Injective stabilization of additive functors. I. Preliminaries

TL;DR

This paper introduces the foundational concepts and operational calculus for injective stabilization of additive functors, focusing on tensor products, with implications for module theory and homological algebra.

Contribution

It establishes a general framework for injective stabilization, connecting it with satellites, exactness, and torsion functors, and extends classical formulas to broader contexts.

Findings

Injective stabilization of tensor product is isomorphic to 1-torsion functor.

Extended Auslander-Reiten formula to arbitrary modules and rings.

Constructed a doubly-infinite exact sequence via injective stabilization.

Abstract

This paper is the first one in a series of three dealing with the concept of injective stabilization of the tensor product and its applications. Its primary goal is to collect known facts and establish a basic operational calculus that will be used in the subsequent parts. This is done in greater generality than is necessary for the stated goal. Several results of independent interest are also established. They include, among other things, connections with satellites, an explicit construction of the stabilization of a finitely presented functor, various exactness properties of the injectively stable functors, a construction, from a functor and a short exact sequence, of a doubly-infinite exact sequence by splicing the injective stabilization of the functor and its derived functors. When specialized to the tensor product with a finitely presented module, the injective stabilization with coefficients in the ring is isomorphic to the 1-torsion functor. The Auslander-Reiten formula is extended to a more general formula, which holds for arbitrary (i.e., not necessarily finite) modules over arbitrary associative rings with identity. Weakening of the assumptions in the theorems of Eilenberg and Watts leads to characterizations of the requisite zeroth derived functors. The subsequent papers, provide applications of the developed techniques. Part~II deals with new notions of torsion module and cotorsion module of a module. This is done for arbitrary modules over arbitrary rings. Part~III introduces a new concept, called the asymptotic stabilization of the tensor product. The result is closely related to different variants of stable homology (these are generalizations of Tate homology to arbitrary rings). A comparison transformation from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be epic.