  We prove several characterizations of Gorenstein rings in terms of vanishings
of derived functors of certain modules or complexes whose scalars are
restricted via contracting endomorphisms. These results can be viewed as
analogues of results of Kunz (in the case of the Frobenius) and
Avramov-Hochster-Iyengar-Yao (in the case of general contracting
endomorphisms).
