相对Gorenstein余可解范畴
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摘要
主要给出相对Gorenstein余可解范畴的定义及同调性质,进而研究相对Gorenstein余可解范畴维数及dimgy(M)≤n的若干等价刻画。证明了对任意n≥1,dimgy(M)≤n当且仅当存在正合列0→Gn→Pn-1→…→P1→P0→M→0(其中Pi∈P(A),G∈GPy(A)),当且仅当对任意非负整数t(0≤t≤n),均存在正合列0→Yn→Yn-1→…→Yt→…→Y1→Y0→M→0(其中,Yt∈GPy(A),Yi≠t∈P(A))。
It was gived the definition and properties of relative Gorenstein coresolving category.Moreover it was defined a resolution dimension relative to the relative Gorenstein coresolving category GPy(A),and giveD some equivalent characterizations of dimgy(M)≤n.It was proved that dimgy(M)≤nif and only if exist exact sequence 0→Gn →Pn-1 → … →P1 →P0 →M →0(eachPi ∈P(A),G∈ GPy(A)),if and only if exist exact sequence 0→Yn →Yn-1 → … →Yt → … →Y1 →Y0 → M →0(each Yt ∈ GPy(A),Yi≠t ∈ P(A))for any t(0≤t≤n).
It was gived the definition and properties of relative Gorenstein coresolving category.Moreover it was defined a resolution dimension relative to the relative Gorenstein coresolving category GPy(A),and giveD some equivalent characterizations of dimgy(M)≤n.It was proved that dimgy(M)≤nif and only if exist exact sequence 0→Gn →Pn-1 → … →P1 →P0 →M →0(eachPi ∈P(A),G∈ GPy(A)),if and only if exist exact sequence 0→Yn →Yn-1 → … →Yt → … →Y1 →Y0 → M →0(each Yt ∈ GPy(A),Yi≠t ∈ P(A))for any t(0≤t≤n).
引文
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[2]Enochs E E,Jenda O M G.Gorenstein injective and projective modules[J].Mathematische Zeitschrift,1995,220(1):611-633.
[3]Holm H.Gorenstein homological dimension[J].Journal of Pure&Applied Algebra,2004,189(1):167-193.
[4]Gao Z H,Xu L Y.Gorenstein coresolving categories[J].Communications in Algebra,2016,45(10):4477-4491.
[5]Benni D,Ouarghi K.X-Gorenstein projective modules[J].Journal of Lanzhou University of Technology,2010,(9-12):487-491.
[6]Zhu X S.Resolving resolution dimension[J].Algebras&Representation Theory,2013,16(4):1165-1191.
[7]Enochs E E,Jenda O M G.Relative homological algebra[M].Berlin:Walter de Gruyter,2000.
[8]Bravo D,Gillespie J.Absolutely clean,level,and Gorenstein AC-injective complexes[J].Communications in Algebra,2014,44(5):2213-2233.
[9]Enochs E,Jenda O G,López-Ramos J A.Covers and envelopes by V-Gorenstein modules[J].Communications in Algebra,2005,33(12):4705-4717.
[10]Geng Y X,Ding N Q.W-Gorenstein modules[J].Journal of Algebra,2011,325(1):132-146.
[11]Huang Z Y.Proper resolutions and Gorenstein categories[J].Journal of Algebra,2013,393(11):142-169.
[12]Meng F,Pan Q.X-Gorenstein projective and Y-Gorenstein injective modules[J].Hacettepe University Bulletin of Natural Sciences&Engineering,2011,40(4):537-554.
[13]Enochs E E,Garcia Rozas J R.Gorenstein injective and projective complexes[J],Communications in Algebra,1998,26(5):1657-1674.
[14]Anderson F W,Fuller K R.Rings and categories of modules[M].New York:Spring Verlag,1992.
[15]Holm H.Gorenstein homological dimensions[J].J Pure Appl Algebra,2004,189(1-3):167-193.
[16]佟文廷.同调代数引论[M].北京:高等教育出版社,1998.
[2]Enochs E E,Jenda O M G.Gorenstein injective and projective modules[J].Mathematische Zeitschrift,1995,220(1):611-633.
[3]Holm H.Gorenstein homological dimension[J].Journal of Pure&Applied Algebra,2004,189(1):167-193.
[4]Gao Z H,Xu L Y.Gorenstein coresolving categories[J].Communications in Algebra,2016,45(10):4477-4491.
[5]Benni D,Ouarghi K.X-Gorenstein projective modules[J].Journal of Lanzhou University of Technology,2010,(9-12):487-491.
[6]Zhu X S.Resolving resolution dimension[J].Algebras&Representation Theory,2013,16(4):1165-1191.
[7]Enochs E E,Jenda O M G.Relative homological algebra[M].Berlin:Walter de Gruyter,2000.
[8]Bravo D,Gillespie J.Absolutely clean,level,and Gorenstein AC-injective complexes[J].Communications in Algebra,2014,44(5):2213-2233.
[9]Enochs E,Jenda O G,López-Ramos J A.Covers and envelopes by V-Gorenstein modules[J].Communications in Algebra,2005,33(12):4705-4717.
[10]Geng Y X,Ding N Q.W-Gorenstein modules[J].Journal of Algebra,2011,325(1):132-146.
[11]Huang Z Y.Proper resolutions and Gorenstein categories[J].Journal of Algebra,2013,393(11):142-169.
[12]Meng F,Pan Q.X-Gorenstein projective and Y-Gorenstein injective modules[J].Hacettepe University Bulletin of Natural Sciences&Engineering,2011,40(4):537-554.
[13]Enochs E E,Garcia Rozas J R.Gorenstein injective and projective complexes[J],Communications in Algebra,1998,26(5):1657-1674.
[14]Anderson F W,Fuller K R.Rings and categories of modules[M].New York:Spring Verlag,1992.
[15]Holm H.Gorenstein homological dimensions[J].J Pure Appl Algebra,2004,189(1-3):167-193.
[16]佟文廷.同调代数引论[M].北京:高等教育出版社,1998.