Homological aspects of the adjoint cotranspose
Volume 150 / 2017
Abstract
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.
