Cohen-Macaulay properties of the Koszul homology
The Koszul homology \(H_\bullet(\underline y,N)\) which is constructed with respect to a sequence \(\underline y\) and a maximal Cohen-Macaulay (CM) module \(N\) over a local CM ring \(A\) admitting a canonical module \(\omega_ A\) will be compared with the Koszul homology \(H_\bullet(\underline y,\Hom_ A(N,\omega_ A))\). If \(R:=A/I\) with \(I=(\underline y)\) is a CM ring, then the canonical \(module\quad \omega_ R\) of R exists and we will mainly show the existence of a natural isomorphism \(H_{\bullet}(\underline y,\Hom_ A(N,\omega_ A))\simeq\Hom_ R(H_{\bullet}(\underline y,N),\omega_ R)\), if \(H_{\bullet}(\underline y,N)\) is a maximal CM module over R. This generalizes a result of \textit{J. Herzog} in his Habilitationsschrift, ``Komplexe, Auflösungen und Dualität in der lokalen Algebra'' (Univ. Regenburg 1974). - Using this isomorphism we are able to compute the graded canonical module of the graded ring \(gr_ I(A)\) in a certain case. In the last part of this paper we define a polynomial \(U^N(\underline y,\underline x)\) associated with the Koszul homology \(H_{\bullet}(\underline y,N)\) similar to the one defined by \textit{C. Huneke} in Commutative algebra: Analytical methods, Conf. Fairfax/Va. 1979, Lect. Notes Pure Appl. Math. 68, 173-199 (1982; Zbl 0499.13008). Huneke proved that \(H_ j(\underline y,N)\) is CM, if \(j<\text{mindeg} (U^N(\underline{y},\underline{x}))\). We will proceed to show that\(H_ j(\underline y,N)\) is CM, if \(j>\text{deg}(U^N(\underline{y},\underline{x}))\).
scientific article; zbMATH DE number 4049757