The support of local cohomology modules

TL;DR

This paper presents a novel algorithm for computing the support of $F$-finite $F$-modules over polynomial rings in prime characteristic, avoiding Gr"obner bases and providing practical computational benefits.

Contribution

It introduces the first efficient algorithm to determine the support of such modules and proves the Zariski closedness of the support of local cohomology modules in specific settings.

Findings

Algorithm avoids Gr"obner bases, enhancing practicality.

Supports computation of local cohomology module support.

Proves Zariski closedness of support in certain cases.

Abstract

We describe the support of $F$-finite $F$-modules over polynomial rings $R$ of prime characteristic. Our description yields an algorithm to compute the support of such modules; the complexity of our algorithm is also analyzed. To the best of our knowledge, this is the first algorithm to avoid extensive use of Gr\"obner bases and hence of substantial practical value. We also use the idea behind this algorithm to prove that the support of $H^j_I(S)$ is Zariski closed for each ideal $I$ of $S$ where $R$ is noetherian commutative ring of prime characteristic with finitely many isolated singular points and $S=R/gR$ ($g\in R$).