The issue of closed support for the local cohomology of Noetherian modules and the related problem of finiteness of the set of associated primes of local cohomology have been intensely studied in the literature. Although it is known that the local cohomology of a hypersurface ring of characteristic $p > 0$ has closed support, it remains an open question whether this property holds for complete intersection rings of higher codimension. For an ideal J generated by a regular sequence of length $c$ in a regular ring $R$ of characteristic $p > 0$, the closed support property for the local cohomology of $R/J$ would follow from known results in the literature if the local cohomology of $J$ itself always had a finite set of associated primes. We give the first example of a module of the form $H^i_I(J)$ with an infinite set of associated primes to demonstrate that this is not necessarily the case. In fact, for $i < 4$ (resp. $i < 5$), we show that if $R/J$ is a domain (resp. factorial), then $\text{Ass}\, H^i_I(J)$ is finite if and only if $\text{Ass}\, H^{i-1}_I(R/J)$ is finite. Our proof of this statement involves a novel generalization of an isomorphism of Hellus. To obtain positive results on closed support, in joint work of the author with Eric Canton, we construct a chain complex consisting of direct sums of Frobenius-stable annihilators in the local cohomology module $H^c_J(R)$. We prove that this complex is exact, and using the exactness property, we show that for an ideal $I$ of $R$ such that $R/I$ is Cohen-Macaulay, the module $H^{ht(I/J)+c}_{I/J}(R/J)$ has closed support.

Monica's advisor is Mel Hochster.

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https://umich.zoom.us/j/99446100740?pwd=SXR4aHFlT000QkZWaFE3cEk5YmNlUT09 Speaker(s): Monica Lewis (UM)