Let A=kQ/I be a finite-dimensional basic algebra over an algebraically closed field k,which is a gentle algebra with the marked ribbon surface(SA,MA,ΓA).It is known that SAcan be divided into some elementary polygons...Let A=kQ/I be a finite-dimensional basic algebra over an algebraically closed field k,which is a gentle algebra with the marked ribbon surface(SA,MA,ΓA).It is known that SAcan be divided into some elementary polygons{Δi|1≤i≤d}byΓA,which has exactly one side in the boundary of SA.Let■(Δi)be the number of sides ofΔibelonging toΓAif the unmarked boundary component of SAis not a side ofΔi;otherwise,■(Δi)=∞,and let f-Δbe the set of all the non-co-elementary polygons and FA(resp.f-FA)be the set of all the forbidden threads(resp.of finite length).Then we have(1)the global dimension of A is max1≤i≤d■(Δi)-1=maxΠ∈FAl(Π),where l(Π)is the length ofΠ;(2)the left and right self-injective dimensions of A are 0,if Q is either a point or an oriented cycle with full relations.masΔi∈f-Δ{1,■(Δi)-1}=max n∈f-F_(A)l(П),otherwise,As a consequence,we get that the finiteness of the global dimension of gentle algebras is invariant under AvellaGeiss(AG)-equivalence.In addition,we get that the number of indecomposable non-projective Gorenstein projective modules over gentle algebras is also invariant under AG-equivalence.展开更多
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
Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we es...Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over ∧, G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective ∧-modules, and G-stable functorially finite torsion classes in the category of finitely generated left ∧-modules. In the case when ∧ is the endomorphism of a G-stable cluster-tilting object T over a Horn-finite 2-Calabi- Yau triangulated category L with a G-action, these are also in bijection with G-stable cluster-tilting objects in L. Moreover, we investigate the relationship between stable support τ-tilitng modules over ∧ and the skew group algebra ∧G.展开更多
Let A be an abelian category and P(A)be the subcategory of A consisting of projective objects.Let C be a full,additive and self-orthogonal subcategory of A with P(A)a generator,and let G(C)be the Gorenstein subcategor...Let A be an abelian category and P(A)be the subcategory of A consisting of projective objects.Let C be a full,additive and self-orthogonal subcategory of A with P(A)a generator,and let G(C)be the Gorenstein subcategory of A.Then the right 1-orthogonal category G(C)^⊥1 of G(C)is both projectively resolving and injectively coresolving in A.We also get that the subcategory SPC(G(C))of A consisting of objects admitting special G(C)-precovers is closed under extensions and C-stable direct summands(*).Furthermore,if C is a generator for G(C)^⊥1,then we have that SPC(G(C))is the minimal subcategory of A containing G(C)^⊥1∪G(C)with respect to the property(*),and that SPC(G(C))is C-resolving in A with a C-proper generator C.展开更多
For a recollement （A,B,C） of abelian categories, we show that torsion pairs in A and C can induce torsion pairs in B; and the converse holds true under certain conditions.
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
基金supported by National Natural Science Foundation of China(Grant Nos.11971225 and 12171207)。
文摘Let A=kQ/I be a finite-dimensional basic algebra over an algebraically closed field k,which is a gentle algebra with the marked ribbon surface(SA,MA,ΓA).It is known that SAcan be divided into some elementary polygons{Δi|1≤i≤d}byΓA,which has exactly one side in the boundary of SA.Let■(Δi)be the number of sides ofΔibelonging toΓAif the unmarked boundary component of SAis not a side ofΔi;otherwise,■(Δi)=∞,and let f-Δbe the set of all the non-co-elementary polygons and FA(resp.f-FA)be the set of all the forbidden threads(resp.of finite length).Then we have(1)the global dimension of A is max1≤i≤d■(Δi)-1=maxΠ∈FAl(Π),where l(Π)is the length ofΠ;(2)the left and right self-injective dimensions of A are 0,if Q is either a point or an oriented cycle with full relations.masΔi∈f-Δ{1,■(Δi)-1}=max n∈f-F_(A)l(П),otherwise,As a consequence,we get that the finiteness of the global dimension of gentle algebras is invariant under AvellaGeiss(AG)-equivalence.In addition,we get that the number of indecomposable non-projective Gorenstein projective modules over gentle algebras is also invariant under AG-equivalence.
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The authors would like to thank Dong Yang and Yuefei Zheng for their helpful discussion. This work was partially supported by the National Natural Science Foundation of China （Grant No. 11571164） and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over ∧, G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective ∧-modules, and G-stable functorially finite torsion classes in the category of finitely generated left ∧-modules. In the case when ∧ is the endomorphism of a G-stable cluster-tilting object T over a Horn-finite 2-Calabi- Yau triangulated category L with a G-action, these are also in bijection with G-stable cluster-tilting objects in L. Moreover, we investigate the relationship between stable support τ-tilitng modules over ∧ and the skew group algebra ∧G.
基金supported by National Natural Science Foundation of China (Grant No. 11571164)Priority Academic Program Development of Jiangsu Higher Education Institutions+1 种基金the University Postgraduate Research and Innovation Project of Jiangsu Province 2016 (Grant No. KYZZ16 0034)Nanjing University Innovation and Creative Program for PhD Candidate (Grant No. 2016011)
文摘Let A be an abelian category and P(A)be the subcategory of A consisting of projective objects.Let C be a full,additive and self-orthogonal subcategory of A with P(A)a generator,and let G(C)be the Gorenstein subcategory of A.Then the right 1-orthogonal category G(C)^⊥1 of G(C)is both projectively resolving and injectively coresolving in A.We also get that the subcategory SPC(G(C))of A consisting of objects admitting special G(C)-precovers is closed under extensions and C-stable direct summands(*).Furthermore,if C is a generator for G(C)^⊥1,then we have that SPC(G(C))is the minimal subcategory of A containing G(C)^⊥1∪G(C)with respect to the property(*),and that SPC(G(C))is C-resolving in A with a C-proper generator C.
基金The authors thank Daniel Juteau for sending us paper [13], and thank Teimuraz Pirashvili and the referees for the useful suggestions. This work was partially 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. KYCX17_0019）.
文摘For a recollement （A,B,C） of abelian categories, we show that torsion pairs in A and C can induce torsion pairs in B; and the converse holds true under certain conditions.
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.
