复形的C-Gorenstein投射维数
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摘要
设R是一个有单位元的结合环,C是一个关于直和封闭且包含所有投射模的左R-模类。介绍左R-模复形的C-Gorenstein投射维数的概念,它是复形的Gorenstein投射维数的一个推广。利用环模理论和同调代数的方法,讨论复形X的C-Gorenstein投射维数C-Gpd(X)与其每个层次上模Xm的C-Gorenstein投射维数C-Gpd(X~m)之间的关系,给出复形X的C-Gorenstein投射维数小于等于n的若干等价刻画。证明了C-Gpd(X)=sup{C-Gpd(X~m) m∈Ζ},且当C-Gpd(X)=n(n≥1)时,存在复形短正合列0→H→G→X→0和0→X→H'→G'→0,其中G,G'为C-Gorenstein投射复形,H的投射维数小于等于n-1且H'的投射维数小于等于n。
Let R be an associative ring with unit cells,C is a class of left R-modules that is closed under direct sums and contains all projective modules. The paper introduces the notion of C-Gorenstein projective dimensions of complexes of left R-modules,it is a generalization of Gorenstein projective dimension of complex. Using theories of rings and modules,and methods of homological algebras,relations between C-Gorenstein projective dimension C-Gpd(X) of complex X and C-Gorenste in projective dimension C-Gpd(X~m) of every term module Xmare discussed,some equivalent characterizations of Gorenstein projective dimension of complex X less than or equal to n are given. It proves C-Gpd(X) = sup{C-Gpd(X~m) | m ∈Z},and there exist short exact sequences of complexes 0 → H → G → X → 0 and 0 → X → H' → G' → 0,with G,G' C-Gorenstein projection complexes,projection dimension of H' is less than or equal to n-1 and projection dimension of H' less than or equal to n,when C-Gpd(X) = n(n≥1).
Let R be an associative ring with unit cells,C is a class of left R-modules that is closed under direct sums and contains all projective modules. The paper introduces the notion of C-Gorenstein projective dimensions of complexes of left R-modules,it is a generalization of Gorenstein projective dimension of complex. Using theories of rings and modules,and methods of homological algebras,relations between C-Gorenstein projective dimension C-Gpd(X) of complex X and C-Gorenste in projective dimension C-Gpd(X~m) of every term module Xmare discussed,some equivalent characterizations of Gorenstein projective dimension of complex X less than or equal to n are given. It proves C-Gpd(X) = sup{C-Gpd(X~m) | m ∈Z},and there exist short exact sequences of complexes 0 → H → G → X → 0 and 0 → X → H' → G' → 0,with G,G' C-Gorenstein projection complexes,projection dimension of H' is less than or equal to n-1 and projection dimension of H' less than or equal to n,when C-Gpd(X) = n(n≥1).
引文
[1]AUSLANDER M,BRIDGER M.Stable module theory[J].Memoirs of the American Mathematical Society,1969.doi:10.1090/memo/0094.
[2]ENOCHS E,JENDA O.Gorenstein injective and projective modules[J].Mathematische Zeitschrift,1995,220(1):611-633.
[3]BENNIS D,OUARGHI K.X-Gorenstein projective modules[J].Journal of Lanzhou University of Technology,2010(9-12):487-491.
[4]YANG X Y,LIU Z K.Gorenstein projective,injective and flat complexes[J].Comm Algebra,2011,39:1705-1721.
[5]ENOCHS E,JENDA O G,LPEZ-RAMOS J A.Covers and envelopes by V-Gorenstein modules[J].Communications in Algebra,2005,33(12):4705-4717.
[6]GENG Y,DING N.$scr W$-G orenstein modules[J].Journal of Algebra,2011,28(3):132-146.
[7]HUANG Z.Proper resolutions and Gorenstein categories[J].Journal of Algebra,2013,393(11):142-169.
[8]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.
[9]ENOCHS E,ROZAS J.Gorenstein injective and projective complexes[J].Communications in Algebra,1998,26(5):1657-1674.
[10]GARCA R J R.Covers and envelopes in the category of complexes of modules[M].Chapman&Hall/CRC Press,1999.
[11]杨刚.Gorenstein投射、投射和平坦复形[J].数学学报:中文版,2011,54(3):451-459.
[12]ZHAO R Y,REN W.VW-Gorenstein complexes[J].Turkish Journal of Mathematics,2017,41(3):537-547.
[13]WEI R.Gorenstein projective modules and Frobenius extensions[J].Science China,2018,61(7):1175-1186.
[14]GILLESPIE J.Gorenstein AC-projective complexes[J].Journal of Homotopy&Related Structures,2017,13(4):769-791.
[15]ESTRADA S,IACOB A,YEOMANS K.Gorenstein projective precovers[J].Mediterranean Journal of Mathematics,2017,14(1):33.
[2]ENOCHS E,JENDA O.Gorenstein injective and projective modules[J].Mathematische Zeitschrift,1995,220(1):611-633.
[3]BENNIS D,OUARGHI K.X-Gorenstein projective modules[J].Journal of Lanzhou University of Technology,2010(9-12):487-491.
[4]YANG X Y,LIU Z K.Gorenstein projective,injective and flat complexes[J].Comm Algebra,2011,39:1705-1721.
[5]ENOCHS E,JENDA O G,LPEZ-RAMOS J A.Covers and envelopes by V-Gorenstein modules[J].Communications in Algebra,2005,33(12):4705-4717.
[6]GENG Y,DING N.$scr W$-G orenstein modules[J].Journal of Algebra,2011,28(3):132-146.
[7]HUANG Z.Proper resolutions and Gorenstein categories[J].Journal of Algebra,2013,393(11):142-169.
[8]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.
[9]ENOCHS E,ROZAS J.Gorenstein injective and projective complexes[J].Communications in Algebra,1998,26(5):1657-1674.
[10]GARCA R J R.Covers and envelopes in the category of complexes of modules[M].Chapman&Hall/CRC Press,1999.
[11]杨刚.Gorenstein投射、投射和平坦复形[J].数学学报:中文版,2011,54(3):451-459.
[12]ZHAO R Y,REN W.VW-Gorenstein complexes[J].Turkish Journal of Mathematics,2017,41(3):537-547.
[13]WEI R.Gorenstein projective modules and Frobenius extensions[J].Science China,2018,61(7):1175-1186.
[14]GILLESPIE J.Gorenstein AC-projective complexes[J].Journal of Homotopy&Related Structures,2017,13(4):769-791.
[15]ESTRADA S,IACOB A,YEOMANS K.Gorenstein projective precovers[J].Mediterranean Journal of Mathematics,2017,14(1):33.