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 investigate the maximal Cohen-Macaulay property of tensor products modules, and then give criteria for projectivity modules in terms vanishing Ext modules. One applications shows that Auslander-Reiten conjecture holds normal rings.

Let [Formula: see text] where be a complete regular local ring of dimension text], for some and an MCM text]-module with then we prove that text]. If is hypersurface infinite residue field let or Our paper the first systematic study depth associated graded modules over rings.

In this survey paper we first present the main properties of sequentially Cohen-Macaulay modules. Some basic examples are provided to help reader with quickly getting acquainted topic. We then discuss two generalizations notion sequential Cohen-Macaulayness which inspired by a theorem J\"urgen Herzog and third author.

We give the functorial moduli construction of pure parabolic sheaves in sense Álvarez-Cónsul and A. King by using filtered Kronecker modules which we introduced our earlier work. also use a version S. G. Langton’s result to deduce projectivity maximal dimension. As an application construction, can get morphisms at level stacks.