On the projective dimension of tensor products of modules

In this paper, we consider finitely generated modules over commutative Noetherian rings whose tensor products have finite projective dimension. We construct examples of modules of infinite projective dimension (and also of infinite Gorenstein dimension) whose tensor products nonetheless have finite projective dimension. Furthermore, we establish nontrivial conditions under which such examples cannot arise. For example, we prove that if the tensor product of two nonzero modules–at least one of which is totally reflexive–has finite projective dimension, then both modules in question must have finite projective dimension.