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$D$-module and $F$-module length of local cohomology modules | | Mordechai Katzman
; Linquan Ma
; Ilya Smirnov
; Wenliang Zhang
; | | Date: |
6 Sep 2016 | | 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, in characteristic $p$, the length of $H_I^c(R)$ with $c=ht I$ in the
category of $F$-modules. We produce a general upper bound for $H^j_I(R)$ in the
category of $D$-modules when $R$ is a polynomial ring. In characteristic $p>0$,
if the dimension of the non-$F$-rational locus of $R/I$ is small, we also
obtain upper bounds on the $D$-module length in terms of the dimensions of the
Frobenius stable parts of certain local cohomology modules of $R/I$ and its
localizations. Our bounds are sharp in many cases, for example, when $R/I$ has
an isolated singularity. When $R/I$ is $F$-pure, we also obtain sharp lower
bounds on the $F$-module length of $H_I^j(R)$ in terms of the number of special
primes of $H_m^{n-j}(R/I)$, and when $R/I$ is Cohen-Macaulay we can explicitly
write down an $F$-module filtration of $H_I^c(R)$ that is maximal when $R/I$ is
Gorenstein. We compute the Frobenius stable part of the top local cohomology
module of the Fermat hypersurface $k[x_0,x_1,dots,x_d]/(x_0^n+x_1^n+cdots
+x_d^n)$ explicitly in terms of the number of solutions to a system of
equations on remainders. We also construct an example of a local cohomology
module of $R$ such that, with its natural structure, its $D$-module length is
strictly greater than its $F$-module length. | | Source: | arXiv, 1609.1643 | | Services: | Forum | Review | PDF | Favorites | |
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