## Some little remarks on big Hecke algebras

Edit (7/22): The second paragraph in the proof of the proposition below is total nonsense.  I’ll put up a second version of this post soon, but I’m leaving the original as a warning to myself to not write blog posts when I’m sleep-deprived in airports. (/Edit)

Fix ${L/\mathbf{Q}_{p}}$ finite with ring of integers ${\mathcal{O}}$ and uniformizer ${\varpi}$. Let ${\mathbf{G}/\mathbf{Q}}$ be connected reductive, ${K^{p}\subset\mathbf{G}(\mathbf{A}_{f}^{p})}$ open compact, ${\Sigma}$ a finite set of places away from ${p}$ such that ${K^{p}}$ may be written as ${K_{\Sigma}^{p}\prod_{\ell\notin\Sigma\cup\{p\}}K_{\ell}^{p}}$ with ${K_{\Sigma}^{p}\subset\prod_{\ell\in\Sigma}\mathbf{G}(\mathbf{Q}_{\ell})}$ open compact and with ${\mathbf{G}/\mathbf{Q}_{\ell}}$ unramified and ${K_{\ell}^{p}:=K^{p}\cap\mathbf{G}(\mathbf{Q}_{\ell})}$ hyperspecial for all ${\ell\notin\Sigma\cup\{p\}}$. Let ${\mathcal{T}_{\Sigma}=\otimes'_{\ell\notin\Sigma\cup\{p\}}\mathcal{C}_{c}^{\infty}(\mathbf{G}(\mathbf{Q}_{\ell})//K_{\ell}^{p},\mathcal{O})}$ be the abstract spherical Hecke algebra at places away from ${p\cup\Sigma}$. In section 2.1.4 of his ICM article, Emerton defines a big p-adic Hecke algebra ${\mathbf{T}_{\Sigma}}$ acting on the cohomology of arithmetic groups, whose definition we recall. Consider the profinite ring

$\displaystyle R=\prod_{i}\prod_{K_{p}}\prod_{\mathcal{W}}\mathrm{End}_{\mathcal{O}}\left(H^{i}(Y(K^{p}K_{p}),\mathcal{W})\right)$

where ${i}$ runs over all cohomological degrees, ${K_{p}\subset\mathbf{G}(\mathbf{Q}_{p})}$ runs over all open compact subgroups, and ${\mathcal{W}}$ runs over all representations of ${K_{p}}$ on finitely generated torsion ${\mathcal{O}}$-modules. (Here ${Y(K^{p}K_{p})}$ is the usual locally symmetric space.) There’s a natural ${\mathcal{O}}$-algebra map ${\mathcal{T}_{\Sigma}\rightarrow R}$, and Emerton defines ${\mathbf{T}_{\Sigma}}$ as the closure of the ${\mathcal{O}}$-subalgebra ${\mathrm{im}\mathcal{T}_{\Sigma}\subset R}$.

The (easy) point of this post is a slightly more pleasing definition of ${\mathbf{T}_{\Sigma}}$.

Proposition. The ring ${\mathbf{T}_{\Sigma}}$ coincides with the profinite completion of the subring of

$\displaystyle S=\prod_{i}\mathrm{End}_{\mathcal{O}}\left(\tilde{H}_{i}(K^{p})\right)$

generated by the image of the obvious map ${\mathcal{T}_{\Sigma}\rightarrow S}$. Here

$\displaystyle \tilde{H}_{i}(K^{p})=\lim_{\{1\}\leftarrow K_{p}}H_{i}(Y(K^{p}K_{p}),\mathcal{O})$

denotes completed homology in degree ${i}$.

Proof (slightly pedantic). First of all, there’s a natural action of ${\mathbf{T}_{\Sigma}}$ on ${\tilde{H}_{i}(K^{p})}$. Indeed, we write

$\displaystyle \tilde{H}_{i}(K^{p})=\lim_{\leftarrow K_{p},s}H_{i}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s})$

and then observe that

$\displaystyle H_{i}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s})\cong\mathrm{Hom}_{\mathcal{O}/\varpi^{s}}\left(H^{i}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s}),\mathcal{O}/\varpi^{s}\right)$

(remember that ${\mathcal{O}/\varpi^{s}}$ is injective as a module over itself, even though it isn’t as an abelian group! took me a minute to remember this), so

$\displaystyle \mathrm{End}_{\mathcal{O}}\left(H_{i}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s})\right)\cong\mathrm{End}_{\mathcal{O}}\left(H^{i}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s})\right)$

and by definition we have natural maps from ${\mathbf{T}_{\Sigma}}$ into the right-hand side which compile into the desired action as ${s}$ and ${K_{p}}$ vary.

