Czechoslovak Mathematical Journal

, Volume 68, Issue 3, pp 741–754 | Cite as

Restricted homological dimensions over local homomorphisms and Cohen-Macaulayness

  • Fangdi Kong
  • Dejun Wu


We define and study restricted projective, injective and flat dimensions over local homomorphisms. Some known results are generalized. As applications, we show that (almost) Cohen-Macaulay rings can be characterized by restricted homological dimensions over local homomorphisms.


Cohen factorization restricted homological dimension Cohen-Macaulay ring 

MSC 2010

13D02 13D05 


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  1. [1]
    M. Auslander, M. Bridger: Stable Module Theory. Memoirs of the American Mathematical Society 94, American Mathematical Society, Providence, 1969.Google Scholar
  2. [2]
    L. L. Avramov, H.-B. Foxby: Homological dimensions of unbounded complexes. J. Pure Appl. Algebra 71 (1991), 129–155.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    L. L. Avramov, H.-B. Foxby, B. Herzog: Structure of local homomorphisms. J. Algebra 164 (1994), 124–145.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    L. L. Avramov, S. Iyengar, C. Miller: Homology over local homomorphisms. Am. J. Math. 128 (2006), 23–90.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    W. Bruns, J. Herzog: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993.Google Scholar
  6. [6]
    L. W. Christensen: Semi-dualizing complexes and their Auslander categories. Trans. Am. Math. Soc. 353 (2001), 1839–1883.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    L. W. Christensen, H.-B. Foxby, A. Frankild: Restricted homological dimensions and Cohen-Macaulayness. J. Algebra 251 (2002), 479–502.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    H.-B. Foxby, S. Iyengar: Depth and amplitude for unbounded complexes. Commutative algebra. Interactions with algebraic geometry (L. L. Avramov, M. Chardin, M. Morales, C. Polini, eds.). Contemporary Mathematics 331, American Mathematical Society, Providence, 2003, pp. 119–137.Google Scholar
  9. [9]
    S. Iyengar: Depth for complexes, and intersection theorems. Math. Z. 230 (1999), 545–567.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    S. Iyengar, S. Sather-Wagstaff: G-dimension over local homomorphisms. Applications to the Frobenius endomorphism. Ill. J. Math. 48 (2004), 241–272.zbMATHGoogle Scholar
  11. [11]
    D. Wu: Gorenstein dimensions over ring homomorphisms. Commun. Algebra 43 (2015), 2005–2028.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D. Wu, Z. Liu: On restricted injective dimensions of complexes. Commun. Algebra 41 (2013), 462–470.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.College of Electrical and Information EngineeringLanzhou University of TechnologyLanzhou, GansuChina
  2. 2.Department of Applied MathematicsLanzhou University of TechnologyLanzhou, GansuChina

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