Geodesic
On a homological characterization of a certain class of local rings
Sbornik. Mathematics, Tome 38 (1981) no. 3, pp. 421-425 
A. F. Ivanov. On a homological characterization of a certain class of local rings. Sbornik. Mathematics, Tome 38 (1981) no. 3, pp. 421-425. http://geodesic.mathdoc.fr/item/SM_1981_38_3_a6/
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     title = {On a~homological characterization of a~certain class of local rings},
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is shown that the nondegeneracy of the Yoneda product $\operatorname{Ext}_A^p(k,M)\times\operatorname{Ext}_A^{n-p}(M,k)$ ($M$ is either a noetherian module or a complex of finite projective dimension, and $k$ is the residue field) characterizes the regularity of the ring $A$, whereas the isomorphism $\operatorname{Ext}_A^p(k,M)\approx\operatorname{Ext}_A^{n-p} (M,k)$ characterizes the fact that $A$ is Gorenstein. Bibliography: 8 titles.

[1] J.-P. Serre, “Algèbre locale”, Multiplicités, Lecture Notes in Math., no. 11, Springer-Verlag, 1965

[2] M. Auslander, D. A. Buchsbaum, “Homological dimension in local rings”, Trans. Arner. Math. Soc., 85:2 (1957), 390–405 | DOI | MR | Zbl

[3] R. Hartshorne, “Residues and duality”, Lecture Notes in Math., no. 20, Springer-Verlag, 1966 | MR | Zbl

[4] A. Kartan, S. Eilenberg, Gomologicheskaya algebra, IL, Moskva, 1960

[5] H. Bass, “On the ubiquity of Gorenstein rings”, Math. Z., 22:1 (1963), 8–28 | DOI | MR

[6] S. Eilenberg, “Homological dimension and syzygies”, Ann. Math., 64:2 (1956), 328–336 | DOI | MR | Zbl

[7] J. Tate, “Homology of noetherian and local rings”, Illinois J. Math., 1:1 (1957), 14–27 | MR | Zbl

[8] S. Maklein, Gomologiya, izd-vo “Mir”, Moskva, 1966