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On Gorenstein Projective Dimensions of Unbounded Complexes

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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.

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References

  1. Avramov, L. L. and Foxby, H.-B., Homological dimensions of unbounded complexes, J. Pure Appl. Algebra, 71, 1991, 129–155.

    Article  MathSciNet  Google Scholar 

  2. Bennis, D., Rings over which the class of Gorenstein flat modules is closed under extensions, Comm. Algebra, 37, 2009, 855–868.

    Article  MathSciNet  Google Scholar 

  3. Christensen, L. W., Gorenstein Dimensions, Lecture Notes in Mathematics, 1747, Springer-Verlag, Berlin, 2000.

    Google Scholar 

  4. Christensen, L. W., Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc., 353, 2001, 1839–1883.

    Article  MathSciNet  Google Scholar 

  5. 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.

    Article  MathSciNet  Google Scholar 

  6. Christensen, L. W. and Holm, H., Ascent properties of Auslander categories, Canad. J. Math., 61, 2009, 76–108.

    Article  MathSciNet  Google Scholar 

  7. 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.

    Article  MathSciNet  Google Scholar 

  8. Christensen, L. W. and Sather-Wagstaff, S., Transfer of Gorenstein dimensions along ring homomorphisms, J. Pure Appl. Algebra, 214, 2010, 982–989.

    Article  MathSciNet  Google Scholar 

  9. Ding, N. Q., Li, Y. L. and Mao, L. X., Strongly Gorenstein flat modules, J. Aust. Math. Soc., 86, 2009, 323–338.

    Article  MathSciNet  Google Scholar 

  10. Enochs, E. E. and Jenda, O. M. G., Relative homological algebra, de Gruyter Expositions in Mathematics, 30, Walter de Gruyter, Berlin, 2000.

    Book  Google Scholar 

  11. Esmkhani, M. A. and Tousi, M., Gorenstein homological dimensions and Auslander categories, J. Algebra, 308, 2007, 321–329.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Holm, H., Gorenstein homological dimensions, J. Pure Appl. Algebra, 189, 2004, 167–193.

    Article  MathSciNet  Google Scholar 

  14. Iacob, A., Gorenstein flat dimension of complexes, J. Math. Kyoto Univ., 49, 2009, 817–842.

    Article  MathSciNet  Google Scholar 

  15. Iyengar, S. and Sather-Wagstaff, S., G-dimension over local homomorphisms, applications to the Frobenius endomorphism, Illinois J. Math., 48, 2004, 241–272.

    Article  MathSciNet  Google Scholar 

  16. Liu, Z. K. and Ren, W., Transfer of Gorenstein dimensions of unbounded complexes along ring homomorphisms, Comm. Algebra, 42, 2014, 3325–3338.

    Article  MathSciNet  Google Scholar 

  17. Mahdou, N. and Tamekkante, M., Strongly Gorenstein flat modules and dimensions, Chin. Ann. Math. Ser. B, 32, 2011, 533–548.

    Article  MathSciNet  Google Scholar 

  18. Veliche, O., Gorenstein projective dimension for complexes, Trans. Amer. Math. Soc., 358, 2006, 1257–1283.

    Article  MathSciNet  Google Scholar 

  19. Wang, Z. P. and Liu, Z. K., Stability of strongly Gorenstein flat modules, Vietnam J. Math., 42, 2014, 171–178.

    Article  MathSciNet  Google Scholar 

  20. Wang, Z. P. and Liu, Z. K., Strongly Gorenstein flat dimensions of complexes, Comm. Algebra, 44, 2016, 1390–1410.

    Article  MathSciNet  Google Scholar 

  21. Wu, D. J., Gorenstein dimensions over ring homomorphisms, Comm. Algebra, 43, 2015, 2005–2028.

    Article  MathSciNet  Google Scholar 

  22. Zhang, C. X. and Liu, Z. K., Rings with finite Ding homological dimensions, Turkish J. Math., 39, 2015, 37–48.

    Article  MathSciNet  Google Scholar 

  23. 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.

    Article  MathSciNet  Google Scholar 

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Acknowledgement

The authors are grateful to the referee for many useful suggestions that improve significantly the exposition of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

    Zhongkui Liu & Zhanping Wang

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  1. Zhongkui Liu
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  2. Zhanping Wang
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Correspondence to Zhanping Wang.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11261050, 11561061).

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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

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  • DOI: https://doi.org/10.1007/s11401-020-0232-7

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