Tate (co)homology via pinched complexes

Abstract

For complexes of modules we study two new constructions which we call the pinched tensor product and the pinched Hom. They provide new methods for computing Tate homology T o r ^ \operatorname {\widehat {Tor}} and Tate cohomology E x t ^ \operatorname {\widehat {Ext}} , which lead to conceptual proofs of balancedness of Tate (co)homology for modules over associative rings. Another application we consider is in local algebra. Under conditions of the vanishing of Tate (co)homology, the pinched tensor product of two minimal complete resolutions yields a minimal complete resolution.