In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applicati...In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applications are given.展开更多
We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the 'resolution theorem' in this ...We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the 'resolution theorem' in this context due to Quillen. We characterize the Gorenstein algebraic K-groups by two different Mgebraic K-groups and by the idempotent completeness of the Gorenstein singularity category of the ring. We compute the Gorenstein algebraic K-groups along a recollement of the bounded Gorenstein derived categories of CM-finite Gorenstein algebras.展开更多
We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C^m is Gorenstein projective in R-Mod for all m E Z. Basing on this we show that if R is a ...We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C^m is Gorenstein projective in R-Mod for all m E Z. Basing on this we show that if R is a right coherent left perfect ring then Gpd(C) = sup{Gpd(C^m)|m ∈ Z} where Gpd(-) denotes Gorenstein projective dimension.展开更多
As we know,a complex P is projective if and only if P is exact with Z_n(P)projective in R-Mod for each n∈Z and any morphism f:P→C is null homotopic for any complex C.In this article,we study the notion of DG-Gorenst...As we know,a complex P is projective if and only if P is exact with Z_n(P)projective in R-Mod for each n∈Z and any morphism f:P→C is null homotopic for any complex C.In this article,we study the notion of DG-Gorenstein projective complexes.We show that a complex G is DG-Gorenstein projective if and only if G is exact with Z_n(G)Gorenstein projective in R-Mod for each n∈Z and any morphism f:G→Q is null homotopic whenever Q is a DG-projective complex.展开更多
Let Λ be an Artin algebra and let Gprj-Λ denote the class of all the finitely generated Gorenstein projective Λ-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain sub...Let Λ be an Artin algebra and let Gprj-Λ denote the class of all the finitely generated Gorenstein projective Λ-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category S(Gprj-Λ) containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category Gprj. In particular, for the finite components, we show that under certain mild conditions,their cardinalities are divisible by 3. We see that this three-periodicity phenomenon reoccurs several times in the paper.展开更多
In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each Xm is a Ding projective R-mod...In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each Xm is a Ding projective R-module for all m ∈ Z.展开更多
For a given class of modules A,let A be the class of exact complexes having all cycles in A,and dw(A)the class of complexes with all components in A.Denote by GL the class of Gorenstein injective modules.We prove that...For a given class of modules A,let A be the class of exact complexes having all cycles in A,and dw(A)the class of complexes with all components in A.Denote by GL the class of Gorenstein injective modules.We prove that the following are equivalent over any ring R:every exact complex of injective modules is totally acyclic;every exact complex of Gorenstein injective modules is in every complex in dw(GL)is dg-Gorenstein injective.The analogous result for complexes of flat and Gorenstein flat modules also holds over arb计rary rings.If the ring is n-perfect for some integer n≥0,the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules.We also prove the following characterization of Gorenstein rings.Let R be a commutative coherent ring;then the following are equivalent:(1)every exact complex of FP-injective modules has all its cycles Ding injective modules;(2)every exact complex of flat modules is F-totally acyclic,and every R-modulc M such that M^(+)is Gorenstein flat is Ding injective;(3)every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that is Gorenstein flat is Ding injective.If R has finite Krull dimension,statements(1)-(3)are equivalent to(4)R is a Gorenstein ring(in the sense of Iwanaga).展开更多
Let R be a domain.In this paper,we show that if R is one dimensional,then R is a Noetherian Warfield domain if and only if every maximal ideal of R is 2-generated and for every maximal ideal M of R,M is divisorial in ...Let R be a domain.In this paper,we show that if R is one dimensional,then R is a Noetherian Warfield domain if and only if every maximal ideal of R is 2-generated and for every maximal ideal M of R,M is divisorial in the ring(M:M).We also prove that a Noetherian domain R is a Noetherian Warfield domain if and only if for every maximal ideal M of R,M^(2) can be generated by two elements.Finally,we give a sufficient condition under which all ideals of R are strongly Gorenstein projective.展开更多
基金Supported by the National Natural Science Foundation of China(11361051) Supported by the Program for New Century Excellent the Talents in University(NCET-13-0957)
文摘In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applications are given.
