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 finite with ring of integers and uniformizer . Let be connected reductive, open compact, a finite set of places away from such that may be written as with open compact and with unramified and hyperspecial for all . Let be the abstract spherical Hecke algebra at places away from . In section 2.1.4 of his ICM article, Emerton defines a big p-adic Hecke algebra acting on the cohomology of arithmetic groups, whose definition we recall. Consider the profinite ring
where runs over all cohomological degrees, runs over all open compact subgroups, and runs over all representations of on finitely generated torsion -modules. (Here is the usual locally symmetric space.) There’s a natural -algebra map , and Emerton defines as the closure of the -subalgebra .
The (easy) point of this post is a slightly more pleasing definition of .
Proposition. The ring coincides with the profinite completion of the subring of
generated by the image of the obvious map . Here
denotes completed homology in degree .
Proof (slightly pedantic). First of all, there’s a natural action of on . Indeed, we write
and then observe that
(remember that is injective as a module over itself, even though it isn’t as an abelian group! took me a minute to remember this), so
and by definition we have natural maps from into the right-hand side which compile into the desired action as and vary.
Now, for any and as above, there’s a spectral sequence of -modules
In particular, if an element kills all the ‘s, then it kills all the -terms and thus kills the whole abutment for any , so it maps to zero in . But the action of on the -page factors through the action of on the ‘s, so we’ve just shown that is a faithful -module.
All that remains is to observe that the natural inclusion of
in is dense in the profinite topology on the latter. This is trivially equivalent to showing that for a fixed separating sequence of finite quotients
of , given any and any we can find some with in . We do this as follows: choose a cofinal sequence of normal open compacts, and observe that the images , of , in the (finite) ring
trivially coincide for any (by seeing this object inside the original definition of ). Now
by the faithfulness proved above, so the ‘s are separating, and all the maps are surjective. Thus given with reduction we may choose any lifting and then take .
Here’s another nice thing coming out of this. Let be a totally real or CM field over , and take . Let be the image of in
satisfying the usual compatibility of Frobenius and Hecke eigenvalues. (I’ll abbreviate for the correct notation in the context of -valued determinants out of a group .) Now suppose we’re given a finite set and a collection of finite torsion -modules for with -actions. Given any , let denote the image of in . It’s easy to see that for any subsets one has surjections whose kernels interset trivially. In particular, if you’re given -dimensional determinants for all , an easy induction on using the gluability of determinants gives a unique determinant projecting to each under . Now suppose Scholze’s result holds with , i.e. with (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 above (so with e.g. and for ), we’d get honest determinants for each compatible with varying and so fitting together into a universal -dimensional determinant
If is a maximal ideal of with finite residue field and the associated representation is absolutely irreducible, then writing for the localization of at , the associated determinant actually comes from a “universal p-adic automorphic deformation”
(use Theorem B in Chenevier’s article.) Now an appropriate global deformation problem for should give rise to a Galois deformation ring and , together with a surjection with . In particular, since is Noetherian, we get (finally!) that 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.