Support of local cohomology modules over hypersurfaces and rings with FFRT

Let $(R, m, K)$ be a local ring and $I$ be an ideal. It is conjectured that the local cohomology modules $H^i_I (R)$ have closed support on $\textrm{Spec }R$. We establish this conjecture for hypersurface rings in positive characteristic. In addition, we prove that the associated primes of $H^i_I (R)$ are finite for rings with finite $F$-representation type. As a consequence, $H^i_I (R)$ has finitely many associated primes for a Stanley–Reisner ring, $R$, over any field of positive characteristic. In addition, we present results in a similar direction for certain determinantal rings.

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The first author was partially supported by the NSF Grant 1401384.

The second author was partially supported by the NSF Grant 1502282.

Received 21 February 2016

Accepted 21 November 2016

Published 24 July 2017