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