Tate (co)homology via pinched complexes

TL;DR

This paper introduces the pinched tensor product and pinched Hom constructions, offering new methods for computing Tate (co)homology and proving their balancedness, with applications in local algebra and minimal resolutions.

Contribution

The paper presents novel constructions called pinched tensor product and pinched Hom, advancing the computation and understanding of Tate (co)homology.

Findings

New methods for computing Tate (co)homology

Conceptual proofs of balancedness of Tate (co)homology

Application to minimal complete resolutions in local algebra

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 and Tate cohomology, 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 vanishing of Tate (co)homology, the pinched tensor product of two minimal complete resolutions yields a minimal complete resolution.