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

Author: Idun Reiten
Journal: Proc. Amer. Math. Soc. 32 (1972), 417-420
MSC: Primary 13H10
DOI: https://doi.org/10.1090/S0002-9939-1972-0296067-7
MathSciNet review: 0296067
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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.

[1] Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 0153708, https://doi.org/10.1007/BF01112819
[2] Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
[3] Tor H. Gulliksen, Massey operations and the Poincaré series of certain local rings, J. Algebra 22 (1972), 223–232. MR 0306190, https://doi.org/10.1016/0021-8693(72)90143-3
[4] Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. MR 0099360
[5] Bruno J. Müller, On Morita duality, Canad. J. Math. 21 (1969), 1338–1347. MR 0255597, https://doi.org/10.4153/CJM-1969-147-7
[6] Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR 0155856
[7] Rodney Y. Sharp, The Cousin complex for a module over a commutative Noetherian ring., Math. Z. 112 (1969), 340–356. MR 0263800, https://doi.org/10.1007/BF01110229
[8] Rodney Y. Sharp, Gorenstein modules, Math. Z. 115 (1970), 117–139. MR 0263801, https://doi.org/10.1007/BF01109819
[9] -, On Gorenstein modules over a complete Cohen-Macaulay ring, Oxford Quart. J. Math. Oxford Ser. (2) 22 (1971).