A characterization of Auslander category. (Q2862256)

The Auslander and Bass classes with respect to a semidualizing module over a commutative Noetherian ring \(R\) have been introduced by \textit{H.-B. Foxby} [Math. Scand. 31(1972), 267-284 (1973; Zbl 0272.13009)] and \textit{L. L. Avramov} and \textit{H.-B. Foxby} [Proc. Lond. Math. Soc., III. Ser. 75, No. 2, 241-270 (1997; Zbl 0901.13011)]. The definitions of these classes were extended to arbitrary associative rings by \textit{H. Holm} and \textit{D. White} [J. Math. Kyoto Univ. 47, No. 4, 781-808 (2007; Zbl 1154.16007)].NEWLINENEWLINE The study of the Auslander and Bass classes with respect to a semidualizing module over an associative ring is continued in this paper. One of the main results is a characterization of the Bass class with respect to a semidualizing module \(_SC_R\) as the right orthogonal subcategory of some right \(R\)-module and the Auslander class as the left orthogonal subcategory of the character module of some left \(S\)-module \(M\).NEWLINENEWLINE A semidualizing module is defined to be minimal if it has no proper direct summand that is semidualizing. Let \(C\) be an \(R\)-module and let \(S=\text{End}_RC\). If \(C\) is a full subcategory of \(\mathrm{Mod }R\), let \(\mathrm{gen}^*C\) be the subcategory of all modules \(N\) for which an exact sequence NEWLINE\[NEWLINE\cdots@>f_2>>M^1@>f_1>>M^0@>f_0>>N\to 0NEWLINE\]NEWLINE of modules \(M^i\) in \(C\) exists with \(\text{Ext}_R^1(C,\text{Ker }f_i)=0\). As a second main result, one-to-one correspondences are found between isomorphism classes of minimal semidualizing \(R\)-modules and maximal classes among the coresolving preenvelope classes of \(\text{Mod }R\) with the same \(\mathrm{Ext}\)-projective generators in \(\mathrm{gen}^*R\) (resp. the resolving precover classes of \(\mathrm{Mod }S\) with the same \(\mathrm{Ext}\)-injective cogenerators in \(\mathrm{gen}^*S\)).