Morita equivalence was established by Morita in the late 1950’s. In part one most of the recent developments in this theory on the categories of modules over rings were reviewed and some results were presented. Here ...Morita equivalence was established by Morita in the late 1950’s. In part one most of the recent developments in this theory on the categories of modules over rings were reviewed and some results were presented. Here this paper discusses those on categories of comodules over coalgebras, these results are due to Ling, Takeuchi and Wang.展开更多
Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S,R,<sub>S</sub>P<sub>R</sub>,<sub>R</sub>Q<sub>S</s...Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S,R,<sub>S</sub>P<sub>R</sub>,<sub>R</sub>Q<sub>S</sub>,) with and surjective.For a factorisable semigroup S,we denote ζ<sub>S</sub>={(s<sub>1</sub>,s<sub>2</sub>)∈S×S|ss<sub>1</sub>=ss<sub>2</sub>,<sub>S</sub>∈S},S′=S/ζ<sub>S</sub> and US-FAct={<sub>S</sub>M∈ S-Act|SM=M and SHom<sub>S</sub>(S,M)≌M}.We show that,for factorisable semigroups S and R,the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S′ and R′ are strongly Morita equivalent.Some conditions for a factorisable semigroup to be strongly Morita equivalent to a sandwich semigroup,local units semigroup,monoid and group separately are also given.Moreover,we show that a seinigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I,SSHom<sub>S</sub>(S,<sub>i∈I</sub>S)→<sub>i∈I</sub>S,st·f(st)f is an S-isomorphism.展开更多
Morita equivalence was established by Morita in the late 1950's. Here most of the recent developments in this theory are reviewed and some results are presented.
Let H be a Hopf π-coalgebra over a commutative ring k with bijective antipode S, and A and B right π-H-comodulelike algebras. We show that the pair of adjoint functors (F3 = A Bop A□ HBop -,G3 = (-)coH) betwee...Let H be a Hopf π-coalgebra over a commutative ring k with bijective antipode S, and A and B right π-H-comodulelike algebras. We show that the pair of adjoint functors (F3 = A Bop A□ HBop -,G3 = (-)coH) between the categories A□HBopM and AMπB-H is a pair of inverse equivalences, when A is a faithfully flat π-H-Galois extension. Furthermore, the categories Moritaπ-H(A,B) and Morita □π-H(AcoH,BcoH) are equivalent, if A and B are faithfully flat π-H-Galois extensions.展开更多
The aim of this article is to study some invariants of associative algebras under stable equivalences of Morita type. First of all, we show that, if two finite-dimensional selfinjective k-algebras are stably equivalen...The aim of this article is to study some invariants of associative algebras under stable equivalences of Morita type. First of all, we show that, if two finite-dimensional selfinjective k-algebras are stably equivalent of Morita type, then their orbit algebras are isomorphic. Secondly, it is verified that the quasitilted property of an algebra is invariant under stable equivalences of Morita type. As an application of this result, it is obtained that if an algebra is of finite representation type, then its tilted property is invariant under stable equivalences of Morita type; the other application to partial tilting modules is given in Section 4. Finally, we prove that when two finite-dimensional k-algebras are stably equivalent of Morita type, their repetitive algebras are also stably equivalent of Morita tvDe under cert..in conditions.展开更多
Let (K, O, k) be a p-modular system and G be a finite group. We prove that block A of RG and block B of RH are nalurally Morita equivalent of degree n if and only if A≌B+…+B}→n^2 as right R[H×H]-modules an...Let (K, O, k) be a p-modular system and G be a finite group. We prove that block A of RG and block B of RH are nalurally Morita equivalent of degree n if and only if A≌B+…+B}→n^2 as right R[H×H]-modules and A and B have the same defect(where R∈{k,O}), which is a generalization of the result of Külshammer Burkhard in a p-modular system for an arbitrary subgroup H of G. It is proved that naturally Morita equivalent blocks are equivalent blocks and Morita equivalent via a bimodule with trivial source.展开更多
In this paper we define and study chain conditions for Hilbert C*-modules through their C*-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that thes...In this paper we define and study chain conditions for Hilbert C*-modules through their C*-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C*-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C*-algebra with a faithful conditional expectation.展开更多
Let X={X_(p)}p∈Pbe a product system over a lattice ordered group(G,P)with coefficients in a C*-algebra A.In this paper,we study the reduced crossed product of the gauge coactionδof G on the Cuntz-Pimsner algebra NO_...Let X={X_(p)}p∈Pbe a product system over a lattice ordered group(G,P)with coefficients in a C*-algebra A.In this paper,we study the reduced crossed product of the gauge coactionδof G on the Cuntz-Pimsner algebra NO_(X)^(r).When X is a product system of Morita equivalence bimodules,we show that the reduced crossed product of the gauge coaction is Morita equivalent to the C*-algebra A.展开更多
We define the ttochschild and cyclic (co)homology groups for superadditive categories and show that these (co)homology groups are graded Morita invariants. We also show that the Hochschild and cyclic homology are ...We define the ttochschild and cyclic (co)homology groups for superadditive categories and show that these (co)homology groups are graded Morita invariants. We also show that the Hochschild and cyclic homology are compatible with the tensor product of superadditive categories.展开更多
文摘Morita equivalence was established by Morita in the late 1950’s. In part one most of the recent developments in this theory on the categories of modules over rings were reviewed and some results were presented. Here this paper discusses those on categories of comodules over coalgebras, these results are due to Ling, Takeuchi and Wang.
