Universal eigenvarieties, trianguline Galois representations, and
p-adic Langlands functoriality

David Hansen; James Newton

doi:10.1515/crelle-2014-0130

Article

Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality

David Hansen and James Newton

Published/Copyright: March 31, 2015

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From the journal Journal für die reine und angewandte Mathematik Volume 2017 Issue 730

Abstract

Using the overconvergent cohomology modules introduced by Ash–Stevens, we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces, building on work of Ash–Stevens, Urban, and others. We also formulate a precise modularity conjecture linking trianguline Galois representations with overconvergent cohomology classes. In the course of giving evidence for this conjecture, we establish several new instances of p-adic Langlands functoriality. Our main technical innovations are a family of universal coefficients spectral sequences for overconvergent cohomology and a generalization of Chenevier’s interpolation theorem.

Funding statement: This work was carried out at Boston College and l’Institut de Mathématiques de Jussieu, and I am pleased to acknowledge the hospitality of these institutions. The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013) / ERC Grant agreement no. 290766 (AAMOT).

A Some commutative algebra

In this appendix, we collect some results relating the projective dimension of a module M and its localizations, the nonvanishing of certain Tor and Ext groups, and the heights of the associated primes of M. We also briefly recall the definition of a perfect module, and explain their basic properties. These results are presumably well known to experts, but they are not given in our basic reference [61].

Throughout this section, R is a commutative Noetherian ring and M is a finite R-module. Our notations follow [61], with one addition: we write mSupp⁢(M) for the set of maximal ideals in Supp⁡(M).

Proposition A.1.

There is an equivalence

projdimR⁡(M)≥n⇔ExtRn⁢(M,N)≠0⁢ for some ⁢N∈ModR.

See e.g. [61, p. 280] for a proof.

Proposition A.2.

The equality

projdimR⁡(M)=sup𝔪∈mSupp⁢(M)⁡projdimR𝔪⁡(M𝔪)

holds.

Proof.

Any projective resolution of M localizes to a projective resolution of M𝔪, so

projdimR𝔪⁡(M𝔪)≤projdimR⁡(M) for all 𝔪.

On the other hand, if projdimR⁡(M)≥n, then ExtRn⁢(M,N)≠0 for some N, so

ExtRn⁢(M,N)𝔪≠0 for some 𝔪;

but ExtRn⁢(M,N)𝔪≃ExtR𝔪n⁢(M𝔪,N𝔪), so

projdimR𝔪⁡(M𝔪)≥n for some 𝔪

by Proposition A.1. ∎

Proposition A.3.

For M any finite R-module, the equality

projdimR⁡(M)=sup𝔪∈mSupp⁢(M)⁡sup⁡{i:ToriR⁢(M,R/𝔪)≠0}

holds. If furthermore projdimR⁡(M)<∞, then the equality

projdimR⁡(M)=sup⁡{i:ExtRi⁢(M,R)≠0}

holds as well.

Proof.

The module ToriR⁢(M,R/𝔪) is a finite-dimensional R/𝔪-vector space, so localization at 𝔪 leaves it unchanged, yielding

ToriR⁢(M,R/𝔪)≃ToriR⁢(M,R/𝔪)𝔪

≃ToriR𝔪⁢(M𝔪,R𝔪/𝔪).

Since the equality projdimS⁡(N)=sup⁡{i:ToriS⁢(N,S/𝔪S)≠0} holds for any local ring S and any finite S-module N (see e.g. [61, Lemma 19.1.ii]), the first claim now follows from Proposition A.2.

For the second claim, we first note that if S is a local ring and N is a finite S-module with projdimS⁡(N)<∞, then projdimS⁡(N)=sup⁡{i:ExtSi⁢(N,S)≠0} by [61, Lemma 19.1.iii]. Hence by Proposition A.2 we have

projdimR⁡(M)=sup𝔪∈mSupp⁢(M)⁡sup⁡{i:ExtR𝔪i⁢(M𝔪,R𝔪)≠0}

=sup⁡{i:ExtRi⁢(M,R)𝔪≠0⁢ for some ⁢𝔪}

=sup⁡{i:ExtRi⁢(M,R)≠0},

as desired. ∎

Proposition A.4.

