بُعدهاي انژكتيو گورنشتاين و كوهن - مكالي بودن

Gorenstein Injective Dimension and Cohen-Macaulayness

فرض كنيم (R,m) حلقه‌اي موضعي، نوتري و جا به ‌جايي باشد. در اين مقاله وجود مدول‌هاي متناهي مولد از بُعد انژكتيو گورنشتاين متناهي روي حلقه‌هاي كوهن-مكالي را بررسي مي‌كنيم. در ابتدا بُعدهاي انژكتيو گورنشتاين كوهمولوژي موضعي هم‌ بافت‌ها را بررسي مي‌كنيم.

Throughout this paper, (R, m) is a commutative Noetherian local ring with the maximal ideal m. The following conjecture proposed by Bass [1], has been proved by Peskin and Szpiro [2] for almost all rings: (B) If R admits a finitely generated R-module of finite injective dimension, then R is Cohen-Macaulay. The problems treated in this paper are closely related to the following generalization of Bass conjecture which is still wide open: (GB) If R admits a finitely generated R-module of finite Gorenstein-injective dimension, then R is Cohen-Macaulay. Our idea goes back to the first steps of the solution of Bass conjecture given by Levin and Vasconcelos in 1968 [3] when R admits a finitely generated R-module of injective dimension 1. Levin and Vasconcelos indicate that if is a non-zerodivisor, then for every finitely generated R/xR-module M, there is . Using this fact, they construct a finitely generated R-module of finite injective dimension in the case where R is Cohen-Macaulay (the converse of Conjecture B). In this paper we study the Gorenstein injective dimension of local cohomology. We also show that if R is Cohen-Macaulay with minimal multiplicity, then every finitely generated module of finite Gorenstein injective dimension has finite injective dimension. We prove that a Cohen-Macaulay local ring has a finitely generated module of finite Gorenstein injective dimension.