Now, for any ${K_{p}}$ and ${\mathcal{W}}$ as above, there’s a spectral sequence of ${\mathcal{T}_{\Sigma}}$-modules

$\displaystyle E_{2}^{i,j}=\mathrm{Ext}_{\mathcal{O}[[K_{p}]]}^{i}(\tilde{H}_{j}(K^{p}),\mathcal{W})\Rightarrow H^{i+j}(Y(K^{p}K_{p}),\mathcal{W}).$

In particular, if an element ${t\in\mathcal{T}_{\Sigma}}$ kills all the ${\tilde{H}_{j}}$‘s, then it kills all the ${E_{2}}$-terms and thus kills the whole abutment for any ${\mathcal{W}}$, so it maps to zero in ${\mathbf{T}_{\Sigma}}$. But the action of ${\mathcal{T}_{\Sigma}}$ on the ${E_{2}}$-page factors through the action of ${\mathbf{T}_{\Sigma}}$ on the ${\tilde{H}_{j}(K^{p})}$‘s, so we’ve just shown that ${\prod_{j}\tilde{H}_{j}(K^{p})}$ is a faithful ${\mathbf{T}_{\Sigma}}$-module.

All that remains is to observe that the natural inclusion of

$\displaystyle \mathrm{T}_{\Sigma}=\mathrm{im}\mathcal{T}_{\Sigma}\subset\prod_{j}\mathrm{End}_{\mathcal{O}}\left(\tilde{H}_{j}(K^{p})\right)$

in ${\mathbf{T}_{\Sigma}}$ is dense in the profinite topology on the latter. This is trivially equivalent to showing that for a fixed separating sequence of finite quotients

$\displaystyle \cdots\twoheadrightarrow\mathbf{T}_{\Sigma}^{3}\twoheadrightarrow\mathbf{T}_{\Sigma}^{2}\twoheadrightarrow\mathbf{T}_{\Sigma}^{1}$

of ${\mathbf{T}_{\Sigma}}$, given any ${0\neq t\in\mathrm{\mathbf{T}_{\Sigma}}}$ and any ${n>0}$ we can find some ${t_{n}\in\mathrm{T}_{\Sigma}}$ with ${\overline{t}=\overline{t_{n}}}$ in ${\mathbf{T}_{\Sigma}^{n}}$. We do this as follows: choose a cofinal sequence ${K_{p}^{1}\supset K_{p}^{2}\supset K_{p}^{3}\supset\cdots}$ of normal open compacts, and observe that the images ${\mathrm{T}_{\Sigma}^{n}}$, ${\mathbf{T}_{\Sigma}^{n}}$ of ${\mathrm{T}_{\Sigma}}$, ${\mathbf{T}_{\Sigma}}$ in the (finite) ring

$\displaystyle \prod_{j}\prod_{0\leq m\leq n}\prod_{0\leq s\leq n}\mathrm{End}_{\mathcal{O}}\left(H_{j}(Y(K^{p}K_{p}^{m}),\mathcal{O}/\varpi^{s})\right)$

trivially coincide for any ${n>0}$ (by seeing this object inside the original definition of ${\mathbf{T}_{\Sigma}}$). Now

$\displaystyle \mathbf{T}_{\Sigma}\cong\lim_{\leftarrow n}\mathbf{T}_{\Sigma}^{n}$

by the faithfulness proved above, so the ${\mathbf{T}_{\Sigma}^{n}}$‘s are separating, and all the maps ${\mathcal{T}_{\Sigma}\rightarrow\mathbf{T}_{\Sigma}^{n}}$ are surjective. Thus given ${t\in\mathbf{T}_{\Sigma}}$ with reduction ${\overline{t}\in\mathbf{T}_{\Sigma}^{n}}$ ${ }$we may choose any ${\tilde{t}\in\mathcal{T}_{\Sigma}}$ lifting ${\overline{t}}$ and then take ${t_{n}=\mathrm{im}\tilde{t}\in\mathrm{T}_{\Sigma}}$. ${\square}$

Here’s another nice thing coming out of this. Let ${F}$ be a totally real or CM field over ${\mathbf{Q}}$, and take ${\mathbf{G}=\mathrm{Res}_{F/\mathbf{Q}}\mathrm{GL}_{d}}$. Let ${\mathbf{T}(K_{p},s)}$ be the image of ${\mathcal{T}_{\Sigma}}$ in

$\displaystyle \prod_{j}\mathrm{End}_{\mathcal{O}}\left(H^{j}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s})\right).$

