Injective stabilization of additive functors. III. Asymptotic stabilization of the tensor product

TL;DR

This paper explores the asymptotic stabilization of the tensor product via an iterative injective process, linking it to stable cohomology, and establishes new connections with Buchweitz cohomology and functor categories.

Contribution

It introduces the asymptotic stabilization of the tensor product, connecting it to Buchweitz's stable cohomology and Triulzi's J-completion, with new results on functor categories and homological properties.

Findings

The asymptotic stabilization is isomorphic to Triulzi's J-completion of Tor.

A comparison map from Vogel homology to the stabilization is always epic.

The category of finitely presented functors is complete and (co)complete.

Abstract

The injective stabilization of the tensor product is subjected to an iterative procedure that utilizes its bifunctor property. The limit of this procedure, called the asymptotic stabilization of the tensor product, provides a homological counterpart of Buchweitz's asymptotic construction of stable cohomology. The resulting connected sequence of functors is isomorphic to Triulzi's $J$-completion of the Tor functor. A comparison map from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be always epic. The category of finitely presented functors is shown to be complete and (co)complete. As a consequence, the inert injective stabilization of the tensor product with fixed variable a finitely generated module over an artin algebra is shown to be finitely presented. A description of its defect and all right-derived functors is given. A surprising connection with Buchweitz cohomology based on injectives is established