Let T be a triangulated category with a proper classξof triangles.We introduce the notions of left Frobenius pairs,left(n-)cotorsion pairs and left(weak)Auslander-Buchweitz contexts with respect toξin T.We show how ...Let T be a triangulated category with a proper classξof triangles.We introduce the notions of left Frobenius pairs,left(n-)cotorsion pairs and left(weak)Auslander-Buchweitz contexts with respect toξin T.We show how to construct left cotorsion pais from left n-cotorsion pairs,and establish a one-to-one correspondence between left Frobenius pairs and left(weak)Auslander-Buchweitz contexts.Some applications are given in the Gorenstein homological theory of triangulated categories.展开更多
In this paper,we generalize the idea of Song,Zhao and Huang[Czechoslov.Math.J.,70,483±504(2020)]and introduce the notion of right(left)Gorenstein subcategory rg(l,∂)(lg(l,D)),relative to two additive full subcate...In this paper,we generalize the idea of Song,Zhao and Huang[Czechoslov.Math.J.,70,483±504(2020)]and introduce the notion of right(left)Gorenstein subcategory rg(l,∂)(lg(l,D)),relative to two additive full subcategoriesφand∂of an abelian category A.Under the assumption thatφ⊆∂,we prove that the right Gorenstein subcategory rg(l,D)possesses many nice properties that it is closed under extensions,kernels of epimorphisms and direct summands.Whenφ⊆Dandφ⊥D,we show that the right Gorenstein subcategory rg(l,D)admits some kind of stability.Then we discuss a resolution dimension for an object in A,called rg(l,D)-projective dimension.Finally,we prove that if(U,V)is a hereditary cotorsion pair with kernelφhas enough injectives,such that U⊆Dand U⊥∂,then(rg(l,D),φφ)is a weak Auslander±Buchweitz context,whereφis the subcategory of A consisting of objects with finiteφ-projective dimension.展开更多
基金supported by the NSF of China(12001168,11901341,11971225,12171207)Henan University of Engineering(DKJ2019010)+1 种基金the Key Research Project of Education Department of Henan Province(21A110006)Youth Innovation Team of Universities of Shandong Province(2022KJ314).
文摘Let T be a triangulated category with a proper classξof triangles.We introduce the notions of left Frobenius pairs,left(n-)cotorsion pairs and left(weak)Auslander-Buchweitz contexts with respect toξin T.We show how to construct left cotorsion pais from left n-cotorsion pairs,and establish a one-to-one correspondence between left Frobenius pairs and left(weak)Auslander-Buchweitz contexts.Some applications are given in the Gorenstein homological theory of triangulated categories.
基金Supported by National Natural Science Foundation of China(Grant No.11971225)。
文摘In this paper,we generalize the idea of Song,Zhao and Huang[Czechoslov.Math.J.,70,483±504(2020)]and introduce the notion of right(left)Gorenstein subcategory rg(l,∂)(lg(l,D)),relative to two additive full subcategoriesφand∂of an abelian category A.Under the assumption thatφ⊆∂,we prove that the right Gorenstein subcategory rg(l,D)possesses many nice properties that it is closed under extensions,kernels of epimorphisms and direct summands.Whenφ⊆Dandφ⊥D,we show that the right Gorenstein subcategory rg(l,D)admits some kind of stability.Then we discuss a resolution dimension for an object in A,called rg(l,D)-projective dimension.Finally,we prove that if(U,V)is a hereditary cotorsion pair with kernelφhas enough injectives,such that U⊆Dand U⊥∂,then(rg(l,D),φφ)is a weak Auslander±Buchweitz context,whereφis the subcategory of A consisting of objects with finiteφ-projective dimension.
