Let R and S be rings and RCS 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.

1. Introduction. The study of semidualizing modules in commutative rings was initiated by Foxby [10] and Golod [12]. Then Holm and White [16] extended it to arbitrary associative rings. Many authors have studied the properties of semidualizing modules and related modules; see for example [10],[12],[15]–[16],[25],[28],[32]–[40] and the references therein. Among various research areas, one basic theme is to extend the “absolute” classical results in homological algebra to the “relative” setting with respect to semidualizing modules. One of the motivations of this paper comes from a classical result due to Jensen [23, Proposition 6], which states that any flat left R-module has finite projective dimension over a ring R with finite left finitistic dimension. Simson [29, Theorem 2.7] extended this result to skeletally small additive categories. Another motivation comes from Emmanouil and Talelli’s work [7], in which the relations between the supremum of the projective lengths of injective left R-modules, the supremum of the injective lengths of projective left R-modules, the finitistic dimension and the left self-injective dimension of a ring R were established. We are interested in whether these results have relative counterparts with respect to semidualizing modules. The paper is organized as follows.