On a theorem of Knörrer concerning Cohen-Macaulay modules
Let \(K\) be a field, \(X = X_1,\ldots,X_n\) and \(Y = Y_1,\ldots,Y_r\) sets of indeterminates. Let \(f \in KX\) be a non-zero formal power series with \(f \in (X)\) and \(s_1,\ldots,s_r\) positive integers with each \(s_i \geq 2\) which are not multiple of the characteristic of \(K\). Put \(R = KX,Y/(f+\sum_{i=1}^rY_i^{s_i})\), \(R_1= KX/(f)\) and \(\widetilde{R} = R/(Y_1^{s_1-1},\ldots,Y_r^{s_r-1})\). The main result of this paper shows that for a maximal Cohen-Macaulay \(R\)-module \(N\) with no free direct summands, we have (i) \(\widetilde N := \widetilde R\otimes_RN\) is a deformation of the maximal Cohen-Macaulay \(R_1\)-module \(N/YN\) to \(\widetilde R\), (ii) \(\Omega_R^r(\widetilde N) \cong (N\oplus \Omega_R^1(N))^{2^{r-1}}\). This generalizes a result of \textit{H. Knörrer} in the case \(r=1\) and \(g = Y_1^2\) [Invent. Math. 88, 153-164 (1987; Zbl 0617.14033)]. The authors also find resolutions of the higher order syzygy modules which are minimal in certain situation, and express them in terms of matrix factorizations. The approach simplifies and extends that of \textit{J. Herzog} and \textit{D. Popescu} [Math. Ann. 309, No. 4, 677-700 (1997; Zbl 0902.13010)] by using iterated mapping cones.
scientific article; zbMATH DE number 1517309