Homological aspects of the adjoint cotranspose

Xi Tang¹ ; Zhaoyong Huang²

¹ College of Science Guilin University of Technology 541004 Guilin, Guangxi, P.R. China
² Department of Mathematics Nanjing University 210093 Nanjing, Jiangsu, P.R. China

Colloquium Mathematicum, Tome 150 (2017) no. 2, pp. 293-311

Résumé

Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We introduce and study the adjoint cotransposes of modules and adjoint $n$-$\omega$-cotorsionfree modules. We show that the Auslander class with respect to $_R\omega_S$ is the intersection of the class of adjoint $\infty$-$\omega$-cotorsionfree modules and the right $\operatorname{Tor}$-orthogonal class of $\omega_S$. As a consequence, the classes of adjoint $\infty$-$\omega$-cotorsionfree modules and of $\infty$-$\omega$-cotorsionfree modules are equivalent under Foxby equivalence if and only if they coincide with the Auslander and Bass classes with respect to $\omega$ respectively. Moreover, we give some equivalent characterizations when the left and right projective dimensions of $_R\omega_S$ are finite in terms of the properties of (adjoint) $\infty$-$\omega$-cotorsionfree modules.

Détail

DOI : 10.4064/cm7121-12-2016

Mots-clés : rings omega semidualizing bimodule introduce study adjoint cotransposes modules adjoint n omega cotorsionfree modules auslander class respect omega intersection class adjoint infty omega cotorsionfree modules right operatorname tor orthogonal class omega nbsp consequence classes adjoint infty omega cotorsionfree modules infty omega cotorsionfree modules equivalent under foxby equivalence only coincide auslander bass classes respect omega respectively moreover equivalent characterizations right projective dimensions omega finite terms properties adjoint infty omega cotorsionfree modules