Let A be a finite-dimensional local algebra over an algebraically closed field,let J be the radical of A.The modules we are interested in are the finitely generated left A-modules.Projectivemodules are always reflexiv...Let A be a finite-dimensional local algebra over an algebraically closed field,let J be the radical of A.The modules we are interested in are the finitely generated left A-modules.Projectivemodules are always reflexive,and an algebra is self-injective iff allmodules are reflexive.We discuss the existence of non-projective reflexive modules in case A is not self-injective.We assume that A is short(this means that J^(3)=0).In a joint paper with Zhang Pu,it has been shown that 6 is the smallest possible dimension of A that can occur and that in this case the following conditions have to be satisfied:J^(2)is both the left socle and the right socle of A and there is no uniform ideal of length 3.The present paper is devoted to showing the converse.展开更多
Let R be a commutative Noetherian ring, I and J be two ideals of R, and M be an R-module. We study the cofiniteness and finiteness of the local cohomology module HiI,J（M） and give some conditions for the finiteness ...Let R be a commutative Noetherian ring, I and J be two ideals of R, and M be an R-module. We study the cofiniteness and finiteness of the local cohomology module HiI,J（M） and give some conditions for the finiteness of HomR（R/I, HsI,J（M）） and Ext1R（R/I, HsI,J（M））. Also, we get some results on the attached primes of HdimMI,J （M）.展开更多
Let I, J be ideals of a commutative Noetherian local ring （R, m） and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that H_I（M） is not Artinian. In this paper we show th...Let I, J be ideals of a commutative Noetherian local ring （R, m） and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that H_I（M） is not Artinian. In this paper we show that inf{f-depth（a, M） 丨a ∈ W（I, J）} is the least integer such that the local cohomology module with respect to a pair of ideals I, J is not Artinian. As a consequence, it follows that H_I,J（M） is （I, J）-cofinite for all i 〈 inf{f-depth（a, M） 丨a ∈ W（I, J）}. In addition, we show that for a Serre subcategory S, if H_I,J（M） belongs to S for all i 〉 n and if b is an ideal of R such that H^n_I,J（M/bM） belongs to S, then the module H^n_I,J（M）/bH^n_I,J（M） belongs to S.展开更多
Let (R, m) be a relative Cohen–Macaulay local ring with respect to an ideal a of R and set c to be ht a. We investigate some properties of the Matlis dual of the R-module H^ca(R), and we show that such modules behave...Let (R, m) be a relative Cohen–Macaulay local ring with respect to an ideal a of R and set c to be ht a. We investigate some properties of the Matlis dual of the R-module H^ca(R), and we show that such modules behave like canonical modules over Cohen–Macaulay local rings. Moreover, we provide some duality and equivalence results with respect to the module H^ca(R)^∨, and these results lead us to achieve generalizations of some known results, such as the local duality theorem, which have been provided over a Cohen–Macaulay local ring admiting a canonical R-module.展开更多
In this paper, the module coradical is introduced. From this, we show that any Hopf module coalgebra which is local finite can be uniquely decomposed into a direct sum of its indecomposable components. This result is ...In this paper, the module coradical is introduced. From this, we show that any Hopf module coalgebra which is local finite can be uniquely decomposed into a direct sum of its indecomposable components. This result is a generalization of the decompostion theorem of coalgebra.展开更多
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文摘Let A be a finite-dimensional local algebra over an algebraically closed field,let J be the radical of A.The modules we are interested in are the finitely generated left A-modules.Projectivemodules are always reflexive,and an algebra is self-injective iff allmodules are reflexive.We discuss the existence of non-projective reflexive modules in case A is not self-injective.We assume that A is short(this means that J^(3)=0).In a joint paper with Zhang Pu,it has been shown that 6 is the smallest possible dimension of A that can occur and that in this case the following conditions have to be satisfied:J^(2)is both the left socle and the right socle of A and there is no uniform ideal of length 3.The present paper is devoted to showing the converse.
基金The NSF(BK2011276) of Jiangsu Provincethe NSF(10KJB110007,11KJB110011) for Colleges and Universities in Jiangsu Provincethe Research Foundation(Q3107803) of Pre-research Project of Soochow University
文摘Let R be a commutative Noetherian ring, I and J be two ideals of R, and M be an R-module. We study the cofiniteness and finiteness of the local cohomology module HiI,J（M） and give some conditions for the finiteness of HomR（R/I, HsI,J（M）） and Ext1R（R/I, HsI,J（M））. Also, we get some results on the attached primes of HdimMI,J （M）.
文摘Let I, J be ideals of a commutative Noetherian local ring （R, m） and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that H_I（M） is not Artinian. In this paper we show that inf{f-depth（a, M） 丨a ∈ W（I, J）} is the least integer such that the local cohomology module with respect to a pair of ideals I, J is not Artinian. As a consequence, it follows that H_I,J（M） is （I, J）-cofinite for all i 〈 inf{f-depth（a, M） 丨a ∈ W（I, J）}. In addition, we show that for a Serre subcategory S, if H_I,J（M） belongs to S for all i 〉 n and if b is an ideal of R such that H^n_I,J（M/bM） belongs to S, then the module H^n_I,J（M）/bH^n_I,J（M） belongs to S.
文摘Let (R, m) be a relative Cohen–Macaulay local ring with respect to an ideal a of R and set c to be ht a. We investigate some properties of the Matlis dual of the R-module H^ca(R), and we show that such modules behave like canonical modules over Cohen–Macaulay local rings. Moreover, we provide some duality and equivalence results with respect to the module H^ca(R)^∨, and these results lead us to achieve generalizations of some known results, such as the local duality theorem, which have been provided over a Cohen–Macaulay local ring admiting a canonical R-module.
文摘In this paper, the module coradical is introduced. From this, we show that any Hopf module coalgebra which is local finite can be uniquely decomposed into a direct sum of its indecomposable components. This result is a generalization of the decompostion theorem of coalgebra.
