Abstract
Let R → S be a ring homomorphism and X be a complex of R-modules. Then the complex of S-modules S ⊗LR X in the derived category D(S) is constructed in the natural way. This paper is devoted to dealing with the relationships of the Gorenstein projective dimension of an R-complex X (possibly unbounded) with those of the S-complex S ⊗LR X. It is shown that if R is a Noetherian ring of finite Krull dimension and ϕ: R → S is a faithfully flat ring homomorphism, then for any homologically degree-wise finite complex X, there is an equality GpdRX = GpdS(S ⊗LR X). Similar result is obtained for Ding projective dimension of the S-complex S ⊗LR X.
Similar content being viewed by others
References
Avramov, L. L. and Foxby, H.-B., Homological dimensions of unbounded complexes, J. Pure Appl. Algebra, 71, 1991, 129–155.
Bennis, D., Rings over which the class of Gorenstein flat modules is closed under extensions, Comm. Algebra, 37, 2009, 855–868.
Christensen, L. W., Gorenstein Dimensions, Lecture Notes in Mathematics, 1747, Springer-Verlag, Berlin, 2000.
Christensen, L. W., Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc., 353, 2001, 1839–1883.
Christensen, L. W., Frankild, A. and Holm, H., On Gorenstein projective, injective and flat dimensions—A functorial description with applications, J. Algebra, 302, 2006, 231–279.
Christensen, L. W. and Holm, H., Ascent properties of Auslander categories, Canad. J. Math., 61, 2009, 76–108.
Christensen, L. W., Koksal, F. and Liang, L., Gorenstein dimensions of unbounded complexes and change of base (with an appendix by Driss Bennis), Sci. China Math., 60, 2017, 401–420.
Christensen, L. W. and Sather-Wagstaff, S., Transfer of Gorenstein dimensions along ring homomorphisms, J. Pure Appl. Algebra, 214, 2010, 982–989.
Ding, N. Q., Li, Y. L. and Mao, L. X., Strongly Gorenstein flat modules, J. Aust. Math. Soc., 86, 2009, 323–338.
Enochs, E. E. and Jenda, O. M. G., Relative homological algebra, de Gruyter Expositions in Mathematics, 30, Walter de Gruyter, Berlin, 2000.
Esmkhani, M. A. and Tousi, M., Gorenstein homological dimensions and Auslander categories, J. Algebra, 308, 2007, 321–329.
Gillespie, J., Model structures on modules over Ding-Chen rings, Homology, Homotopy Appl., 12, 2010, 61–73.
Holm, H., Gorenstein homological dimensions, J. Pure Appl. Algebra, 189, 2004, 167–193.
Iacob, A., Gorenstein flat dimension of complexes, J. Math. Kyoto Univ., 49, 2009, 817–842.
Iyengar, S. and Sather-Wagstaff, S., G-dimension over local homomorphisms, applications to the Frobenius endomorphism, Illinois J. Math., 48, 2004, 241–272.
Liu, Z. K. and Ren, W., Transfer of Gorenstein dimensions of unbounded complexes along ring homomorphisms, Comm. Algebra, 42, 2014, 3325–3338.
Mahdou, N. and Tamekkante, M., Strongly Gorenstein flat modules and dimensions, Chin. Ann. Math. Ser. B, 32, 2011, 533–548.
Veliche, O., Gorenstein projective dimension for complexes, Trans. Amer. Math. Soc., 358, 2006, 1257–1283.
Wang, Z. P. and Liu, Z. K., Stability of strongly Gorenstein flat modules, Vietnam J. Math., 42, 2014, 171–178.
Wang, Z. P. and Liu, Z. K., Strongly Gorenstein flat dimensions of complexes, Comm. Algebra, 44, 2016, 1390–1410.
Wu, D. J., Gorenstein dimensions over ring homomorphisms, Comm. Algebra, 43, 2015, 2005–2028.
Zhang, C. X. and Liu, Z. K., Rings with finite Ding homological dimensions, Turkish J. Math., 39, 2015, 37–48.
Zhang, C. X., Wang, L. M. and Liu, Z. K., Ding projective modules with respect to a semidualizing bimodule, Rocky Mountain J. Math., 45, 2015, 1389–1411.
Acknowledgement
The authors are grateful to the referee for many useful suggestions that improve significantly the exposition of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (Nos. 11261050, 11561061).
Rights and permissions
About this article
Cite this article
Liu, Z., Wang, Z. On Gorenstein Projective Dimensions of Unbounded Complexes. Chin. Ann. Math. Ser. B 41, 761–772 (2020). https://doi.org/10.1007/s11401-020-0232-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-020-0232-7