文摘We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the 'resolution theorem' in this context due to Quillen. We characterize the Gorenstein algebraic K-groups by two different Mgebraic K-groups and by the idempotent completeness of the Gorenstein singularity category of the ring. We compute the Gorenstein algebraic K-groups along a recollement of the bounded Gorenstein derived categories of CM-finite Gorenstein algebras.
基金Supported by National' Natural Science Foundation of China (Grant No. 10961021), TRAPOYT and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China
文摘We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C^m is Gorenstein projective in R-Mod for all m E Z. Basing on this we show that if R is a right coherent left perfect ring then Gpd(C) = sup{Gpd(C^m)|m ∈ Z} where Gpd(-) denotes Gorenstein projective dimension.
基金Supported by the National Natural Science Foundation of China(2061061)Fundamental Research Funds for the Central Universities(31920190054)+1 种基金Funds for Talent Introduction of Northwest Minzu University(XBMUYJRC201406)First-Rate Discipline of Northwest Minzu University。
文摘As we know,a complex P is projective if and only if P is exact with Z_n(P)projective in R-Mod for each n∈Z and any morphism f:P→C is null homotopic for any complex C.In this article,we study the notion of DG-Gorenstein projective complexes.We show that a complex G is DG-Gorenstein projective if and only if G is exact with Z_n(G)Gorenstein projective in R-Mod for each n∈Z and any morphism f:G→Q is null homotopic whenever Q is a DG-projective complex.
基金supported by National Natural Science Foundation of China (Grant No. 12101316)。
文摘Let Λ be an Artin algebra and let Gprj-Λ denote the class of all the finitely generated Gorenstein projective Λ-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category S(Gprj-Λ) containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category Gprj. In particular, for the finite components, we show that under certain mild conditions,their cardinalities are divisible by 3. We see that this three-periodicity phenomenon reoccurs several times in the paper.
基金Supported by National Natural Science Foundation of China(Grant Nos.11561039 and 11761045)Natural Science Foundation of Gansu Province of China(Grant No.17JR5RA091)
文摘In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each Xm is a Ding projective R-module for all m ∈ Z.
基金S.Estrada was partly supported by grant MTM2016-77445-PFEDER funds and by grant 19880/GERM/15 from the Fundacion Seneca-Agencia de Ciencia y Tecnologfa de la Region de Murcia.
文摘For a given class of modules A,let A be the class of exact complexes having all cycles in A,and dw(A)the class of complexes with all components in A.Denote by GL the class of Gorenstein injective modules.We prove that the following are equivalent over any ring R:every exact complex of injective modules is totally acyclic;every exact complex of Gorenstein injective modules is in every complex in dw(GL)is dg-Gorenstein injective.The analogous result for complexes of flat and Gorenstein flat modules also holds over arb计rary rings.If the ring is n-perfect for some integer n≥0,the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules.We also prove the following characterization of Gorenstein rings.Let R be a commutative coherent ring;then the following are equivalent:(1)every exact complex of FP-injective modules has all its cycles Ding injective modules;(2)every exact complex of flat modules is F-totally acyclic,and every R-modulc M such that M^(+)is Gorenstein flat is Ding injective;(3)every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that is Gorenstein flat is Ding injective.If R has finite Krull dimension,statements(1)-(3)are equivalent to(4)R is a Gorenstein ring(in the sense of Iwanaga).
基金This work was partially supported by the Department of Mathematics in Kyungpook National University and National Natural Science Foundation of China(Grant No.11671283)The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education,Science and Technology(2017R1C1B1008085),Korea.
文摘Let R be a domain.In this paper,we show that if R is one dimensional,then R is a Noetherian Warfield domain if and only if every maximal ideal of R is 2-generated and for every maximal ideal M of R,M is divisorial in the ring(M:M).We also prove that a Noetherian domain R is a Noetherian Warfield domain if and only if for every maximal ideal M of R,M^(2) can be generated by two elements.Finally,we give a sufficient condition under which all ideals of R are strongly Gorenstein projective.