Results on Generalization of Burch’s Inequality and the Depth of Rees Algebra and Associated Graded Rings of an Ideal with Respect to a Cohen-Macaulay Module

Let  be a local Cohen-Macaulay ring with infinite residue field,  an Cohen - Macaulay module and  an ideal of  Consider  and , respectively, the Rees Algebra and associated graded ring of , and denote by  the analytic spread of  Burch’s inequality says that  and equality holds if  is Cohen-Macaulay. Thus, in that case one can compute the depth of associated graded ring of  as  In this paper we extend results in case of rings and ideals to the case of modules and we show that for associated graded module of  with respect to i.e, , such an equality is also valid when  is not necessarily Cohen-Macaulay, and we extend Burch’s inequality to modules. Also, we compute the Rees Algebra and associated graded ring of generically complete intersection of an ideal with respect to module  in local Cohen - Macaulay ring and we obtain positive results for ideals with analytic deviation less or equal than one and reduction number at most two with respect to module