The principle "Every result in classical homological algebra should have a
counterpart in Gorenstein homological algebra" is given in [3]. There is a
remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps
one of the most glaring exceptions is provided by the fact that tensor products
of Gorenstein projective modules need not be Gorenstein projective, even over
Gorenstein rings. So perhaps it is surprising that tensor products of
Gorenstein injective modules over Gorenstein rings of finite Krull dimension
are Gorenstein injective.
Our main result is in support of the principle. Over commutative, noetherian
rings injective modules have direct sum decompositions into indecomposable
modules. We will show that Gorenstein injective modules over Gorenstein rings
of finite Krull dimension have filtrations analogous to those provided by these
decompositions. This result will then provide us with the tools to prove that
all tensor products of Gorenstein injective modules over these rings are
Gorenstein injective.