基金The research is partially supported by a UGC(HK) grant ≠2160092Project is supported by the National Natural Science Foundation of China
文摘Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S,R,<sub>S</sub>P<sub>R</sub>,<sub>R</sub>Q<sub>S</sub>,) with and surjective.For a factorisable semigroup S,we denote ζ<sub>S</sub>={(s<sub>1</sub>,s<sub>2</sub>)∈S×S|ss<sub>1</sub>=ss<sub>2</sub>,<sub>S</sub>∈S},S′=S/ζ<sub>S</sub> and US-FAct={<sub>S</sub>M∈ S-Act|SM=M and SHom<sub>S</sub>(S,M)≌M}.We show that,for factorisable semigroups S and R,the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S′ and R′ are strongly Morita equivalent.Some conditions for a factorisable semigroup to be strongly Morita equivalent to a sandwich semigroup,local units semigroup,monoid and group separately are also given.Moreover,we show that a seinigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I,SSHom<sub>S</sub>(S,<sub>i∈I</sub>S)→<sub>i∈I</sub>S,st·f(st)f is an S-isomorphism.
文摘Morita equivalence was established by Morita in the late 1950's. Here most of the recent developments in this theory are reviewed and some results are presented.
基金Supported by the Key Programs of Jiaxing University (Grant No. 70110X03BL)Scientific Research Foundation of Jiaxing University (Grant No.70509015)
文摘Let H be a Hopf π-coalgebra over a commutative ring k with bijective antipode S, and A and B right π-H-comodulelike algebras. We show that the pair of adjoint functors (F3 = A Bop A□ HBop -,G3 = (-)coH) between the categories A□HBopM and AMπB-H is a pair of inverse equivalences, when A is a faithfully flat π-H-Galois extension. Furthermore, the categories Moritaπ-H(A,B) and Morita □π-H(AcoH,BcoH) are equivalent, if A and B are faithfully flat π-H-Galois extensions.
基金Project supported by the National Natural Science Foundation of China(10871170)the Zhejiang Provincial Natural Science Foundation of China(D7080064)supported by the National Natural Science Foundation of China(10801117)
文摘The aim of this article is to study some invariants of associative algebras under stable equivalences of Morita type. First of all, we show that, if two finite-dimensional selfinjective k-algebras are stably equivalent of Morita type, then their orbit algebras are isomorphic. Secondly, it is verified that the quasitilted property of an algebra is invariant under stable equivalences of Morita type. As an application of this result, it is obtained that if an algebra is of finite representation type, then its tilted property is invariant under stable equivalences of Morita type; the other application to partial tilting modules is given in Section 4. Finally, we prove that when two finite-dimensional k-algebras are stably equivalent of Morita type, their repetitive algebras are also stably equivalent of Morita tvDe under cert..in conditions.
基金Supported by the National Programfor the BasicScience Researches of China(G19990751)
文摘Let (K, O, k) be a p-modular system and G be a finite group. We prove that block A of RG and block B of RH are nalurally Morita equivalent of degree n if and only if A≌B+…+B}→n^2 as right R[H×H]-modules and A and B have the same defect(where R∈{k,O}), which is a generalization of the result of Külshammer Burkhard in a p-modular system for an arbitrary subgroup H of G. It is proved that naturally Morita equivalent blocks are equivalent blocks and Morita equivalent via a bimodule with trivial source.
文摘In this paper we define and study chain conditions for Hilbert C*-modules through their C*-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C*-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C*-algebra with a faithful conditional expectation.
基金Wang was supported in part by NSF of China(Grant Nos.11871303,11971463,11671133)NSF of Shandong Province(Grant No.ZR2019MA039)Yuan was supported in part by NSF of China(Grant Nos.11871303,11871127,11971463)。
文摘Let X={X_(p)}p∈Pbe a product system over a lattice ordered group(G,P)with coefficients in a C*-algebra A.In this paper,we study the reduced crossed product of the gauge coactionδof G on the Cuntz-Pimsner algebra NO_(X)^(r).When X is a product system of Morita equivalence bimodules,we show that the reduced crossed product of the gauge coaction is Morita equivalent to the C*-algebra A.
基金Supported by National Natural Science Foundation of China(Grant No.11101037)
文摘We define the ttochschild and cyclic (co)homology groups for superadditive categories and show that these (co)homology groups are graded Morita invariants. We also show that the Hochschild and cyclic homology are compatible with the tensor product of superadditive categories.