A note on a question of Bass
Author:
Sankar P. Dutta
Journal:
Trans. Amer. Math. Soc. 290 (1985), 691700
MSC:
Primary 13D05; Secondary 13D03, 13H10
MathSciNet review:
792820
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Abstract: In this paper we study the following question raised by Bass: Is a local ring with a finitely generated module of finite injective dimension CohenMacaulay? We prove that the answer is in the affirmative when a certain local cohomology of the ring is either decomposible or cyclic. We also study the above question in some special cases and some of its implications.
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0269685 (42 #4580)
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Hyman
Bass, On the ubiquity of Gorenstein rings, Math. Z.
82 (1963), 8–28. MR 0153708
(27 #3669)
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Robert
Fossum, HansBjørn
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Griffith, and Idun
Reiten, Minimal injective resolutions with applications to
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Sci. Publ. Math. 45 (1975), 193–215. MR 0396529
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II, Math. Scand. 41 (1977), no. 1, 19–44.
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0476801 (57 #16355)
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Rodney
Y. Sharp, The construction of a module of finite projective
dimension from a finitely generated module of finite injective
dimension, Comment. Math. Helv. 50 (1975),
15–26. MR
0369344 (51 #5577)
 [AB]
 M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. No. 94 (1969). MR 0269685 (42:4580)
 [B]
 H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 828. MR 0153708 (27:3669)
 [FFGR]
 R. Fossum, H. B. Foxby, P. Griffith and I. Reiten, Minimal injective resolutions with applications to dualizing modules and Gorenstein modules, I.H.E.S. Publ. Math. No. 45, 1975, pp. 193215. MR 0396529 (53:392)
 [F]
 H. B. Foxby, On the in a minimal injective resolution. II, Math. Scand. 41 (1977), 1944. MR 0476801 (57:16355)
 [Ha]
 R. Hartshorne, Residues and duality, vol. 20, SpringerVerlag, Berlin, Heideberg and New York, 1966. MR 0222093 (36:5145)
 [H]
 M. Hochster, Topics in the homological theory of modules over commutative rings, C.B.M.S. Regional Conf. Ser. Math., No. 24, Amer. Math. Soc., Providence, R. I., 1975. MR 0371879 (51:8096)
 [M]
 M. P. Murphy, A note on factorial rings, Arch. Math. 15 (1964), 418420. MR 0173695 (30:3905)
 [PS]
 C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, I.H.E.S. Publ. Math. No. 42, 1973, pp. 49119.
 [S]
 R. Y. Sharp, The construction of a module of finite projective dimension from a finitely generated module of finite injective dimension, Comment. Math. Helv. 50 (1975), 1526. MR 0369344 (51:5577)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507928205
PII:
S 00029947(1985)07928205
Article copyright:
© Copyright 1985
American Mathematical Society
