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

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 such that the completion of the quotient A = R/I is a domain. It is shown that the local cohomology module HcI (R) contains a unique simple DR-submodule L (A, R). This should be viewed as an analog of the Kashiwara-Brylinski DR-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 a concrete construction as a certain dual of the tight closure of zero in Hdm (A). In order to prove this result the theory of R[ Finfinity]-modules and techniques from the theory of tight closure play a crucial role. A key result tying R( Finfinity]-modules to DR-modules is that if k is an uncountable algebraically closed field then a finitely generated simple unit R[F infinity] is simple as a DR-module. The given construction of L (A, R) yields a precise DR-simplicity criterion for HcI (R), namely HcI (R) is DR-simple if and only if the tight closure of zero in Hdm (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.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/127222/2/3029301.pd