Bulletin of the Korean Mathematical Society (대한수학회보)

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GORENSTEIN DIMENSIONS OF UNBOUNDED COMPLEXES UNDER BASE CHANGE

  • Wu, Dejun (Department of Mathematics Shanghai Jiao Tong University)
  • Received : 2015.04.29
  • Published : 2016.05.31
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Abstract

Transfer of homological properties under base change is a classical field of study. Let $R{\rightarrow}S$ be a ring homomorphism. The relations of Gorenstein projective (or Gorenstein injective) dimensions of unbounded complexes between $U{\otimes}^L_RX$(or $RHom_R(X,U)$) and X are considered, where X is an R-complex and U is an S-complex. In addition, some sufficient conditions are given under which the equalities $G-dim_S(U{\otimes}^L_RX)=G-dim_RX+pd_SU$ and $Gid_S(RHom_R(X,U))=G-dim_RX+id_SU$ hold.

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References

  1. J. Asadollahi and S. Salarian, Gorenstein injective dimension for complexes and Iwanaga-Gorenstein rings, Comm. Algebra 34 (2006), no. 8, 3009-3022. https://doi.org/10.1080/00927870600639815
  2. L. L. Avramov and H-B. Foxby, Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71 (1991), no. 2-3, 129-155. https://doi.org/10.1016/0022-4049(91)90144-Q
  3. L. W. Christensen, Gorenstein Dimensions, Lecture Notes in Math. 1747. Springer, Berlin, 2000.
  4. L. W. Christensen, A. Frankild, and H. Holm, On Gorenstein projective, injective and flat dimensions-A functorial description with applications, J. Algebra 302 (2006), no. 1, 231-279. https://doi.org/10.1016/j.jalgebra.2005.12.007
  5. L. W. Christensen and H. Holm, Ascent properties of Auslander categories, Canad. J. Math. 61 (2009), no. 1, 76-108. https://doi.org/10.4153/CJM-2009-004-x
  6. L. W. Christensen and D. A. Jorgensen, Vanishing of Tate homology and depth formulas over local rings, J. Pure Appl. Algebra 219 (2015), no. 3, 464-481. https://doi.org/10.1016/j.jpaa.2014.05.005
  7. L. W. Christensen and S. Sather-Wagstaff, Transfer of Gorenstein dimensions along ring homomorphisms, J. Pure Appl. Algebra 214 (2010), no. 6, 982-989. https://doi.org/10.1016/j.jpaa.2009.09.007
  8. H.-B. Foxby and S. Iyengar, Depth and amplitude for unbounded complexes, Commutative algebra (Grenoble/Lyon, 2001), 119-137, Contemp. Math., 331, Amer. Math. Soc., Providence, RI, 2003.
  9. Z. Liu and Wei Ren, Transfer of Gorenstein dimensions of unbounded complexes along ring homomorphisms, Comm. Algebra 42 (2014), no. 8, 3325-3338. https://doi.org/10.1080/00927872.2013.783039
  10. O. Veliche, Gorenstein projective dimension for complexes, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1257-1283. https://doi.org/10.1090/S0002-9947-05-03771-2
  11. D. Wu, Gorenstein dimensions over ring homomorphisms, Comm. Algebra 43 (2015), no. 5, 2005-2028. https://doi.org/10.1080/00927872.2014.881836
  12. D. Wu and Z. Liu, Vanishing of Tate cohomology and Gorenstein injective dimension, Comm. Algebra 42 (2014), no. 5, 2181-2194. https://doi.org/10.1080/00927872.2013.791303