We introduce and study （pre）resolving subcategories of a triangulated category and the homological dimension relative to these subcategories. We apply the obtained properties to relative Gorenstein categories.
In Enochs'relative homological dimension theory occur the(co)resolvent and(co)proper dimensions,which are defined by proper and coproper resolutions constructed by precovers and preenvelopes,respectively.Recently,...In Enochs'relative homological dimension theory occur the(co)resolvent and(co)proper dimensions,which are defined by proper and coproper resolutions constructed by precovers and preenvelopes,respectively.Recently,some authors have been interested in relative homological dimensions defined by just exact sequences.In this paper,we contribute to the investigation of these relative homological dimensions.First we study the relation between these two kinds of relative homological dimensions and establish some transfer results under adjoint pairs.Then relative global dimensions are studied,which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories.At the end of this paper,relative derived functors are studied and generalizations of some known results of balance for relative homology are established.展开更多
Let A be an abelian category,C an additive,full and self-orthogonal subcategory of A closed under direct summands,rG(C)the right Gorenstein subcategory of A relative to C,and⊥C the left orthogonal class of C.For an o...Let A be an abelian category,C an additive,full and self-orthogonal subcategory of A closed under direct summands,rG(C)the right Gorenstein subcategory of A relative to C,and⊥C the left orthogonal class of C.For an object A in A,we prove that if A is in the right 1-orthogonal class of rG(C),then the C-projective and rG(C)-projective dimensions of A are identical;if the rG(C)-projective dimension of A is finite,then the rG(C)-projective and⊥C-projective dimensions of A are identical.We also prove that the supremum of the C-projective dimensions of objects with finite C-projective dimension and that of the rG(C)-projective dimensions of objects with finite rG(C)-projective dimension coincide.Then we apply these results to the category of modules.展开更多
Let A be a small abelian category.For a closed subbifunctor F of Ext_A^1（-,-）,Buan has generalized the construction of Verdier’s quotient category to get a relative derived category,where he localized with respect ...Let A be a small abelian category.For a closed subbifunctor F of Ext_A^1（-,-）,Buan has generalized the construction of Verdier’s quotient category to get a relative derived category,where he localized with respect to F-acyclic complexes.In this paper,the homological properties of relative derived categories are discussed,and the relation with derived categories is given.For Artin algebras,using relative derived categories,we give a relative version on derived equivalences induced by F-tilting complexes.We discuss the relationships between relative homological dimensions and relative derived equivalences.展开更多
For two classes of right R-modules W, X such that P W X, where P is the class of projective right R-modules, we show that there is an Avramov-Martsinkovsky type exact sequence with generalized Tate homology func...For two classes of right R-modules W, X such that P W X, where P is the class of projective right R-modules, we show that there is an Avramov-Martsinkovsky type exact sequence with generalized Tate homology functor Tor^X,W, relative homology functors Tor^W and Tor^X. Many results in Iacob [Comm. Algebra, 35, 1589-1606 （2007）] and Liang [Algebr. Represent. Theory, 16, 1541-1560 （2013）] are generalized and improved.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11571164)a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions,Postgraduate Research and Practice Innovation Program of Jiangsu Province(Grant No.KYZZ16 0034)Nanjing University Innovation and Creative Program for PhD candidate(Grant No.2016011)
文摘We introduce and study （pre）resolving subcategories of a triangulated category and the homological dimension relative to these subcategories. We apply the obtained properties to relative Gorenstein categories.
基金The second and fourth authors were partially supported by the grant MTM2014-54439-P from Ministerio de Economia y CompetitividadThe third author was partially supported by NSFC(11771202).
文摘In Enochs'relative homological dimension theory occur the(co)resolvent and(co)proper dimensions,which are defined by proper and coproper resolutions constructed by precovers and preenvelopes,respectively.Recently,some authors have been interested in relative homological dimensions defined by just exact sequences.In this paper,we contribute to the investigation of these relative homological dimensions.First we study the relation between these two kinds of relative homological dimensions and establish some transfer results under adjoint pairs.Then relative global dimensions are studied,which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories.At the end of this paper,relative derived functors are studied and generalizations of some known results of balance for relative homology are established.
基金This research was partially supported by NSFC(Grant Nos.11571164,11971225,11901341)the NSF of Shandong Province(Grant No.ZR2019QA015)。
文摘Let A be an abelian category,C an additive,full and self-orthogonal subcategory of A closed under direct summands,rG(C)the right Gorenstein subcategory of A relative to C,and⊥C the left orthogonal class of C.For an object A in A,we prove that if A is in the right 1-orthogonal class of rG(C),then the C-projective and rG(C)-projective dimensions of A are identical;if the rG(C)-projective dimension of A is finite,then the rG(C)-projective and⊥C-projective dimensions of A are identical.We also prove that the supremum of the C-projective dimensions of objects with finite C-projective dimension and that of the rG(C)-projective dimensions of objects with finite rG(C)-projective dimension coincide.Then we apply these results to the category of modules.
基金Supported by National Natural Science Foundation of China(Grant No.11201022)the Fundamental Research Funds for the Central Universities(Grant No.2015JBM101)
文摘Let A be a small abelian category.For a closed subbifunctor F of Ext_A^1（-,-）,Buan has generalized the construction of Verdier’s quotient category to get a relative derived category,where he localized with respect to F-acyclic complexes.In this paper,the homological properties of relative derived categories are discussed,and the relation with derived categories is given.For Artin algebras,using relative derived categories,we give a relative version on derived equivalences induced by F-tilting complexes.We discuss the relationships between relative homological dimensions and relative derived equivalences.
基金Supported by National Natural Science Foundation of China(Grant Nos.11301240,11401475)Natural Science Foundation of Chongqing(cstc2017jcyjAX0298)
文摘For two classes of right R-modules W, X such that P W X, where P is the class of projective right R-modules, we show that there is an Avramov-Martsinkovsky type exact sequence with generalized Tate homology functor Tor^X,W, relative homology functors Tor^W and Tor^X. Many results in Iacob [Comm. Algebra, 35, 1589-1606 （2007）] and Liang [Algebr. Represent. Theory, 16, 1541-1560 （2013）] are generalized and improved.
