## Are Galois deformation rings Cohen-Macaulay?

Hyman Bass once wrote a paper on the ubiquity of Gorenstein rings. The first time they arose in the context of Hecke algebras, however, was Barry’s Eisenstein ideal paper, where he proves (at prime level) that the completions $\mathbf{T}_{\mathfrak{m}}$ are Gorenstein for all non-Eisenstein maximal ideals $\mathfrak{m}$ of $\mathbf{T}$ except possibly those which are ordinary of residual characteristic two. He also shows that the completions at Eisenstein primes are also Gorenstein, although this is trickier and makes fundamental use of the assumption that the level is prime. The Gorenstein property of various Hecke at non-Eisenstein maximal ideals was crucially used by Wiles to deduce non-minimal modularity lifting theorems. In the late 90’s, including around the time I started graduate school, it seemed as though all Hecke algebras in weight two were going to be Gorenstein (localized at non-Eisenstein ideals). One case remained, however, namely when $\mathrm{char}(k) = 2$, and

$\overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(k)$

has the property that $\overline{\rho}$ is unramified at $2$ and, moreover, the image of Frobenius at $2$ is a scalar. (The other cases having been dealt with by results of Mazur, Wiles, Ribet, and Buzzard.) But then it turned out, amazingly, that $\mathbf{T}$ was not always Gorenstein. Lloyd Kilford found a counter-example at level $N = 431$. The natural place to look, of course, is at $\mathrm{GL}_2(\mathbf{F}_2) = S_3$-representations. They have to come from a quadratic field $K$ with class number divisible by three and such that $2$ splits completely in the corresponding unramified degree three extension of $K$. It also makes sense to work at prime level, because this will make computing the integral Hecke ring easier. The condition that $2$ splits in $K$ forces $\Delta_K$ to be congruent to $1 \mod 8$, which certainly means the class number is odd. The condition that $2$ split in the corresponding cubic field is more subtle; if the class number of the field was $3$, then this would be equivalent to the primes in $K$ above $2$ splitting principally in $K$, but this can’t happen for norm reasons. So one has to start with a quadratic field $K$ with $\Delta_K \equiv 1 \mod 8$ and class number $h = 3h'$ for some $h' > 1$, and such that the class given by $[\mathfrak{p}]$ for the prime above $2$ does not generate the 3-Sylow subgroup. The smallest prime number with this property is … $N = 431$. So it fails at the first opportunity! As Kevin once joked to me in a statement that sums up the best of all attitudes towards advising: “If I had known it was going to be that easy, I would have done it meself!”

Nowadays we know, at least in the analogous context when $p$ is odd and we are in weight $p$, that the appropriate Hecke algebras are Cohen-Macaulay. But we understand that the reason that these global Hecke algebras have these properties is because the *local* Hecke algebras have nice properties. The idea of deducing facts about the global Hecke algebra in the process of proving modularity lifting theorems started with Diamond, who found the first improvement to the Taylor-Wiles method. Essentially, given an $R=\mathbf{T}$ theorem, one has a presentation of $\mathbf{T}$ as a quotient of a (power series over a) local deformation ring by a sequence of parameters. If the local deformation rings are nice (Complete Intersections, Gorenstein, Cohen-Macaulay, etc.) then so is the global Hecke ring. Now this is only true in the contexts where $\ell_0 = 0$; otherwise one is taking the quotient by “too many” relations (that is, not a sequence of parameters), and so there’s no longer any reason to expect that $\mathbf{T}$ has those nice properties unless $\ell_0 = 1$ and $\mathbf{T}$ is finite.

So now we come to the question: are all local deformation rings Cohen-Macaulay? Well, perhaps there is not really any reason to suppose that they are. Perhaps even worse, there is a paper by Fabian Sander, a student of Vytas, proving that a certain deformation ring is not Cohen-Macaulay. But I am not deterred. My issue is that one has to take the correct deformation ring. And the correct deformation ring is the one that should include the extra data corresponding to the local Hecke operators which may not come (at an integral level) from the Galois representation.

To take a well known example, consider ordinary $p$-adic representations of weight $p$. From a characteristic zero ordinary representation, one can always recover the (unique) eigenvalue of Frobenius on the unramified quotient. But this is not possible at the integral level, because (for example) $\overline{\rho}$ could be locally trivial. This exactly corresponds to the fact that in weight $p$, the Hecke operator $T_p$ does not have to lie in the algebra generated by the other Hecke operators (the “anemic” Hecke algebra — was that term coined by Ken Ribet?). In order to prove modularity theorems, it usually suffices to work with the anemic Hecke algebra, but when one does include data which captures $T_p$ (or $U_p$) the local deformation ring is (in this case) Cohen-Macaulay, as was shown by Snowden. So, for example, I would conjecture that the ordinary deformation ring (in any dimension) which includes the local Galois information corresponding to *all* the Hecke operators is Cohen-Macaulay.

Is there any real evidence for this guess besides the fact that it would be useful? Well, not really. But it would provide a systematic local Galois explanation for why deformation rings are torsion free, which is consistent with the guess that, appropriately defined, one should latex $R = \mathbf{T}$ theorems on the nose, not just after looking at (say) $\mathrm{MaxSpec}$. Of course, all of this is in the residually globally irreducible setting. Note that one reason to care about integral modularity statements is that most of the time, one would expect both $R$ and $\mathbf{T}$ to be torsion anyway.