Complete cohomology for complexes with finite Gorenstein AC-projective dimension

doi:10.1007/s11464-016-0564-5

Front. Math. China

2016, Vol. 11

Issue (4) : 901-920 https://doi.org/10.1007/s11464-016-0564-5

RESEARCH ARTICLE

Complete cohomology for complexes with finite Gorenstein AC-projective dimension

Jiangsheng HU^1,²,Yuxian GENG^2,^*(

),Qinghua JIANG³

¹. School of Mathematics Sciences, Nanjing Normal University, Nanjing 210046, China
². School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China
³. School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China

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Abstract

We study complete cohomology of complexes with finite Gorenstein AC-projective dimension. We show first that the class of complexes admitting a complete level resolution is exactly the class of complexes with finite Gorenstein AC-projective dimension. This lets us give some general techniques for computing complete cohomology of complexes with finite Gorenstein ACprojective dimension. As a consequence, the classical relative cohomology for modules of finite Gorenstein AC-projective dimension is extended. Finally, the relationships between projective dimension and Gorenstein AC-projective dimension for complexes are given.

Keywords Gorenstein AC-projective complete level resolution complete cohomology

Corresponding Author(s): Yuxian GENG

Issue Date: 30 August 2016

Cite this article:

Jiangsheng HU,Yuxian GENG,Qinghua JIANG. Complete cohomology for complexes with finite Gorenstein AC-projective dimension[J]. Front. Math. China, 2016, 11(4): 901-920.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0564-5
https://academic.hep.com.cn/fmc/EN/Y2016/V11/I4/901

1	Asadollahi J, Salarian Sh. Cohomology theories for complexes. J Pure Appl Algebra, 2007, 210: 771–787 https://doi.org/10.1016/j.jpaa.2006.11.014
2	Avramov L L, Foxby H-B. Homological dimensions of unbounded complexes. J Pure Appl Algebra, 1991, 71: 129–155 https://doi.org/10.1016/0022-4049(91)90144-Q
3	Avramov L L, Foxby H-B, Halperin S. Differential Graded Homological Algebra. Preprint, 2009
4	Avramov L L, Martsinkovsky A. Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension. Proc Lond Math Soc, 2002, 85: 393–440 https://doi.org/10.1112/S0024611502013527
5	Benson D J, Carlson J F. Products in negative cohomology. J Pure Appl Algebra, 1992, 82: 107–130 https://doi.org/10.1016/0022-4049(92)90116-W
6	Bravo D, Hovey M, Gillespie J. The stable module category of a general ring. arXiv:1405.5768
7	Butler M C R, Horrocks G. Classes of extensions and resolutions. Philos Trans R Soc Lond Ser A, 1961/1962, 254: 155–222 https://doi.org/10.1098/rsta.1961.0014
8	Cartan H, Eilenberg S. Homological Algebra. Princeton; Princeton Univ Press, 1956
9	Christensen L W. Gorenstein Dimensions. Lecture Notes in Math, Vol 1747. Berlin:Springer-Verlag, 2000 https://doi.org/10.1007/BFb0103980
10	Christensen L W, Foxby H-B, Holm H. Derived Category Methods in Commutative Algebra. Preprint, 2012
11	Christensen L W, Frankild A, Holm H. On Gorenstein projective, injective and flat dimensions—a functorial description with applications. J Algebra, 2006, 302: 231–279 https://doi.org/10.1016/j.jalgebra.2005.12.007
12	Eilenberg S, Moore J C. Foundations of Relative Homological Algebra. Mem Amer Math Soc, No 55. Providence: Amer Math Soc, 1965
13	Eklof P, Trlifaj J. How to make Ext vanish. Bull Lond Math Soc, 2001, 33: 41–51 https://doi.org/10.1112/blms/33.1.41
14	Enochs E E, Jenda O M G. Gorenstein injective and projective modules. Math Z, 1995, 220: 611–633 https://doi.org/10.1007/BF02572634
15	Enochs E E, Jenda O M G. Relative Homological Algebra. Berlin-New York: Walter de Gruyter, 2000 https://doi.org/10.1515/9783110803662
16	Gillespie J. The flat model structure on Ch(R). Trans Amer Math Soc, 2004, 356:3369–3390 https://doi.org/10.1090/S0002-9947-04-03416-6
17	Gillespie J. Model structures on modules over Ding-Chen rings. Homology, Homotopy Appl, 2010, 12: 61–73 https://doi.org/10.4310/HHA.2010.v12.n1.a6
18	Goichot F. Homologie de Tate-Vogel équivariante. J Pure Appl Algebra, 1992, 82:39–64 https://doi.org/10.1016/0022-4049(92)90009-5
19	Holm H. Gorenstein derived functors. Proc Amer Math Soc, 2004, 132: 1913–1923 https://doi.org/10.1090/S0002-9939-04-07317-4
20	Hovey M. Cotorsion pairs and model categories. Contemp Math, 2007, 436: 277–296 https://doi.org/10.1090/conm/436/08413
21	Hu J S, Ding N Q. A model structure approach to the Tate-Vogel cohomology. J Pure Appl Algebra, 2016, 220(6): 2240–2264 https://doi.org/10.1016/j.jpaa.2015.11.004
22	Mislin G. Tate cohomology for arbitrary groups via satellites. Topology Appl, 1994, 56: 293–300 https://doi.org/10.1016/0166-8641(94)90081-7
23	Salce L. Cotorsion theories for abelian groups. Symposia Math, 1979, 23: 11–32
24	Sather-Wagstaff S, Sharif T, White D. Gorenstein cohomology in abelian categories. J Math Kyoto Univ, 2008, 48: 571–596
25	Sather-Wagstaff S, Sharif T, White D. AB-contexts and stability for Gorenstein flat modules with respect to semidualizing modules. Algebr Represent Theory, 2011, 14:403–428 https://doi.org/10.1007/s10468-009-9195-9
26	Veliche O. Gorenstein projective dimension for complexes. Trans Amer Math Soc, 2006, 358: 1257–1283 https://doi.org/10.1090/S0002-9947-05-03771-2
27	Yang G, Liu Z K. Cotorsion pairs and model structures on Ch(R). Proc Edinb Math Soc, 2011, 54: 783–797 https://doi.org/10.1017/S0013091510000489
28	Yang X Y, Ding N Q. On a question of Gillespie. Forum Math, 2015, 27(6): 3205–3231 https://doi.org/10.1515/forum-2013-6014

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