Cominimax modules and generalized local cohomology modules

The local cohomology theory plays an important role in commutative algebra and algebraic geometry. The I-cofiniteness of local cohomology modules is one of interesting properties which has been studied by many mathematicians. The I-cominimax modules is an extension of I-cofinite modules which was introduced by Hartshorne.
Nội dung trích xuất từ tài liệu:
Cominimax modules and generalized local cohomology modulesScience & Technology Development Journal, 23(1):479-483 Open Access Full Text Article Research ArticleCominimax modules and generalized local cohomology modulesBui Thi Hong Cam, Nguyen Minh Tri* ABSTRACT The local cohomology theory plays an important role in commutative algebra and algebraic ge- ometry. The I-cofiniteness of local cohomology modules is one of interesting properties whichUse your smartphone to scan this has been studied by many mathematicians. The I-cominimax modules is an extension of I-cofiniteQR code and download this article modules which was introduced by Hartshorne. An R -module M is I-cominimax if SuppR M ⊆ V (I) and ExtiR (R/I, M) is minimax for all i ≥ 0. The aim of this paper is to show some conditions such that the generalized local cohomology module HI′ (M, N) is I-cominimax for all i ≥ 0. We prove that HIi (M, K) if is I-cofinite for all i ≥ 0. We prove that if HIi (M, K) is I-cofinite for all i < t and all finitely generated R-module K, then HIi (M, N) is I-cominimax for all i < t and all minimax R-module N. If M is a finitely generated R-module, N is a minimax R-module and t is a non-negative integer such that dim SuppR HIi (M, N) ≤ 1 for all i < t , then HIi (M, N) is I-cominimax for all i < t . When dim R/I ≤ 1 and HIi (N) is I-cominimax for all i ≥ 0, then HI′ (M, N) is I-cominimax for all i ≥ 0. Key words: Generalized local cohomology, I-cominimax INTRODUCTION j-th generalized local cohomology module of M and N with respect to I is defined by Let R be a local Noetherian ring, I an ideal of R and M a finitely generated R -module. It is well known that ( ) HI (M, N) ∼ j j = lim ExtR (M/I n M, N) the local cohomology modules HIi (M) are not gen- ⃗n erally finitely generated for i > 0. In a 1970 paper j j We see that if M = R, then HI (M, N) = HI (N) the Hartshorne 1 gave the concept of I-cofinite modules. usual local cohomology module of Grothendieck 8 . An R-module K to be I-cofinite if SuppR K ⊆ V (I) j and ExtR (R/I, K) is finitely generated for all j ≥ 0. Another similar question is: When is the module jDepartment of Natural Science Hartshorne asked which rings R and ideals I the mod- HI (M, N)I-cofinite for all j ≥ 0?Education, Dong Nai University, DongNai, Vietnam ules HIi (M) were I-cofinite for all i and all finitely gen- In 2001, Yassemi [ 9 , Theorem 2.8] showed that in a j erated modules M. Gorenstein ring, HI (M, N) is I -cofinite for all j ≥ 0Correspondence where I is non-zero principal ideal. In 2004, Divaani- In 1, if (R , m) is a complete regular local ring andNguyen Minh Tri, Department of Natural M is a finitely generated R-module, then HIi (M) is I Aazar and Sazeedeh [ 10 , Theorem 2.8 and TheoremScience Education, Dong Nai University,Dong Nai, Vietnam -cofinite in two cases: 2.9] have eliminated the Gorenstein hypothesis andEmail: nguyenminhtri@dnpu.edu.vn showed that if either • I is a nonzero principal ideal, orHistory • I is a prime ideal with dim R/I = 1 1. I is principal, or• Received: 2019-07-11• Accepted: 2020-02-11 ...