Complete cohomology for complexes with finite Gorenstein AC-projective dimension

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

References

[1] Asadollahi J, Salarian Sh. Cohomology theories for complexes. J Pure Appl Algebra, 2007, 210: 771–787

[2] Avramov L L, Foxby H-B. Homological dimensions of unbounded complexes. J Pure Appl Algebra, 1991, 71: 129–155

[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

[5] Benson D J, Carlson J F. Products in negative cohomology. J Pure Appl Algebra, 1992, 82: 107–130

[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

[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

[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

[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

[14] Enochs E E, Jenda O M G. Gorenstein injective and projective modules. Math Z, 1995, 220: 611–633

[15] Enochs E E, Jenda O M G. Relative Homological Algebra. Berlin-New York: Walter de Gruyter, 2000

[16] Gillespie J. The flat model structure on Ch(R). Trans Amer Math Soc, 2004, 356:3369–3390

[17] Gillespie J. Model structures on modules over Ding-Chen rings. Homology, Homotopy Appl, 2010, 12: 61–73

[18] Goichot F. Homologie de Tate-Vogel équivariante. J Pure Appl Algebra, 1992, 82:39–64

[19] Holm H. Gorenstein derived functors. Proc Amer Math Soc, 2004, 132: 1913–1923

[20] Hovey M. Cotorsion pairs and model categories. Contemp Math, 2007, 436: 277–296

[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

[22] Mislin G. Tate cohomology for arbitrary groups via satellites. Topology Appl, 1994, 56: 293–300

[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

[26] Veliche O. Gorenstein projective dimension for complexes. Trans Amer Math Soc, 2006, 358: 1257–1283

[27] Yang G, Liu Z K. Cotorsion pairs and model structures on Ch(R). Proc Edinb Math Soc, 2011, 54: 783–797

[28] Yang X Y, Ding N Q. On a question of Gillespie. Forum Math, 2015, 27(6): 3205–3231

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Higher Education Press and Springer-Verlag Berlin Heidelberg