  What is generally known as the "Bloch--Srinivas method" consists of
decomposing the diagonal of a smooth projective variety, and then considering
the action of correspondences in cohomology. In this note, we observe that this
same method can also be extended to singular and quasi--projective varieties.
We give two applications of this observation: the first is a version of
Mumford's theorem, the second is concerned with the Hodge conjecture for
singular varieties.
