Commutative Algebra Seminar
Monica A. Lewis
University of Michigan
The local cohomology of a parameter ideal with respect to an arbitrary ideal
Abstract: Let S be a complete intersection presented as R/J for R a regular ring and J a parameter ideal. Let I be an ideal containing J. It is well known that the set of associated primes of H^i_I(S) can be infinite, but far less is known about the set of minimal primes. In 2017, Hochster and Núñez-Betancourt showed that if R has prime characteristic p > 0, then the finiteness of Ass H^i_I(J) implies the finiteness of Min H^{i-1}_I(S), raising the following question: is Ass H^i_I(J) always finite? We give a positive answer when i=2 but provide a counterexample when i=3. The counterexample crucially requires Ass H^2_I(S) to be infinite. The following question, to the best of our knowledge, is open: (under suitable hypotheses on R) does the finiteness of Ass H^{i-1}_I(S) imply the finiteness of Ass H^i_I(J)? When S is a domain, we give a positive answer when i=3. When S is locally factorial, we extend this to i=4. Finally, if R has prime characteristic p > 0 and S is regular, we give a complete answer by showing that Ass H^i_I(J) is finite for all values of i.
Wednesday October 30, 2019 at 4:00 PM in 1227 SEO