the Iwasawa dimension of M is equal to d,dthen we have that dim H 0 (Gr , V ) ∼ λ · [G0 : Gr ] dim G0 , for some λ > 0,dwhile dim H i (Gr , V ) = O([G0 : Gr ] dim G0 ) for all i > 0.
Remark 1.19 If G is (the group of Qp -points of) a reductive groupover Qp , if V is an irreducible admissible smooth representation of G
over C, and if {Gr }r≥0 is a neighbourhood basis of the identity of G

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15

consisting of compact open subgroups, then dim H 0 (Gr , V ) ∼ λ · [G0 :dGKGr ] dim G , for some λ > 0, where dGK is the Gelfand–Kirillov dimensionof V . Theorem 1.18 thus suggests an analogy between Gelfand–Kirillov
dimensions of admissible smooth representations in characteristic zero,and Iwasawa dimensions of duals to admissible smooth representations
in characteristic p. It remains to be seen how fruitful this analogy is.
Turning now to the context of Sections 1.5 and 1.6, we believe thatthe analogue of Conjecture 1.5 above, with codimensions replaced by Fp BM
codimensions, should hold for H̃•,Fp and H̃•,F. In fact, Conjecture 1.5p
and its mod p analogue are far from independent. The following result
gives one example of how they are related.
Proposition 1.20 If H̃q0 is p-torsion-free, then H̃q0 has codimension
equal to l0 if and only if H̃q0 ,Fp has Fp -codimension equal to l0 .
Proof This follows immediately from part (5) of Theorem 1.16, together
with Lemma 1.17.
We also have the following interpretation of (a somewhat weakenedform) of the mod p analogue of Conjecture 1.5 in terms of the rate of
growth of Fp -cohomology up the congruence tower Xr .
Proposition 1.21

The following are equivalent:

1. H̃i,Fp has codimension > l0 for i < q0 , and H̃q0 ,Fp has codimension
l0 ;
l0 +1
1− dimG
2. dim H i (Xr , Fp ) = O V (Xr )if i < q0 , whilel0

dim H q0 (Xr , Fp )  V (Xr )1− dim G .
Proof The equivalence of the two conditions follows from Theorem 1.18
together with part (4) of Theorem 1.16.
Of course, there is an analogous statement relating the codimension
BMof the modules H̃i,F
to the rate of growth of compactly supported Fp p
cohomology up the tower.
Remark 1.22 If we assume that not only the equivalent conditions (1)and (2) hold, but that the analogous statements also hold for each Levifactor of G, then we may use a Poincaré duality argument, togetherBMwith a comparison of H̃•,F
and H̃•,Fp , to deduce that furthermore
p

1−((l
0 +1)/d)dim H i (Xr , Fp ) = O V (Xr )if i > l0 + q0 .

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Calegari and Emerton

1.8 Heuristics related to the p-adic Langlands
programme∼

Throughout this section we fix an isomorphism ı : C −→ Qp . If G
is a reductive group over Q, then the Langlands programme [13], whencoupled with the conjecture of Fontaine–Mazur [10], posits the existenceof a correspondence of the form
{algebraic automorphic Hecke eigenforms on G(A)}←→{continuous representations GQ → GL (Qp ) which are unramified
outside finitely many primes and are de Rham at p},(1.5)
where GQ denotes the absolute Galois group of Q, and GL denotes the
L-group of G (thought of as a reductive group over Q). There are many
subtleties involved in making a conjecture of this form precise, beginningperhaps with the precise definition of algebraic in the context of automorphic forms, and we won’t attempt to discuss them here, or to state aprecise conjecture, referring instead to the recent paper [3], which gives
this issue the careful consideration that it deserves.
We content ourselves by remarking that the matching (such as it exists) between automorphic forms and Galois representations is supposed
to be made in the following manner (which depends on our chosen iso∼morphism ı : C −→ Qp ): an algebraic automorphic Hecke eigenform f
and a p-adic Galois representation ρ match if, for some sufficiently large
finite set of primes S, containing p together with all the primes dividing
either the conductor of f or the conductor of ρ, we have that for eachprime ` 6∈ S, the `th Langlands parameter computed by applying theSatake isomorphism (suitably normalized) to the system of Hecke eigenvalues at the prime ` attached to f , which is a semi-simple conjugacy∼
class in GL (C), coincides (under the isomorphism GL (C) −→ GL (Qp )induced by ı) with the semi-simple part of the conjugacy class in GL (C)obtained by applying ρ to the conjugacy class of Frobenius elements at` in GQ . We refer to [3, Conj. 3.2.1] and the subsequent remarks for amore precise statement and discussion.
We note in addition that to an algebraic automorphic eigenform we
can associate a collection of integral “Hodge numbers” ([3, Rem. 3.2.3];
these are constructed using the Langlands parameter of the representation of G(R) generated by f , or alternatively, using the infinitesimalcharacter of f ), and that if f and ρ match in the above sense, thenCompleted cohomology – a survey

