Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give ...Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod R and the Auslander-Reiten functors, triangles in C, respectively. Furthermore, if T is a cluster-tilting subcategory of C and mod T- is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod T- corresponding to the ones in C. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Probenius, then its stable category is （2d-1）-Calabi-Yau. This result was first proved by Keller and Reiten in the case d= 2, and then by Dugas in the general case, using different methods. 2010 Mathematics Subject Classification： 16G20, 16G70展开更多
Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projecti...Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projective （respectively, injective） middle terms. It is proved that Mp （respectively, MI） is contravariantly finite （respectively, covariantly finite） in ε. As an application, it is shown that Mp = MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective. Keywords category of exact sequences, contravariantly finite subcategory, functorially finite subcategory Auslander-Reiten sequences, selfinjective algebra展开更多
文摘Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod R and the Auslander-Reiten functors, triangles in C, respectively. Furthermore, if T is a cluster-tilting subcategory of C and mod T- is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod T- corresponding to the ones in C. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Probenius, then its stable category is （2d-1）-Calabi-Yau. This result was first proved by Keller and Reiten in the case d= 2, and then by Dugas in the general case, using different methods. 2010 Mathematics Subject Classification： 16G20, 16G70
基金supported by National Natural Science Foundation of China(Grant No.11271257)National Science Foundation of Shanghai Municiple(Granted No.13ZR1422500)
文摘Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projective （respectively, injective） middle terms. It is proved that Mp （respectively, MI） is contravariantly finite （respectively, covariantly finite） in ε. As an application, it is shown that Mp = MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective. Keywords category of exact sequences, contravariantly finite subcategory, functorially finite subcategory Auslander-Reiten sequences, selfinjective algebra
