## Homological invariants related to semidualizing bimodules

### Volume 156 / 2019

#### Abstract

\looseness -1Let $R$ and $S$ be rings and $_RC_S$ a semidualizing bimodule. We show that the supremum of the $C$-projective dimensions of $C$-flat left $R$-modules is less than or equal to that for left $R$-modules with finite $C$-projective dimension, and the latter is less than or equal to the supremum of the $C$-injective dimensions of projective (or flat) left $S$-modules. We also show that the supremum of the $C$-projective dimensions of injective left $R$-modules and that of the $C$-injective dimensions of projective left $S$-modules are identical provided that both of them are finite. Finally, we show that the supremum of the $C$-projective dimensions of $C$-flat left $R$-modules (a relative homological invariant) and that of the projective dimensions of flat left $S$-modules (an absolute homological invariant) coincide.