If R is a Cohen–Macaulay ring, M is a finite R-module of finite projective dimension, and p is an associated prime of M, then

ht⁢𝔭=projdimR𝔭⁡(M𝔭).

In particular, ht⁢p≤projdimR⁡(M).

Proof.

Supposing 𝔭 is an associated prime of M, there is an injection R/𝔭↪M; this localizes to an injection R𝔭/𝔭↪M𝔭, so depthR𝔭⁢(M𝔭)=0. Now we compute

ht⁢𝔭=dim⁡(R𝔭)

=depthR𝔭⁢(R𝔭)(by the CM assumption)

=depthR𝔭⁢(M𝔭)+projdimR𝔭⁡(M𝔭)(by the Auslander–Buchsbaum formula)

=projdimR𝔭⁡(M𝔭),

whence the result. ∎

Now we single out an especially nice class of modules, which are equidimensional in essentially every sense of the word. Recall the grade of a module M, written gradeR⁢(M), is the annR⁡(M)-depth of R; by [61, Theorems 16.6 and 16.7],

gradeR⁢(M)=inf⁡{i:ExtRi⁢(M,R)≠0},

so quite generally gradeR⁢(M)≤projdimR⁡(M).

Definition A.5.

A finite R-module M is perfect if gradeR⁢(M)=projdimR⁡(M)<∞.

Proposition A.6.

Let R be a Noetherian ring, and let M be a perfect R-module, with gradeR⁢(M)=projdimR⁡(M)=d. Then for any p∈Supp⁡(M) we have

gradeR𝔭⁢(M𝔭)=projdimR𝔭⁡(M𝔭)=d.

If furthermore R is Cohen–Macaulay, then M is Cohen–Macaulay as well, and every associated prime of M has height d.

Proof.

The grade of a module can only increase under localization (as evidenced by the Ext definition above), while the projective dimension can only decrease; on the other hand, gradeR⁢(M)≤projdimR⁡(M) for any finite module over any Noetherian ring. This proves the first claim.

For the second claim, Theorems 16.6 and 17.4.i of [61] combine to yield the formula

dim⁡(M𝔭)+gradeR𝔭⁢(M𝔭)=dim⁡(R𝔭)

for any 𝔭∈Supp⁢(M). The Auslander–Buchsbaum formula reads

depthR𝔭⁢(M𝔭)+projdimR𝔭⁡(M𝔭)=depthR𝔭⁢(R𝔭).

But

dim⁡(R𝔭)=depthR𝔭⁢(R𝔭)

by the Cohen–Macaulay assumption, and

gradeR𝔭⁢(M𝔭)=projdimR𝔭⁡(M𝔭)

by the first claim. Hence depthR𝔭⁢(M𝔭)=dim⁡(M𝔭) as desired. The assertion regarding associated primes is immediate from the first claim and Proposition A.4. ∎

B The dimension of irreducible components By James Newton at London

In this appendix, we use the results of the above article to give some additional evidence for Conjecture 4. In the notation and terminology of Section 1 above, we prove

Proposition B.1.

Any irreducible component of XG,Kp containing a given point x has dimension at least dim⁡(WKp)-l⁢(x).

Note that [72, Proposition 5.7.4] implies that at least one of these components has dimension at least dim⁡(𝒲Kp)-l⁢(x). This is stated without proof in that reference, and is due to G. Stevens and E. Urban. We learned the idea of the proof of this result from E. Urban—in this appendix we adapt that idea and make essential use of Theorem 1 (in particular the “Tor spectral sequence”) to provide a fairly simple proof of Proposition B.1.

