While the Yoneda embedding and its generalizations have been studied extensively in the literature,the so-called tensor embedding has only received a little attention.In this paper,we study the tensor embedding for cl...While the Yoneda embedding and its generalizations have been studied extensively in the literature,the so-called tensor embedding has only received a little attention.In this paper,we study the tensor embedding for closed symmetric monoidal categories and show how it is connected to the notion of geometrically purity,which has recently been investigated in the works of Enochs et al.(2016)and Estrada et al.(2017).More precisely,for a Grothendieck cosmos,i.e.,a bicomplete Grothendieck category V with a closed symmetric monoidal structure,we prove that the geometrically pure exact category(V,ε■)has enough relative injectives;in fact,every object has a geometrically pure injective envelope.We also show that for some regular cardinalλ,the tensor embedding yields an exact equivalence between(V,ε■)and the category ofλ-cocontinuous V-functors from Presλ(V)to V,where the former is the full V-subcategory ofλ-presentable objects in V.In many cases of interest,λcan be chosen to be■0 and the tensor embedding identifies the geometrically pure injective objects in V with the(categorically)injective objects in the abelian category of V-functors from fp(V)to V.As we explain,the developed theory applies,e.g.,to the category Ch(R)of chain complexes of modules over a commutative ring R and to the category Qcoh(X)of quasi-coherent sheaves over a(suitably nice)scheme X.展开更多
Motivated by the classical Gorenstein homological theory and structure of Tate cohomology, we develop a theory of Gorenstein projective objects and Tate cohomology in an exact category A with enough projectives. We st...Motivated by the classical Gorenstein homological theory and structure of Tate cohomology, we develop a theory of Gorenstein projective objects and Tate cohomology in an exact category A with enough projectives. We study some properties of Gorenstein projective objects and establish Tate cohomology of objects with finite Gorenstein projective dimension.展开更多
This paper aims at developing a "local-global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applic...This paper aims at developing a "local-global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. We first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. We prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary(the "good" prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells.展开更多
on Double Vector Bundles Zhuo CHEN Zhang Ju LIU Yun He SHENG Abstract In this paper,we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector b...on Double Vector Bundles Zhuo CHEN Zhang Ju LIU Yun He SHENG Abstract In this paper,we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles.Moreover,operations on double vector bundles can be transferred to operations on the corresponding short exact sequences.In particular,we study the duality theory of double vector bundles in term展开更多
After defining Hom(chi (A), eta (B)) and chi (A) circle times eta (B) in the fuzzy modular category Fm, the sufficient conditions of the existence for exact Hom functors Hom(delta (M),), and Hom(, delta (M)), as well ...After defining Hom(chi (A), eta (B)) and chi (A) circle times eta (B) in the fuzzy modular category Fm, the sufficient conditions of the existence for exact Hom functors Hom(delta (M),), and Hom(, delta (M)), as well as exact Tensor functors delta (M)circle times and circle times delta (M) are given in this paper. Finally the weak isomorphisms relations between Horn functors and Tensor functors are displayed.展开更多
基金supported by CONICYT/FONDECYT/INICIACIOóN(Grant No.11170394)。
文摘While the Yoneda embedding and its generalizations have been studied extensively in the literature,the so-called tensor embedding has only received a little attention.In this paper,we study the tensor embedding for closed symmetric monoidal categories and show how it is connected to the notion of geometrically purity,which has recently been investigated in the works of Enochs et al.(2016)and Estrada et al.(2017).More precisely,for a Grothendieck cosmos,i.e.,a bicomplete Grothendieck category V with a closed symmetric monoidal structure,we prove that the geometrically pure exact category(V,ε■)has enough relative injectives;in fact,every object has a geometrically pure injective envelope.We also show that for some regular cardinalλ,the tensor embedding yields an exact equivalence between(V,ε■)and the category ofλ-cocontinuous V-functors from Presλ(V)to V,where the former is the full V-subcategory ofλ-presentable objects in V.In many cases of interest,λcan be chosen to be■0 and the tensor embedding identifies the geometrically pure injective objects in V with the(categorically)injective objects in the abelian category of V-functors from fp(V)to V.As we explain,the developed theory applies,e.g.,to the category Ch(R)of chain complexes of modules over a commutative ring R and to the category Qcoh(X)of quasi-coherent sheaves over a(suitably nice)scheme X.
文摘Motivated by the classical Gorenstein homological theory and structure of Tate cohomology, we develop a theory of Gorenstein projective objects and Tate cohomology in an exact category A with enough projectives. We study some properties of Gorenstein projective objects and establish Tate cohomology of objects with finite Gorenstein projective dimension.
基金supported by a 2017 University of New South Wales Science Goldstar Grant(Jie Du)the Simons Foundation(Grant Nos. #359360(Brian Parshall) and #359363 (Leonard Scott))
文摘This paper aims at developing a "local-global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. We first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. We prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary(the "good" prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells.
文摘on Double Vector Bundles Zhuo CHEN Zhang Ju LIU Yun He SHENG Abstract In this paper,we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles.Moreover,operations on double vector bundles can be transferred to operations on the corresponding short exact sequences.In particular,we study the duality theory of double vector bundles in term
文摘After defining Hom(chi (A), eta (B)) and chi (A) circle times eta (B) in the fuzzy modular category Fm, the sufficient conditions of the existence for exact Hom functors Hom(delta (M),), and Hom(, delta (M)), as well as exact Tensor functors delta (M)circle times and circle times delta (M) are given in this paper. Finally the weak isomorphisms relations between Horn functors and Tensor functors are displayed.