Scholze’s amazing work says there is some ${N\geq1}$ depending only on ${[F:\mathbf{Q}]}$ and ${d}$ and a nilpotent ideal ${\mathfrak{a}=\mathfrak{a}(K_{p},s)\subset\mathbf{T}(K_{p},s)}$ with ${\mathfrak{a}^{N}=0}$ together with a universal ${d}$-dimensional determinant

$\displaystyle D:G_{F}\rightarrow\mathbf{T}(K_{p},s)/\mathfrak{a}$

satisfying the usual compatibility of Frobenius and Hecke eigenvalues. (I’ll abbreviate ${D:G\rightarrow A}$ for the correct notation ${D:A[G]\rightarrow A}$ in the context of ${A}$-valued determinants out of a group ${G}$.) Now suppose we’re given a finite set ${I}$ and a collection of finite torsion ${\mathcal{O}}$-modules ${M_{i}}$ for ${i\in I}$ with ${\mathcal{T}_{\Sigma}}$-actions. Given any ${I'\subset I}$, let ${\mathbf{T}^{I'}}$ denote the image of ${\mathcal{T}_{\Sigma}}$ in ${\prod_{i\in I'}\mathrm{End}_{\mathcal{O}}(M_{i})}$. It’s easy to see that for any subsets ${I_{1},I_{2}}$ one has surjections ${\mathbf{T}^{I_{1}\cup I_{2}}\twoheadrightarrow\mathbf{T}^{I_{1}},\mathbf{T}^{I_{2}}}$ whose kernels interset trivially. In particular, if you’re given ${d}$-dimensional determinants ${D_{i}:\mathcal{T}_{\Sigma}\rightarrow\mathbf{T}^{\{i\}}}$ for all ${i}$, an easy induction on ${|I'|}$ using the gluability of determinants gives a unique determinant ${D:G_{F}\rightarrow\mathbf{T}^{I}}$ projecting to each ${D_{i}}$ under ${\mathbf{T}^{I}\rightarrow\mathbf{T}^{\{i\}}}$. Now suppose Scholze’s result holds with ${N=1}$, i.e. with ${\mathfrak{a}=0.}$ (My spies tell me at least two different people know how to actually prove this, although nothing is public yet.) Then applying this argument in the context of the definition of ${\mathbf{T}_{\Sigma}^{n}}$ above (so with e.g. ${I=\{(m,s)\in[0,n]^{2}\}}$ and ${M_{i}=\prod_{j}H_{j}(Y(K^{p}K_{p}^{m}),\mathcal{O}/\varpi^{s})}$ for ${i=(m,s)}$), we’d get honest determinants ${D_{n}:G_{F}\rightarrow\mathbf{T}_{\Sigma}^{n}}$ for each ${n}$ compatible with varying ${n}$ and so fitting together into a universal ${d}$-dimensional determinant

$\displaystyle D:G_{F}\rightarrow\mathbf{T}_{\Sigma}.$

If ${\mathfrak{m}}$ is a maximal ideal of ${\mathbf{T}_{\Sigma}}$ with finite residue field ${k}$ and the associated representation ${r_{\mathfrak{m}}:G_{F}\rightarrow\mathrm{GL}_{d}(k)}$ is absolutely irreducible, then writing ${\mathbf{T}_{\mathfrak{m}}}$ for the localization of ${\mathbf{T}_{\Sigma}}$ at ${\mathfrak{m}}$, the associated determinant actually comes from a “universal p-adic automorphic deformation”

$\displaystyle \tilde{r}_{\mathfrak{m}}:G_{F}\rightarrow\mathrm{GL}_{d}(\mathbf{T}_{\mathfrak{m}})$

(use Theorem B in Chenevier’s article.) Now an appropriate global deformation problem for ${r_{\mathfrak{m}}}$ should give rise to a Galois deformation ring ${R_{\mathfrak{m}}^{\mathrm{univ}}}$ and ${r_{\mathfrak{m}}^{\mathrm{univ}}:G_{F}\rightarrow\mathrm{GL}_{d}(R_{\mathfrak{m}}^{\mathrm{univ}})}$, together with a surjection ${f:R_{\mathfrak{m}}^{\mathrm{univ}}\twoheadrightarrow\mathbf{T}_{\mathfrak{m}}}$ with ${f\circ r_{\mathfrak{m}}^{\mathrm{univ}}\simeq\tilde{r}_{\mathfrak{m}}}$In particular, since ${R_{\mathfrak{m}}^{\mathrm{univ}}}$ is Noetherian, we get (finally!) that ${\mathbf{T}_{\mathfrak{m}}}$ is Noetherian. There really seems to be no idea for a purely automorphic proof of this fact, since (for example) we’re entirely lacking in a meaningful deformation theory for automorphic representations.