17

the Hodge numbers associated to f should match with the Hodge–Tateweights of the de Rham representation ρ |GQp .
Now the property of being algebraic is defined in terms of these “Hodge
numbers” being integral [3] (the precise definition of algebraic then depending on exactly how the Hodge numbers are normalized [3]). It isthen possible to enlarge the automorphic side of the conjectured correspondence by considering automorphic forms that are non-algebraic, i.e.
whose associated “Hodge numbers” are not integral. It is also possible toenlarge the Galois side of the conjectured correspondence by considering
automorphic forms that are non-de Rham. These representations haveassociated Hodge–Sen–Tate weights which are typically non-integral.
Unfortunately, it seems that when we enlarge the two sides of the correspondence in this manner, the enlarged sets are not related to eachother. To understand why, note that on the automorphic side we areenlarging the set of allowed Hodge numbers in an archimedean manner, while on the Galois side, we are doing so in a p-adic manner, andthere is no reason (that we know of) to think that these two kinds ofenlargements should have any relationship to one another.
What we would like, in order to extend the Langlands correspondence
so as to allow non-de Rham Galois representations to be included, is
to introduce a notion of p-adic automorphic form which generalizes thenotion of algebraic automorphic form, whose associated “Hodge numbers” (whatever they might be) are p-adic rather than archimedean. We
believe that p-adically completed cohomology provides a partial (andslightly indirect) solution to the problem of defining such a notion of
p-adic automorphic form.
We put ourselves in the context of Section 1.5. As noted there, thecompleted homology modules H̃n admit an action of a completed Hecke
algebra T.
Franke’s theorem [11] describing the cohomology of congruence quotients in terms of automorphic forms shows that any system of Hecke
eigenvalues appearing in some degree of cohomology (with C-coefficients,
or more generally with coefficients in some local system associated to an
algebraic representation of G/C ) of G(Q)\G(A)/A∞ K∞ Gr K ∞,p , is alsoa system of Hecke eigenvalues arising from some algebraic automorphic
Hecke eigenform (more precisely, a C-algebraic eigenform in the senseof [3]) that is invariant under K ∞,p . We call such automorphic Hecke
eigenforms cohomological Hecke eigenforms of tame level K ∞,p .
Using our fixed isomorphism ı : C → Qp , we may identify cohomology
with coefficients in C (or a local system associated to an algebraic rep-18

Calegari and Emerton

resentation of G/C ) with cohomology with coefficients in Qp (or a local
system associated to an algebraic representation of G/Qp ). One can verify that the Hochschild–Serre-type spectral sequence of Theorem 1.1 (4)(and its generalization for local systems mentioned in Remark 1.2) isHecke equivariant, and so we conclude that each cohomological Hecke
eigenform of tame level K ∞,p gives rise to a point λ ∈ Spec T(Qp ).
We expect that Spec T will be topologically finitely generated as a Zp algebra. The point of the preceding discussion is that every system of
Hecke eigenvalues associated to a cohomological Hecke eigenform of tamelevel K ∞,p appears as a point of Spec T(Qp ). Thus we regard Spec T(Qp )as a space parameterizing “p-adic cohomological systems of Hecke eigenvalues”; points of this space serve as a working model for the notion of(cohomological) p-adic automorphic Hecke eigenvalues.
We then expect that the conjectural correspondence (1.5) (or, more
precisely, its restriction to the cohomological Hecke eigenforms) should
extend to identify T with a space of p-adic Galois representations. Justslightly more precisely, we write the complete ring T as a product T =Q
m Tm of its local factors. Attached to each m, and to a choice of an embedding T/m ,→ Fp , we expect that there should be an associated Galois
representation ρm : GQ → GL (Fp ). (This is quite possibly an oversimplification, as the discussion of [3] shows,2 or perhaps more accurately,
it is an idealization of the actual situation; however, it will serve wellenough for our present purpose, which is only to explain our motivations in the most general way.) Associated to ρm is a deformation ringRρm parameterizing deformations of ρm . (Here we should impose ramification conditions away from p expressing the idea that the prime-to-p
conductor of the deformations considered is bounded by K ∞,p — this isa relatively minor point. The key point is that we impose no ramification
conditions at p. Since we are only discussing heuristics here, we don’t try
to make our discussion more precise; in particular, we don’t attempt toaddress the precise meaning of Rρm in the case when ρm is “reducible”(i.e. factors through a parabolic of GL ).) What we then expect is that∼
there will be an isomorphism Tm −→ Rρm .
Now one can compute the expected dimension of Rρm via the global
Euler characteristic formula [15], and what one finds is that expected2

