Contracting Endomorphisms and Dualizing Complexes

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 ∼ RHom_R(ⁿR,C) for some n > 0; (iii) G_C-dimⁿR < ∞ and C is derived RHom_R(ⁿR,C)-reflexive for some n > 0; and (iv) G_C-dimⁿR < ∞ for infinitely many n > 0.