Ore localization and minimal injective resolutions. (Q392166)

It is well-known that if \(R_S\) denotes the localization at a multiplicatively closed set \(S\) with \(1\) of a commutative Noetherian ring \(R\), and if \(M\) is a finitely generated \(R\)-module, then for any \(R\)-module \(N\) the \(R_S\)-modules \(\mathrm{Ext}^i_R(M,N)_S\) and \(\mathrm{Ext}^i_{R_S}(M_S,N_S)\) are isomorphic for all positive integers \(i\).NEWLINENEWLINE The question whether a similar result holds for non commutative Noetherian rings is meaningful only if at least one of the modules \(M\) and \(N\) is a bimodule, and a positive answer is known only in specific instances. The paper under review presents two such generalizations: (1) Let \(R\) be a Noetherian right FBN ring, let \(X\) be an Ore set in \(R\), let \(M\) be an \(R\)-module, and let \(P\) be a prime ideal of \(R\). Then \(\mathrm{Ext}^i_R(R/P,M)\otimes_RR_X\cong\mathrm{Ext}^i_{R_X}(R_X/P_X,M_X)\) as \(R_X\)-modules for all \(i\); (2) Let \(R\) be a grade-symmetric Auslander-Gorenstein ring, let \(P\) be a right localizable correct prime ideal with clique \(C\) and corresponding Ore set \(X\), and let \(M\) be an \(R\)-module of finite flat dimension. Then for all \(i\), \(\mathrm{Ext}^i_R(R/P,M)\otimes_RR_X\cong\mathrm{Ext}^i_{R_X}(R_X/P_X,M_X)\) as right \(R/P\)-modules. The author applies these results to the study of minimal injective resolutions of modules over Noetherian rings.