  In this paper, we answer a question of Dwyer, Greenlees, and Iyengar by
proving a local ring $R$ is a complete intersection if and only if every
complex of $R$-modules with finitely generated homology is proxy small.
Moreover, we establish that a commutative noetherian ring $R$ is locally a
complete intersection if and only if every complex of $R$-modules with finitely
generated homology is virtually small.
