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HOMOLOGY THEORIES FOR COMPLEXES BASED ON FLATS

Part of: Homological algebra

Published online by Cambridge University Press:  02 December 2019

LI LIANG
LI LIANG*
Affiliation:
School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China e-mail: lliangnju@gmail.com
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Abstract

In this paper, we introduce and study the Gorenstein relative homology theory for unbounded complexes of modules over arbitrary associative rings, which is defined using special Gorenstein flat precovers. We compare the Gorenstein relative homology to the Tate/unbounded homology and get some results that improve the known ones.

MSC classification

Primary: 18G15: Ext and Tor, generalizations, Künneth formula
Secondary: 18G25: Relative homological algebra, projective classes 18G35: Chain complexes
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 1 , January 2021 , pp. 27 - 44
Copyright
© Glasgow Mathematical Journal Trust 2019

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References

REFERENCES

Auslander, M. and Buchweitz, R.-O., The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.) 38 (1989), 5–37.CrossRefGoogle Scholar
Avramov, L. L. and Martsinkovsky, A., Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. 85 (2002), 393–440.CrossRefGoogle Scholar
Bennis, D., Rings over which the class of Gorenstein flat modules is closed under extensions, Comm. Algebra 37 (2009), 855–868.CrossRefGoogle Scholar
Celikbas, O., Christensen, L. W., Liang, L. and Piepmeyer, G., Stable homology over associate rings, Trans. Amer. Math. Soc. 369 (2017), 8061–8086.Google Scholar
Christensen, L. W. and Jorgensen, D. A., Tate (co)homology via pinched complexes, Trans. Amer. Math. Soc. 366 (2014), 667–689.Google Scholar
Enochs, E. E., Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189–209.CrossRefGoogle Scholar
Garcia Rozas, J. R., Covers and envelopes in the category of complexes of modules (CRC Press, Boca Raton-London-New York-Washington, D.C., 1999).Google Scholar
Gillespie, J., The flat model structure on, Ch(R), Trans. Amer. Math. Soc. 356 (2004), 3369–3390.CrossRefGoogle Scholar
Holm, H., Gorenstein homological dimensions, J. Pure and Appl. Algebra 189 (2004), 167–193.CrossRefGoogle Scholar
Holm, H., Gorenstein derived functors, Proc. of Amer. Math. Soc. 132 (2004), 1913–1923.CrossRefGoogle Scholar
Iacob, A., Absolute, Gorenstein, and Tate torsion modules, Comm. Algebra 35 (2007), 1589–1606.Google Scholar
Iacob, A., Gorenstein flat dimension of complexes, J. Math. Kyoto Univ. 49 (2009), 817–842.CrossRefGoogle Scholar
Jensen, C. U., On the vanishing of $\underset{\longleftarrow}{\lim}^{(i)},$ , J. Algebra 15 (1970), 151–166.CrossRefGoogle Scholar
Liang, L., Tate homology of modules of finite Gorenstein flat dimension, Algebr. Represent. Theory 16 (2013), 1541–1560.CrossRefGoogle Scholar
Liang, L., Homology theories and Gorenstein dimensions for complexes, preprint, arXiv:1808.07685v2[math.RT].Google Scholar
Liang, L., Ding, N. Q. and Yang, G., Some remarks on projective generators and injective cogenerators, Acta Math. Sin. (Engl. Ser.) 30 (2014), 2063–2078.CrossRefGoogle Scholar
Liu, Z. K., Relative cohomology of complexes, J. Algebra 502 (2018), 79–97.CrossRefGoogle Scholar
Osofsky, B. L., Homological dimension and cardinality, Trans. Amer. Math. Soc. 151 (1970), 641–649.CrossRefGoogle Scholar
Šaroch, J. and Št’ovíček, J., Singular compactness and definability for Σ-cotorsion and Gorenstein modules, preprint, arXiv:1804.09080v2[math.RT].Google Scholar
Veliche, O., Gorenstein projective dimension for complexes, Trans. Amer. Math. Soc. 358 (2006), 1257–1283.CrossRefGoogle Scholar
Xu, J. Z., Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, (Springer-Verlag, Berlin, 1996).CrossRefGoogle Scholar
Yang, G. and Liu, Z. K., Cotorsion pairs and model structures on Ch(R), Proc. Edinb. Math. Soc. 54 (2011), 783–797.CrossRefGoogle Scholar
Yang, G. and Liu, Z. K., Stability of Gorenstein flat categories, Glasgow Math. J. 54 (2012), 177–191.CrossRefGoogle Scholar
Yang, X. Y. and Liu, Z. K., Gorenstein projective, injective and flat complexes, Comm. Algebra 39 (2011), 1705–1721.CrossRefGoogle Scholar