بُعدهای انژکتیو گورنشتاین و کوهن-مکالی بودن

فرض کنیم (r,m) حلقه‌ای موضعی، نوتری و جابه‌جایی باشد. در این مقاله وجود مدول‌های متناهی مولد از بُعد انژکتیو گورنشتاین متناهی روی حلقه‌های کوهنمکالی را بررسی می‌کنیم. در ابتدا بُعدهای انژکتیو گورنشتاین کوهمولوژی موضعی هم‌بافت‌ها را بررسی می‌کنیم. 

Gorenstein Injective Dimension and Cohen-Macaulayness

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 Rmodule of‎ finite injective dimension‎, ‎then R is CohenMacaulay.‎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 Rmodule of‎ finite Gorensteininjective dimension‎, ‎then R is‎ CohenMacaulay.‎  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 Rmodule of injective dimension 1‎.Levin and Vasconcelos indicate that if‎ is a nonzerodivisor‎, ‎then for‎ every finitely generated R/xRmodule M‎, ‎there is‎ ‎. ‎Using this fact‎, ‎they construct a‎ finitely generated Rmodule of finite injective dimension in‎ the case where R is CohenMacaulay (the converse of Conjecture‎ B)‎. ‎ In this paper we study the Gorenstein injective dimension of local cohomology‎. ‎We also show that if R is CohenMacaulay‎ with minimal multiplicity‎, ‎then every finitely generated module‎ of finite Gorenstein injective dimension has finite injective‎ dimension.‎‎We prove that a CohenMacaulay local ring‎ has a finitely generated module of‎ finite Gorenstein injective dimension.‎./files/site1/files/52/4.pdf