Geodesic
On duality in the homology algebra of a Koszul complex
Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 1, pp. 77-81 
E. S. Golod. On duality in the homology algebra of a Koszul complex. Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 1, pp. 77-81. http://geodesic.mathdoc.fr/item/FPM_2003_9_1_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The homology algebra of the Koszul complex $K(x_1,\ldots,x_n;R)$ of a Gorenstein local ring $R$ has Poincaré duality if the ideal $I=(x_1,\ldots,x_n)$ of $R$ is strongly Cohen–Macaulay (i.e., all homology modules of the Koszul complex are Cohen–Macaulay) and under the assumption that $\dim R-\operatorname{grade}I\leq4$ the converse is also true.

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[2] Auslander M., Bridger M., Stable module theory, Mem. Amer. Math. Soc., 94, 1969 | MR | Zbl

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