Let R be a commutative noetherian local ring. In this paper, we study Gorenstein projective, injective and flat modules with respect to a semidualizing R-module C, and we give some connections between C-Gorenstein hom...Let R be a commutative noetherian local ring. In this paper, we study Gorenstein projective, injective and flat modules with respect to a semidualizing R-module C, and we give some connections between C-Gorenstein homological dimensions and the Auslander categories of R.展开更多
We investigate stable homology of modules over a commutative noetherian ring R with respect to a semidualzing module C, and give some vanishing results that improve/extend the known results. As a consequence, we show ...We investigate stable homology of modules over a commutative noetherian ring R with respect to a semidualzing module C, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of the theory forces C to be trivial and R to be Gorenstein.展开更多
Let C be a semidualizing module for a commutative ring R. It is shown that the ：IC-injective dimension has the ability to detect the regularity of R as well as the Pc-projective dimension. It is proved that if D is d...Let C be a semidualizing module for a commutative ring R. It is shown that the ：IC-injective dimension has the ability to detect the regularity of R as well as the Pc-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that idR（D） = n 〈 ∞, then ：ID-idR（F） ≤ n for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then/TC-idR（M） ≤ IC-idR（N）. This generalizes the result of Enochs and Holm.展开更多
In this paper, we introduce and study GC-flat complexes over a commutative Noetherian ring, where C is a semidualizing module. We prove that Ge-flat complexes are actually the complexes of Go-flat modules. This comple...In this paper, we introduce and study GC-flat complexes over a commutative Noetherian ring, where C is a semidualizing module. We prove that Ge-flat complexes are actually the complexes of Go-flat modules. This complements a result of Yang and Liang. As an application, we get that every complex has a GF-C（C）-cover, where GFC（C） is the class of Ge-flat complexes. We also give a characterization of complexes of modules in HC（FC） that are defined by Sather-Wagstaff, Sharif and White.展开更多
Let A be an abelian category,T a self-orthogonal subcategory of A and each object in T admit finite projective and injective dimensions.If the left Gorenstein subcategory lG(T)equals to the right orthogonal class of T...Let A be an abelian category,T a self-orthogonal subcategory of A and each object in T admit finite projective and injective dimensions.If the left Gorenstein subcategory lG(T)equals to the right orthogonal class of T and the right Gorenstein subcategory rG(T)equals to the left orthogonal class of T,we prove that the Gorenstein subcategory G(T)equals to the intersection of the left orthogonal class of T and the right orthogonal class of T,and prove that their stable categories are triangle equivalent to the relative singularity category of A with respect to T.As applications,let R be a left Noetherian ring with finite left self-injective dimension and _(R)C_(S) a semidualizing bimodule,and let the supremum of the flat dimensions of all injective left R-modules be finite.We prove that if RC has finite injective(or flat)dimension and the right orthogonal class of C contains R,then there exists a triangle-equivalence between the intersection of C-Gorenstein projective modules and Bass class with respect to C,and the relative singularity category with respect to C-projective modules.Some classical results are generalized.展开更多
Let R be a commutative ring and C a semidualizing R-module. We introduce the notion of Dc-projective dimension for homologically bounded below complexes and give some characterizations of this dimension.
