The intersection of homology <italic>D</italic>-module in finite characteristic.

Abstract

Let <italic>R</italic> be a regular, local and <italic>F</italic>-finite ring defined over a field of finite characteristic. Let <italic>I</italic> be an ideal of height <italic>c</italic> such that the completion of the quotient <italic> A</italic> = <italic>R</italic>/<italic>I</italic> is a domain. It is shown that the local cohomology module HcI (<italic>R</italic>) contains a unique simple <italic> D_R</italic>-submodule L (<italic>A, R</italic>). This should be viewed as an analog of the Kashiwara-Brylinski <italic>D_R</italic>-module in characteristic zero which corresponds to the intersection cohomology complex via the Riemann-Hilbert correspondence. Besides the existence of L (<italic>A, R</italic>), more importantly, we give a concrete construction as a certain dual of the tight closure of zero in Hdm (<italic>A</italic>). In order to prove this result the theory of <italic>R</italic>[<italic> F</italic><super>infinity</super>]-modules and techniques from the theory of tight closure play a crucial role. A key result tying <italic>R</italic>(<italic> F</italic><super>infinity</super>]-modules to <italic>D_R</italic>-modules is that if <italic>k</italic> is an uncountable algebraically closed field then a finitely generated simple unit <italic>R</italic>[<italic>F</italic><super> infinity</super>] is simple as a <italic>D_R</italic>-module. The given construction of L (<italic>A, R</italic>) yields a precise <italic>D_R</italic>-simplicity criterion for HcI (<italic>R</italic>), namely HcI (<italic>R</italic>) is <italic>D_R</italic>-simple if and only if the tight closure of zero in Hdm (<italic>A</italic>) is Frobenius nilpotent, in particular this is the case if <italic>A</italic> is <italic>F</italic>-rational. Furthermore, the techniques developed imply a result in tight closure theory, saying that the parameter test module commutes with completion.