  The singularity category of a ring makes only the modules of finite
projective dimension vanish among the modules, so the singularity category is
expected to characterize a homological property of modules of infinite
projective dimension. In this paper, among such modules, we deal with
eventually periodic modules over a left artin ring, and, as our main result, we
characterize them in terms of morphisms in the singularity category. As
applications, we first prove that, for the class of finite dimensional algebras
over a field, being eventually periodic is preserved under singular equivalence
of Morita type with level. Moreover, we determine which finite dimensional
connected Nakayama algebras are eventually periodic when the ground field is
algebraically closed.
