Base change along the Frobenius endomorphism and the Gorenstein property

Let R be a local ring of positive characteristic and X a complex with nonzero finitely generated homology and finite injective dimension. We prove that if the derived base change of X via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then R is Gorenstein. In particular, we give an affirmative answer to a question by Falahola and Marley [7, Question 3.9]. (C) 2021 Elsevier Inc. All rights reserved.