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

Subject Areas : Statistics

محمد Tohidi ¹ , Amir Mafi ^{2 *} , خدیجه Ahmadi ³

1 - Department of Mathematics, Payame Noor University, Tehran, Iran
2 - Department of Math. University of Kurdistan, P.O.Box:416, Sanandaj, Iran
3 - Department of Mathematics, Payame Noor University, Tehran, Iran

Received: 2018-03-04 Accepted : 2018-07-04 Published : 2018-08-01

Keywords: بسط تحلیلی, عدد تقلیل, انحراف تحلیلی, عدد بورخ, مدول‌ مدرج وابسته, همبرش کامل,

Abstract :

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