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Totally reflexive modules with respect to a semidualizing bimodule. (English) Zbl 1281.16015

In the paper totally reflexive modules with respect to a semidualizing bimodule are investigated. Let \(R,S\) be associative rings and \(_SC_R\) be a semidualizing bimodule, i.e., both \(_SC\) and \(C_R\) admit degreewise finitely generated projective resolutions, natural homothety maps \(_SS_S\to\operatorname{Hom}_R(C,C)\) and \(_RR_R\to\operatorname{Hom}(C,C)\) are isomorphisms and \(\text{Ext}_R^i(C,C)=0=\text{Ext}_S^i(C,C)\) for all \(i\geq 1\). A finitely generated right \(R\)-module \(M_R\) is called totally \(C_R\)-reflexive if \(M_R\) admits a degreewise finitely generated \(R\)-projective resolution, the biduality map \(\delta_M^C\colon M\to\operatorname{Hom}_S(\operatorname{Hom}_R(M,C),C)\) is an \(R\)-module isomorphism, \(\operatorname{Hom}_R(M,C)\) admits a degreewise finitely generated \(S\)-projective resolution and \(\text{Ext}_R^i(M,C)=0=\text{Ext}_S^i(\operatorname{Hom}_R(M,C),C)\) for all \(i\geq 1\). Among other results, the class of totally \(C_R\)-reflexive modules is characterized (Theorem 2.3) and it is shown that every totally \(C_R\)-reflexive module with finite projective dimension is already projective (Theorem 2.11).

MSC:

16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16E05 Syzygies, resolutions, complexes in associative algebras
16E10 Homological dimension in associative algebras
16D20 Bimodules in associative algebras
16D40 Free, projective, and flat modules and ideals in associative algebras
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