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Keywords:
Bass classes; contracting endomorphisms; dualizing complex; Frobenius endomorphisms; ${\rm G}_{C}$-dimension; semidualizing complex
Summary:
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) $C\sim {\bold R}{\rm Hom}_R(^nR,C)$ for some $n>0$; (iii) ${\rm G}_C\text {\rm -dim} ^nR <\infty $ and $C$ is derived ${\bold R}{\rm Hom}_R(^nR,C)$-reflexive for some $n>0$; and (iv) ${\rm G}_C\text {\rm -dim} ^nR <\infty $ for infinitely many $n>0$.
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