Commutative Algebra Seminar

Let S be a complete intersection presented as R/J for R a regular ring of prime characteristic p > 0 and J an ideal generated by a regular sequence of length c (the codimension of S). Let I be an ideal containing J. Must the support of every module of the form H^i_I(S) be a Zariski closed set? Independent results of Hochster and Núñez-Betancourt (2017) or Katzman and Zhang (2017) confirm that this is indeed the case when c = 1 (that is, when S is a hypersurface). The question is open in higher codimension, and remains a problem of significant interest. Hochster and Núñez-Betancourt's hypersurface strategy depends on two key ingredients: control over the associated primes of H^{i+1}_I(J), and the finite generation of certain R{F}-modules. Our goal in this talk is twofold. First, we will present our results from 2019 showing that, in many important cases within the c > 1 setting, the associated primes of H^{i+1}_I(J) can be impossible to control independently of Ass H^i_I(S). Second, we will present recent joint work with Eric Canton on the additional structure (including a simplicial complex of R{F}-submodules) that becomes present only through a non-standard Frobenius action on the local cohomology of S, referred to as the Fedder action. When c = 1, we show how this additional structure can be used to recover the well-known closed support result for hypersurfaces. In higher codimension, it leads to a number of fascinating (and currently open) finite generation questions surrounding the Fedder action. A positive answer to these questions would circumvent the need to control Ass H^{i+1}_I(J), and imply a positive answer to the closed support problem in higher codimension. Speaker(s): Monica Lewis (University of Michigan)