Let_(R)C_(S) be a semidualizing(R,S)-bimodule.Then_(R)C_(S) induces an equivalent between the Auslander class A_(C)(S)and the Bass class B_C(R).Let A and B be free normalizing extensions of R and S respectively.In thi...Let_(R)C_(S) be a semidualizing(R,S)-bimodule.Then_(R)C_(S) induces an equivalent between the Auslander class A_(C)(S)and the Bass class B_C(R).Let A and B be free normalizing extensions of R and S respectively.In this paper,we prove that Hom S(_(B)B_(S),_(R)C_(S))is a semidualizing(A,B)-bimodule under some suitable conditions,and so Hom S(_(B)B_(S),_(R)C_(S))induces an equivalence between the Auslander class AHomS (_(B)B_(S),_(R)C_(S))(B). and the Bass class BHomS (BBS,RCS)(A) Furthermore,under a suitable condition on_(R)C_(S),we develop a generalized Morita theory for Auslander categories.展开更多
In this paper we are concerned with absolute,relative and Tate Tor modules.In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory,and o...In this paper we are concerned with absolute,relative and Tate Tor modules.In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory,and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules.In the second part of the paper we consider a depth equality,called the depth formula,which has been initially introduced by Auslander and developed further by Huneke and Wiegand.As an application of our main result,we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension.展开更多
As the dual of the Auslander transpose and the resulting k-torsionfree module,the cotranspose and k-cotorsionfree module with respect to a semidualizing bimodule have been introduced recently.In this paper we first in...As the dual of the Auslander transpose and the resulting k-torsionfree module,the cotranspose and k-cotorsionfree module with respect to a semidualizing bimodule have been introduced recently.In this paper we first investigate the relation between relative k-cotorsionfree modules and relative k-cosyzygy modules.Then we study the extension closure of these two classes of modules.展开更多
The authors introduce and investigate the Tc-Gorenstein projective, Lc- Gorenstein injective and Hc-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenste...The authors introduce and investigate the Tc-Gorenstein projective, Lc- Gorenstein injective and Hc-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the Tc-Gorenstein projective or the Hc-Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the Lc-Gorenstein ＇injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the Tc（R）-projective and Lc（R）-injective dimensions and Tc（R）-precovers and Lc（R）-preenvelopes. Fiually, the authors study the Hc-Gorenstein flat modules and extend the Foxby equivalences.展开更多
Let R and S be associative rings and sVR a semidualizing （S-R）-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a HomR（Zv（R）,-） and HomR（-,Zv（R）） exact exact complex . of V-inject...Let R and S be associative rings and sVR a semidualizing （S-R）-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a HomR（Zv（R）,-） and HomR（-,Zv（R）） exact exact complex . of V-injective modules Ii and Ii,i ∈ N0, such that N We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class ,4v（R） which leads to the fact that V-Gorenstein injective modules admit exact right Iv （R）-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V- Gorenstein injective if and only if N @ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Corenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if Ext Iv （R） （I, N） = 0 for all modules I with finite Iv （R）-injective dimension.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11261050)
文摘Let R be a commutative noetherian local ring. In this paper, we study Gorenstein projective, injective and flat modules with respect to a semidualizing R-module C, and we give some connections between C-Gorenstein homological dimensions and the Auslander categories of R.
文摘We investigate stable homology of modules over a commutative noetherian ring R with respect to a semidualzing module C, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of the theory forces C to be trivial and R to be Gorenstein.
文摘Let C be a semidualizing module for a commutative ring R. It is shown that the ：IC-injective dimension has the ability to detect the regularity of R as well as the Pc-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that idR（D） = n 〈 ∞, then ：ID-idR（F） ≤ n for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then/TC-idR（M） ≤ IC-idR（N）. This generalizes the result of Enochs and Holm.
基金Partially supported by the National Natural Science Foundation of China （No. 11301240）, and the Young Scholars Science Foundation of Lanzhou Jiaotong University （No. 2012020）.
文摘In this paper, we introduce and study GC-flat complexes over a commutative Noetherian ring, where C is a semidualizing module. We prove that Ge-flat complexes are actually the complexes of Go-flat modules. This complements a result of Yang and Liang. As an application, we get that every complex has a GF-C（C）-cover, where GFC（C） is the class of Ge-flat complexes. We also give a characterization of complexes of modules in HC（FC） that are defined by Sather-Wagstaff, Sharif and White.
