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Embedding of modules over Gorenstein rings


Author: Hans-Bjørn Foxby
Journal: Proc. Amer. Math. Soc. 36 (1972), 336-340
MSC: Primary 13H10; Secondary 13C99
DOI: https://doi.org/10.1090/S0002-9939-1972-0309930-5
MathSciNet review: 0309930
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Abstract: Let A be a noetherian commutative ring which is a homomorphic image of a Gorenstein ring. The first result in this note is that A is a Cohen-Macaulay ring if and only if every finitely generated A-module can be embedded in a finitely generated A-module T which is pointwise of finite injective dimension (i.e. all localizations of T are of finite injective dimension). The second result is that if A is Cohen-Macaulay, then every finitely generated A-module can be embedded in a finitely generated A-module of finite projective dimension if and only if A is Gorenstein.


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  • [1] M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. No. 94 (1969). MR 42 #4580. MR 0269685 (42:4580)
  • [2] H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28. MR 27 # 3669. MR 0153708 (27:3669)
  • [3] H. Bass, Algebraic K-theory, Benjamin, New York, 1968. MR 40 #2736. MR 0249491 (40:2736)
  • [4] N. Bourbaki, Éléments de mathématique. Fasc. XXVII. Algèbre commutative. Chap. 2: Localisation, Actualités Sci. Indust., no. 1290, Hermann, Paris, 1961. MR 36 #146.
  • [5] I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54-106. MR 7, 509. MR 0016094 (7:509h)
  • [6] H.-B. Foxby, On the $ {\mu ^i}$ in a minimal injective resolution, Math. Scand. 29 (1972), 175-186. MR 0309919 (46:9023)
  • [7] -, Gorenstein modules and related modules, Math. Scand. (to appear). MR 0327752 (48:6094)
  • [8] G. Levin and W. V. Vasconcelos, Homological dimensions and Macaulay rings, Pacific J. Math. 25 (1968), 315-323. MR 37 #6275. MR 0230715 (37:6275)
  • [9] C. Peskine and L. Szpiro, Modules de type et de dimension injective finie sur un anneau local noethérien, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A1117-A1120. MR 38 #3266. MR 0234952 (38:3266)
  • [10] R. Y. Sharp, Gorenstein modules, Math. Z. 115 (1970), 117-139, MR 41 #8401. MR 0263801 (41:8401)
  • [11] -, Finitely-generated modules of finite injective dimension over certain Cohen-Macaulay rings, Proc. London Math. Soc. 25 (1972), 303-328. MR 0306188 (46:5315)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0309930-5
Keywords: Cohen-Macaulay rings, Gorenstein rings, G-dimension
Article copyright: © Copyright 1972 American Mathematical Society

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