On tensor products of rings and extension conjectures
In recent years several authors have considered a commutative local version of two important conjectures in module and representation theory: i) (Tachikawa) Let \(A\) be a Cohen-Macualy local ring. If \(A\) has a canonical module \(\omega\) and \(\mathrm{Ext}_A^i(\omega , A)=0\) for all \(i>0\), then \(A\) is Gorenstein, i.e. \(\omega = A\); and ii) (Auslander-Reiten) Let \(A\) commutative noetherian local ring, and let \(M\) be a finitely generated \(A\)-module. If \(\mathrm{Ext}e_A^i(M,M\oplus A)=0\) for all \(i>i\), then \(M\) is free. In this paper, the author shows that if two local rings \(R_1\) and \(R_2\) essentially of finite type over the same field \(k\), with maximal ideals \(M_1\) and \(M_2\) respectively, are Cohen-Macualay and the Tachikawa conjecture holds for both of them, then it also holds for \((R_1\otimes_kR_2)_M\), where \(M:=M_1\otimes R_2+R_1\otimes_kM_2\). Moreover, if the Auslander-Reiten conjecture holds for classes \({\mathcal C}_i\) of maximal Cohen-Macaulay modules over Cohen-Macualay rings \(R_i\), \(i=1,2\), then the Auslander-Reiten conjecture holds for \(R\)-modules with a syzygy belonging to the class of all \(R\)-modules wich are finite direct sums of modules of the form \((X_1\otimes X_2)_M\) with \(X_i\in {\mathcal C}_i\), \(i=1,2\). The paper finishes analyzing the last condition.
scientific article; zbMATH DE number 5673550
scientific article