To be just slightly less oblique, we are (in the terminology of [3]) considering
C-algebraic, rather than L-algebraic, automorphic forms, and the discussion of[3], especially Section 5, shows that it may be necessary to replace G by az-extension in order to get a correspondence between C-algebraic automorphic
eigenforms and p-adic Galois representations.

Completed cohomology – a survey

19

dimension of Rρm is 1 + dim B − l0 , where B is a Borel subgroup of
G. Thus we expect that each Tm , and so T itself, should have Krull
dimension 1 + dim B − l0 .
Now consider H̃q0 as T-module. We in fact hope that it will be faithful(in other words, there are no systems of eigenvalues appearing in H̃n for
n 6= q0 which don’t also appear in H̃q0 ), the reason being that q0 is thefirst dimension in which tempered representations of G∞ can contributecohomology [2], and so is the first “interesting dimension of cohomology”. Since it is the first dimension which knows about the interesting
automorphic forms, we also expect that it knows about all the interesting
Galois representations, and we hope that all the Galois representations
are in the Zariski closure of the interesting ones.
Assuming this faithfulness, one can then naively view H̃q0 as a kind ofbundle over Spec T, and one might then guess the following dimension
formula:Iwasawa dim. of H̃q0?

= Krull dim. of Spec T + Iwasawa dim. of the fibres. (1.6)
What do we expect the Iwasawa dimension of a fibre to be? The fibreof H̃q0 over a point λ ∈ Spec T(Qp ) associated to a cohomological Hecke
eigenform is at least morally dual to the λ-eigenspace in cohomology, andthe rough analogy between Iwasawa dimensions and Gelfand–Kirillov dimensions of smooth representations (see Theorem 1.18 and Remark 1.19)
then suggests that the dimension of this fibre might be equal to theGelfand–Kirillov dimension of the λ-eigenspace in cohomology, which inturn will be equal to dim G/B in the generic situation. (An irreducible
admissible smooth representation of G(Qp ) has Gelfand–Kirillov dimension at most equal to dim G/B, with equality precisely when it is generic,i.e. when it admits a Whittaker model.) Combining this final heuristicwith (1.6) and our computation of the expected dimension of T, we areleft with the following guess:?

Iwasawa dim. of H̃q0 = 1 + dim B − l0 + dim G/B = 1 + dim G − l0 .
Equivalently, we guess that the codimension of H̃q0 is equal to l0 , which
is Conjecture 1.5 (2). The preceding discussion provides one of the main
motivations for this conjecture.
Since we expect that the systems of eigenvalues appearing in H̃n for
n < q0 will be quite special, we expect that the support of H̃n in
Spec T will have positive codimension. This helps to motivate Conjec-

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Calegari and Emerton

ture 1.5 (1). As noted in Remark 1.7, if we believe parts (1) and (2) ofConjecture 1.5, we are naturally led to believe the entire conjecture.
We close by remarking that a large part of the framework describedhere has been worked out in detail in the particular case of G = GL2 [9].

References[1] K. Ardakov, K. Brown, Ring-Theoretic Properties of Iwasawa Algebras:a survey, Documenta Math., Extra volume in honour of John Coates’s
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