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arXiv · 1609.01643

$D$-module and $F$-module length of local cohomology modules

Mordechai Katzman·Linquan Ma·Ilya Smirnov·Wenliang Zhang
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Abstract

Let $R$ be a polynomial or power series ring over a field $k$. We study the length of local cohomology modules $H^j_I(R)$ in the category of $D$-modules and $F$-modules. We show that the $D$-module length of $H^j_I(R)$ is bounded by a polynomial in the degree of the generators of $I$. In characteristic $p>0$ we obtain upper and lower bounds on the $F$-module length in terms of the dimensions of Frobenius stable parts and the number of special primes of local cohomology modules of $R/I$. The obtained upper bound is sharp if $R/I$ is an isolated singularity, and the lower bound is sharp when $R/I$ is Gorenstein and $F$-pure. We also give an example of a local cohomology module that has different $D$-module and $F$-module lengths.

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Mordechai Katzman, Linquan Ma, Ilya Smirnov, Wenliang Zhang. 2016-09-06. $D$-module and $F$-module length of local cohomology modules. https://arxiv.org/abs/1609.01643

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