In this paper we will generalize  some of known results on the tight closure of an ideal to the tight closure of an ideal relative to a module .

in this paper we will generalize  some of known results on the tight closure of an ideal to the tight closure of an ideal relative to a module .

We show that for excellent, normal equicharacteristic rings with perfect residue fields the tight closure of an ideal is contained in its axes closure. First we prove this characteristic p. This achieved by using notion special established Huneke and Vraciu. By reduction to positive containment also holds 0. From deduce a ring finite type over C primary inside continuous

We give a survey about some recent work on tight closure and Hilbert-Kunz theory from the viewpoint of vector bundles. This work is based in understanding tight closure in terms of forcing algebras and the cohomological dimension of torsors of syzygy bundles. These geometric methods allowed to answer some fundamental questions of tight closure, in particular the equality between tight closure a...