纯奇点范畴中的Buchweitz定理
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摘要
我们定义纯奇点范畴D_(psg)~b(R)为有界纯导出范畴D_(pur)~b(R)与纯投射模构成的有界同伦范畴K~b(■)的Verdier商,得到了纯奇点范畴D_(psg)~b(R)三角等价于相对纯投射模的Gorenstein范畴的稳定范畴■的一个充分必要条件.同时,还给出三角等价D_(psg)~b(R)≌D_(psg)~b(S)的充分条件,这里R和S都是环.
We define the pure singularity category D_(psg)~b(R)as the Verdier quotient of the bounded pure derived category D_(pur)~b(R)by the triangulated subcategory K~b■ of the bounded homotopy category consisting of pure projective modules,a sufficient and necessary condition under which D_(psg)~b(R)is equivalent to the stable category of the Gorenstein category ■ of pure projective modules is given.Moreover,we give a sufficient condition for the triangle-equivalence D_(psg)~b(R))≌ D_(psg)~b(S),where R and S are rings.
We define the pure singularity category D_(psg)~b(R)as the Verdier quotient of the bounded pure derived category D_(pur)~b(R)by the triangulated subcategory K~b■ of the bounded homotopy category consisting of pure projective modules,a sufficient and necessary condition under which D_(psg)~b(R)is equivalent to the stable category of the Gorenstein category ■ of pure projective modules is given.Moreover,we give a sufficient condition for the triangle-equivalence D_(psg)~b(R))≌ D_(psg)~b(S),where R and S are rings.
引文
[1]Asadollahi J.,Hafezi R.,Vahed R.,Gorenstein derived equivalences and their invariants,J.Pure Appl.Algebra,2014,218(5):888-903.
[2]Bao Y.H.,Du X.N.,Zhao Z.B.,Gorenstein singularity categories,J.Algebra,2015,428:122-137.
[3]Buchweitz R.O.,Matrix Cohen-Macaulay modules and Tate cohomology over Gorenstein rings,Hamburg,1987,155 pp.,unpublished manuscript.
[4]Chen W.J.,Liu Z.K.,Yang X.Y.,Singularity categories with respect to Ding projective modules,Acta Math.Sinica, 2017,33:793-806.
[5]Chen X.W.,Relative singularity categories and Gorenstein projective modules,Math.Nachr,2011,284:199-212.
[6]Enochs E.E., Jenda O.M.G.,López-Ramos J.A.,Covers and envelopes by V-Gorenstein modules,Comm.Algebra,2005,33:4705-4717.
[7]Huang Z.Y.,Proper resolutions and Gorenstein categories,J.Algebra,2013,393:142-169.
[8]Kong F.,Zhang P.,From CM-finite to CM-free,J.Pure Appl.Algebra,2016,220:782-801.
[9]Li H.H.,Huang Z.Y.,Relative singularity categories,J.Pure Appl.Algebra,2015,219:4090-4104.
[10]Orlov D.,Triangulated categories of singularities and D-branes in Landau-Ginzburg models,Proc.Steklov Inst. Math.,2004,246:227-248.
[11]Rickard J.,Derived categories and stable equivalence,J.Pure Appl.Algebra,1989,61:303-317.
[12]Sather-Wagstaff S.,Sharif T.,White D.,Stability of Gorenstein categories,J.Lond.Math.Soc.,2008,77:481-502.
[13]Verdier J.L.,Des Catégories Dérivées Abéliennes,Asterisque,1996,239:xii+253 pp.
[14]Zheng Y.F.,Huang Z.Y.,On pure derived categories,J.Algebra,2016,454:252-272.
[15]Zhang P.,Triangulated Category and Derived Category(in Chinese),Science Press,Beijing,2015.
[16]Zhu S.J.,Left Homotopy Theory and Buchweitz's Theorem,Master Thesis at Shanghai Jiaotong Universty,Shanghai,2011.
[2]Bao Y.H.,Du X.N.,Zhao Z.B.,Gorenstein singularity categories,J.Algebra,2015,428:122-137.
[3]Buchweitz R.O.,Matrix Cohen-Macaulay modules and Tate cohomology over Gorenstein rings,Hamburg,1987,155 pp.,unpublished manuscript.
[4]Chen W.J.,Liu Z.K.,Yang X.Y.,Singularity categories with respect to Ding projective modules,Acta Math.Sinica, 2017,33:793-806.
[5]Chen X.W.,Relative singularity categories and Gorenstein projective modules,Math.Nachr,2011,284:199-212.
[6]Enochs E.E., Jenda O.M.G.,López-Ramos J.A.,Covers and envelopes by V-Gorenstein modules,Comm.Algebra,2005,33:4705-4717.
[7]Huang Z.Y.,Proper resolutions and Gorenstein categories,J.Algebra,2013,393:142-169.
[8]Kong F.,Zhang P.,From CM-finite to CM-free,J.Pure Appl.Algebra,2016,220:782-801.
[9]Li H.H.,Huang Z.Y.,Relative singularity categories,J.Pure Appl.Algebra,2015,219:4090-4104.
[10]Orlov D.,Triangulated categories of singularities and D-branes in Landau-Ginzburg models,Proc.Steklov Inst. Math.,2004,246:227-248.
[11]Rickard J.,Derived categories and stable equivalence,J.Pure Appl.Algebra,1989,61:303-317.
[12]Sather-Wagstaff S.,Sharif T.,White D.,Stability of Gorenstein categories,J.Lond.Math.Soc.,2008,77:481-502.
[13]Verdier J.L.,Des Catégories Dérivées Abéliennes,Asterisque,1996,239:xii+253 pp.
[14]Zheng Y.F.,Huang Z.Y.,On pure derived categories,J.Algebra,2016,454:252-272.
[15]Zhang P.,Triangulated Category and Derived Category(in Chinese),Science Press,Beijing,2015.
[16]Zhu S.J.,Left Homotopy Theory and Buchweitz's Theorem,Master Thesis at Shanghai Jiaotong Universty,Shanghai,2011.