The intersection homology D--module in finite characteristic

TL;DR

This paper constructs a unique simple D-module in finite characteristic local cohomology, linking it to tight closure theory and providing criteria for D-simplicity based on Frobenius nilpotency.

Contribution

It introduces a finite characteristic analogue of the intersection homology D-module, constructing it via tight closure and establishing a D-simplicity criterion.

Findings

Existence of a unique simple D_R--submodule in local cohomology.

D-simplicity characterized by Frobenius nilpotency of tight closure.

Parameter test module commutes with completion.

Abstract

Let R be a regular, local and F-finite ring defined over a field of finite characteristic. Let I be an ideal of height c with normal quotient $A=R/I$. It is shown that the local cohomology module H^c_I(R) contains a unique simple D_R--submodule L(A,R). This should be viewed as a finite characteristic analog of the Kashiwara--Brylinski D_R--module in characteristic zero which corresponds to the intersection cohomology complex via the Riemann--Hilbert correspondence. Besides the existence of L(A,R), more importantly, we give its construction as a certain dual of the tight closure of zero in $H^d_m(A)$. We obtain a precise D_R--simplicity criterion for H^c_I(R), namely H^c_I(R) is D_R--simple if and only if the tight closure of zero in H^d_m(A) is Frobenius nilpotent, in particular this is the case if A is F--rational. Furthermore, the techniques developed imply a result in tight closure theory, saying that the parameter test module commutes with completion.