  In this paper we consider a question of Roger Wiegand, which is about tensor
products of finitely generated modules that have finite projective dimension
over commutative Noetherian rings. We construct modules of infinite projective
dimension (and of infinite Gorenstein dimension) whose tensor products have
finite projective dimension. Furthermore we determine nontrivial conditions
under which such examples cannot occur. For example we prove that, if the
tensor product of two nonzero modules, at least one of which is totally
reflexive (or equivalently Gorenstein-projective), has finite projective
dimension, then both modules in question have finite projective dimension.
