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

Author:
Katzman, Mordechai   Ma, Linquan   Smirnov, Ilya   Zhang, Wenliang   


Journal:
Transactions of the American Mathematical Society


Issue Date:
2018


Abstract(summary):

Let R be a polynomial or power series ring over a field k. We study the length of local cohomology modules H-I(j) (R) in the category of D-modules and F-modules. We show that the D-module length of H-I(j) (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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