The converse to a theorem of Sharp on Gorenstein modules

Reiten, Idun

doi:10.1090/S0002-9939-1972-0296067-7

The converse to a theorem of Sharp on Gorenstein modules
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Proc. Amer. Math. Soc. 32 (1972), 417-420 Request permission

Abstract:

Let A be a commutative local Noetherian ring with identity of Krull dimension n, m its maximal ideal. Sharp has proved that if A is Cohen-Macauley and a homomorphic image of a Gorenstein local ring, then A has a Gorenstein module M with ${\dim _{A/m}}\operatorname {Ext}^n(A/m,M) = 1$. The aim of this note is to prove the converse to this theorem.

References

Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 153708, DOI 10.1007/BF01112819
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Tor H. Gulliksen, Massey operations and the Poincaré series of certain local rings, J. Algebra 22 (1972), 223–232. MR 306190, DOI 10.1016/0021-8693(72)90143-3
Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. MR 99360
Bruno J. Müller, On Morita duality, Canadian J. Math. 21 (1969), 1338–1347. MR 255597, DOI 10.4153/CJM-1969-147-7
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
Rodney Y. Sharp, The Cousin complex for a module over a commutative Noetherian ring, Math. Z. 112 (1969), 340–356. MR 263800, DOI 10.1007/BF01110229
Rodney Y. Sharp, Gorenstein modules, Math. Z. 115 (1970), 117–139. MR 263801, DOI 10.1007/BF01109819

On Gorenstein modules over a complete Cohen-Macaulay ring

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