Infinitesimal module deformations in the Thom-Sebastiani problem
This paper deals with a particular case of the so-called problem of Thom-Sebastiani type for the categories of Cohen-Macaulay modules. The authors give some applications of the results given by \textit{J. Herzog} and \textit{D. Popescu} [Math. Ann. 309, No. 4, 677-700 (1997; Zbl 0902.13010)] and \textit{D. Popescu} [An. Ştiinţ. Univ. ``Ovidius'' Constanţa, Ser. Mat. 2, 112-119 (1994; Zbl 0920.13004)] which show that every maximal Cohen-Macaulay \(R'\)-module \(N\) is a direct summand of the first syzygy \(\Omega_{R'}^{1} (T)\) of \(T\) over \(R'\), where \(R' = Kx,y / (f+ y^{s})\), \(x=(x_{1}, \ldots, x_{n})\), \(y=(y_{1}, \ldots, y_{m})\), \(f \in Kx\), \(T=N / y^{s-1}N\), \(K\) is a field and \(s\) an integer which is not multiple of the characteristic of \(K\). In the paper under review, the case \(s=3\) is considered and it is shown that the correspondence \(T \rightarrow \Omega_{R'}^{1} (T)\) is ``almost'' injective and ``almost'' preserves indecomposability. Moreover, the authors give a precise description of the maximal Cohen-Macaulay modules over \(R'\) (\(f=x^{t}\)) such that \(N/yN \cong P_{i}^{d}\) where \(P_{i} = Kx/(x^{i})\) are all non-free indecomposable modules over \(R =Kx/(x^{t})\) and \(d \geq 1\) is an integer.
scientific article; zbMATH DE number 1145743