Let $R$ be a commutative ring and $𝒞$ a semidualizing $R$-module. We investigate the relations between $𝒞$-flat modules and $𝒞$-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring $R$. Our focus is on homological properties of contracting endomorphisms of $R$, e.g., the Frobenius endomorphism when $R$ contains a field of positive characteristic. For instance, in this case, when $R$ is $F$-finite and $C$ is a semidualizing $R$-complex, we prove that the following conditions are equivalent: (i) $C$ is a dualizing $R$-complex; (ii) $C\sim 𝐑{\mathrm{Hom}}_R{(}^{n}R,C)$ for some $n>0$; (iii) ${\mathrm{G}}_C{\text{-dim}}^{n}R<\infty$ and $C$ is derived $𝐑{\mathrm{Hom}}_R{(}^{n}R,C)$-reflexive...