Covers and envelopes with respect to a semidualizing module
Let \(R\) be a commutative ring. Semidualizing modules were introduced by \textit{H.-B. Foxby} [Math. Scand. 31(1972), 267--284 (1973; Zbl 0272.13009)], \textit{W. V. Vasconcelos} [Divisor theory in module categories. North-Holland Mathematics Studies 14, Amsterdam-Oxford: North-Holland Publishing Comp. (1974; Zbl 0296.13005)] and \textit{E. S. Golod} [Tr. Mat. Inst. Steklova 165, 62--66 (1984; Zbl 0577.13008)] and it is a generalization of the notion of dualizing module of a Cohen-Macaulay ring. An \(R\)-module \(C\) is called semidualizing if \(C\) admits a resolution (possible infinite) by finitely generated projective modules, the natural map \(R\to \Hom_R(C,C)\) is an isomorphism, and \(\mathrm{Ext}_R^i(C,C)=0\) for \(i>0\). A class of modules \({\mathcal A}_R(C)\), the Auslander class, is associated to any semidualing module. This class consists in all the \(R\)-modules \(M\) satisfying i) \(\mathrm{Tor}_i^R(C,M)=0\) and \(\mathrm{Ext}_R^i(C,C\otimes M)=0\) for \(i>0\) and ii) the canonical map \(M\to \Hom_R(C,C\otimes_RM)\) is an isomorphism. In this paper, for a class of modules \(\mathcal Q\), a new class \({\mathcal Q}_C\) is defined by \(M\in {\mathcal Q}_C\) when \(\Hom_R(C,M)\in {\mathcal Q}\). It is shown that if \({\mathcal Q}\) is a subclass of the Auslander class which is a Kaplansky class and closed under direct sums, then the orthogonal class \({\mathcal Q}_C^\perp\) is special preenveloping. As a consequence, the class of modules of with relative to \(C\) projective dimension less or equal to \(n\) is special preenveloping. Similar results are obtained for classes contained in the Bass class associated to a semidualing module.
scientific article; zbMATH DE number 6044051