We place ourselves in the setting of Sections 1 and 4.3. In particular, 𝐆 is a reductive group over 𝐐. Fix an open compact subgroup Kp⊂𝐆⁢(𝐀fp) and a slope datum (Ut,Ω,h). Suppose that 𝔐 is a maximal ideal of 𝐓Ω,h⁢(Kp) corresponding to a point x∈𝒳𝐆,Kp⁢(𝐐p¯). Denote by 𝔪 the contraction of 𝔐 to 𝒪⁢(Ω). Let 𝒫 be a minimal prime of 𝐓Ω,h⁢(Kp) contained in 𝔐. Since H∗⁢(Kp,𝒟Ω)≤h is a finite faithful 𝐓Ω,h⁢(Kp)-module, minimal primes of 𝐓Ω,h⁢(Kp) are in bijection with minimal elements of Supp𝐓Ω,h⁢(Kp)⁡(H∗⁢(Kp,𝒟Ω)≤h); by [61, Theorem 6.5], minimal elements of the latter set are in bijection with minimal elements of Ass𝐓Ω,h⁢(Kp)⁢(H*⁢(Kp,𝒟Ω)≤h).

Definition B.2.

Denote by r the minimal index i such that 𝒫 is in the support of Hi⁢(Kp,𝒟Ω)≤h,𝔐, and by q the minimal index i such that Hi⁢(Kp,𝒟λx)(ker⁡ϕx)≠0.

Let ℘ denote the contraction of 𝒫 to a prime of 𝒪⁢(Ω)𝔪; in particular, ℘ is an associated prime of Hr⁢(Kp,𝒟Ω)≤h,𝔐. The ring 𝒪⁢(Ω)𝔪 is a regular local ring. The localisation 𝒪⁢(Ω)℘ is therefore a regular local ring, with maximal ideal ℘⁢𝒪⁢(Ω)℘. We let (x1,…,xd) denote a regular sequence generating ℘⁢𝒪⁢(Ω)℘. After multiplying the xi by units in 𝒪⁢(Ω)℘, we may assume that the xi are in 𝒪⁢(Ω). Note that (x1,…,xd)⁢𝒪⁢(Ω)𝔪 may be a proper submodule of ℘. Nevertheless, we have

d=dim⁡(𝒪⁢(Ω)℘)=ht⁢(℘).

We will show that d≤l⁢(x).

Denote by Ai the quotient 𝒪⁢(Ω)℘/(x1,…,xi)⁢𝒪⁢(Ω)℘ and denote by Σi the Zariski-closed subspace of Ω defined by the ideal (x1,…,xi)⁢𝒪⁢(Ω). The affinoids Σi may be non-reduced. Note that Ai=𝒪⁢(Σi)℘ and 𝒪⁢(Σi+1)=𝒪⁢(Σi)/xi+1⁢𝒪⁢(Σi).

Lemma B.3.

The space Hr-d⁢(Kp,DΣd)≤h,P is non-zero.

Proof.

By induction, it suffices to prove the following: let i be an integer satisfying 0≤i≤d-1. Suppose Hr-i⁢(Kp,𝒟Σi)≤h,𝒫 is a non-zero Ai-module, with ℘⁢Ai an associated prime, and

Ht⁢(Kp,𝒟Σi)≤h,𝒫=0

for every t<r-i. Then Hr-i-1⁢(Kp,𝒟Σi+1)≤h,𝒫 is a non-zero Ai+1-module, with ℘⁢Ai+1 an associated prime, and

Ht⁢(Kp,𝒟Σi+1)≤h,𝒫=0

for every t<r-i-1.

Note that the hypothesis of this claim holds for i=0, by the minimality of r. Suppose the hypothesis is satisfied for i. It will suffice to show that

Ht⁢(Kp,𝒟Σi+1)≤h,𝒫=0 for every t<r-i-1,
there is an isomorphism of non-zero Ai-modules
ι:Tor1Ai⁢(Hr-i⁢(Kp,𝒟Σi)≤h,𝒫,Ai/xi+1⁢Ai)≅Hr-i-1⁢(Kp,𝒟Σi+1)≤h,𝒫.

