The homomorphismM*⊗N → Hom (M,N)



In this note we prove two theorems. In theorem 1 we prove that if M andN are two non-zero reflexive modules of finite projective dimensions over a Gorenstein local ring, such that Hom (M, N) is a third module of syzygies, then the natural homomorphismM* ⊗N → Hom (M, N) is an isomorphism. This extends the result in [7]. In theorem 2, we prove that projective dimension of a moduleM over a regular local ringR is less than or equal ton if and only if ExtR n (M, R) ⊗M → ExtR n (M, M) is surjective; in which case it is actually bijective. This extends the usual criterion for the projectivity of a module.


Homomorphism Gorenstein local ring isomorphism projectivity 


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  1. [1]
    Auslander M and Bridger M 1969 Stable module theory.Am. Math. Soc., Providence, Rhode IslandGoogle Scholar
  2. [2]
    Auslander M and Goldman O 1960 Maximal orders.Trans. Am. Math. Soc. 97 Google Scholar
  3. [3]
    Bass H 1963 on the ubiquity of Gorenstein rings.Math. Z. 82 Google Scholar
  4. [4]
    Rees D 1957 The grade of an ideal or module.Proc. Cambridge Philos. Soc. 84 Google Scholar
  5. [5]
    Licheten baum S 1966 On the vanishing of Tor in regular local rings.Ill. J. Math. 10 Google Scholar
  6. [6]
    Jinnah M I 1973 Reflexive modules over regular local rings.Arch. Math. 26 Google Scholar
  7. [7]
    Jothilingam P 1975 A note on grade.Nagoya Math. J. 59 Google Scholar
  8. [8]
    Serre J P 1965 Algebre Locale—Multiplicites.Lecture Notes in Mathematics, No. 11 (Berlin: Springer-Verlag)Google Scholar

Copyright information

© Indian Academy of Sciences 1985

Authors and Affiliations

  • S. Bose
    • 1
  1. 1.Department of MathematicsNorth-Eastern Hill UniversityShillongIndia

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