Homological independence of infinite syzygies

Abstract

We consider modules that appear as syzygies of acyclic complexes of flat modules and examine a certain condition on pairs of such modules, that generalizes the vanishing of Tate homology. If this condition is satisfied for two modules over a commutative ring , then (a) the tensor product of the two modules is also a syzygy of an acyclic complex of flat modules and (b) the tensor product of the corresponding acyclic complexes of flat modules is an acyclic complex of flat modules as well, whose syzygies can be expressed in terms of the syzygies of the factor complexes. We also examine the analogous (dual) case of homomorphism groups of infinite syzygies.