Indeed, the left side (which we denote by T) of the isomorphism ι is given by the xi+1-torsion in Hr-i⁢(Kp,𝒟Σi)≤h,𝒫, so a non-zero Ai-submodule of Hr-i⁢(Kp,𝒟Σi)≤h,𝒫 with annihilator ℘⁢Ai immediately gives a non-zero Ai+1-submodule of T with annihilator ℘⁢Ai+1.

Both the claimed facts are shown by studying the localisation at 𝒫 of the spectral sequence

E2s,t:Tor-sA⁢(Σi)⁢(Ht⁢(Kp,𝒟Σi)≤h,A⁢(Σi+1))⇒Hs+t⁢(Kp,𝒟Σi+1)≤h

(cf. Remark 2). After localisation at 𝒫, the spectral sequence degenerates at E2. This is because we have a free resolution

0→Ai→×xi+1Ai→Ai+1→0

of Ai+1 as an Ai-module (we use the fact that xi+1 is not a zero-divisor in Ai), so (E2s,t)𝒫 vanishes whenever s∉{-1,0}. Moreover, since

Ht⁢(Kp,𝒟Σi)≤h,𝒫=0

for every t<r-i, we know that (E2s,t)𝒫 vanishes for t<r-i. The existence of the isomorphism ι and the desired vanishing of Ht⁢(Kp,𝒟Σi+1)≤h,𝒫 are therefore demonstrated by the spectral sequence, since the only non-zero term (E2s,t)𝒫 contributing to (E∞r-i-1)𝒫 is given by s=-1, t=r-i, whilst (E2s,t)𝒫=0 for all (s,t) with s+t<r-i-1. ∎

Corollary B.4.

We have an inequality r-d≥q. Since r≤q+l, we obtain d≤l. In particular, ℘ has height ≤l, so the irreducible component of TΩ,h⁢(Kp) corresponding to P has dimension ≥dim⁡(Ω)-l.

Proof.

It follows from Proposition 2 (with Ω replaced by Σd) that

Hi⁢(Kp,𝒟Σd)≤h,𝔐=0

for i<q. Our lemma therefore implies that r-d≥q. The conclusion on dimensions follows from the observation made in Section 4.5 that 𝐓Ω,h⁢(Kp)/𝒫 has the same dimension as 𝒪⁢(Ω)/℘. ∎

Proposition B.1 follows immediately from the corollary. We have also shown that if d=l, then r=q+l.

Acknowledgements

This article is a revised and expanded version of my 2013 Boston College Ph.D. thesis [‘Overconvergent cohomology: Theory and applications’, Ph.D. thesis, Boston College, 2013]. First and foremost, I heartily thank my advisor, Avner Ash, for his invaluable suggestions, sage advice, patient readings of preliminary drafts, and overall kindness and generosity. I am grateful to be his student. I am indebted to Glenn Stevens for kindly encouraging me to work on overconvergent cohomology, for explaining the utility of slope decompositions to me, and for generally serving as a second advisor. The intellectual debt this article owes to the ideas of Ash and Stevens will be obvious to the reader. I am also grateful to Ben Howard for some helpful conversations and for his detailed remarks on a preliminary draft of my thesis. During the development of this material, I enjoyed stimulating conversations with Joël Bellaïche, John Bergdall, Kevin Buzzard, Przemysław Chojecki, Giovanni Di Matteo, Michael Harris, Eugen Hellmann, Keenan Kidwell, Judith Ludwig, Barry Mazur, James Newton, Jay Pottharst, and Jack Thorne, and it is a pleasure to acknowledge the help and influence of all these mathematicians. I am also especially grateful to Jack for his detailed comments on several preliminary drafts over the years. It is a particular pleasure to thank Barry Mazur for a number of inspiring discussions, for his generous and invaluable encouragement, and for providing financial support during the summer of 2013. Finally, I am grateful to the referee for helpful comments and corrections.

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Received: 2012-9-25

Revised: 2014-10-25

Published Online: 2015-3-31

Published in Print: 2017-9-1

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https://doi.org/10.1515/crelle-2014-0130