Title: Injective stabilization of additive functors. I. Preliminaries,
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.