基金Supported by the Project of Natural Science Foundation of Changzhou College of Information Technology(Grant No.CXZK202204Y)the Project of Youth Innovation Team of Universities of Shandong Province(Grant No.2022KJ314)。
文摘Let A be an abelian category,T a self-orthogonal subcategory of A and each object in T admit finite projective and injective dimensions.If the left Gorenstein subcategory lG(T)equals to the right orthogonal class of T and the right Gorenstein subcategory rG(T)equals to the left orthogonal class of T,we prove that the Gorenstein subcategory G(T)equals to the intersection of the left orthogonal class of T and the right orthogonal class of T,and prove that their stable categories are triangle equivalent to the relative singularity category of A with respect to T.As applications,let R be a left Noetherian ring with finite left self-injective dimension and _(R)C_(S) a semidualizing bimodule,and let the supremum of the flat dimensions of all injective left R-modules be finite.We prove that if RC has finite injective(or flat)dimension and the right orthogonal class of C contains R,then there exists a triangle-equivalence between the intersection of C-Gorenstein projective modules and Bass class with respect to C,and the relative singularity category with respect to C-projective modules.Some classical results are generalized.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1136105211401476)
文摘Let R be a commutative ring and C a semidualizing R-module. We introduce the notion of Dc-projective dimension for homologically bounded below complexes and give some characterizations of this dimension.
基金Supported by the Natural Science Foundation of Anhui Province(Grant No.2008085QA03)。
文摘Let_(R)C_(S) be a semidualizing(R,S)-bimodule.Then_(R)C_(S) induces an equivalent between the Auslander class A_(C)(S)and the Bass class B_C(R).Let A and B be free normalizing extensions of R and S respectively.In this paper,we prove that Hom S(_(B)B_(S),_(R)C_(S))is a semidualizing(A,B)-bimodule under some suitable conditions,and so Hom S(_(B)B_(S),_(R)C_(S))induces an equivalence between the Auslander class AHomS (_(B)B_(S),_(R)C_(S))(B). and the Bass class BHomS (BBS,RCS)(A) Furthermore,under a suitable condition on_(R)C_(S),we develop a generalized Morita theory for Auslander categories.
基金partly supported by the National Natural Science Foundation of China(Grant Nos.12271230,11761045 and 11971388)the Natural Science Foundation of Gansu Province(Grant No.21JR7RA297)。
文摘In this paper we are concerned with absolute,relative and Tate Tor modules.In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory,and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules.In the second part of the paper we consider a depth equality,called the depth formula,which has been initially introduced by Auslander and developed further by Huneke and Wiegand.As an application of our main result,we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension.
基金This research was partially supported by NSFC(11401147,11531002)Natural Science Foundation of Shandong Province of China(ZR2017BA028).
文摘As the dual of the Auslander transpose and the resulting k-torsionfree module,the cotranspose and k-cotorsionfree module with respect to a semidualizing bimodule have been introduced recently.In this paper we first investigate the relation between relative k-cotorsionfree modules and relative k-cosyzygy modules.Then we study the extension closure of these two classes of modules.
基金Project supported by the National Natural Science Foundation of China(No.10971090)
文摘The authors introduce and investigate the Tc-Gorenstein projective, Lc- Gorenstein injective and Hc-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the Tc-Gorenstein projective or the Hc-Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the Lc-Gorenstein ＇injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the Tc（R）-projective and Lc（R）-injective dimensions and Tc（R）-precovers and Lc（R）-preenvelopes. Fiually, the authors study the Hc-Gorenstein flat modules and extend the Foxby equivalences.
文摘Let R and S be associative rings and sVR a semidualizing （S-R）-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a HomR（Zv（R）,-） and HomR（-,Zv（R）） exact exact complex . of V-injective modules Ii and Ii,i ∈ N0, such that N We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class ,4v（R） which leads to the fact that V-Gorenstein injective modules admit exact right Iv （R）-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V- Gorenstein injective if and only if N @ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Corenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if Ext Iv （R） （I, N） = 0 for all modules I with finite Iv （R）-injective dimension.
