Deconstructing Auslander's formulas, I. Fundamental sequences associated with additive functors

TL;DR

This paper generalizes Auslander's formulas by constructing long sequences linking derived functors, satellites, and stabilizations for additive functors, applicable to arbitrary rings and modules, and extends to universal coefficient theorems.

Contribution

It introduces a unified formalism connecting derived functors, satellites, and stabilizations, broadening Auslander's classical results to more general settings.

Findings

Sequences are exact for half-exact functors.

Classical formulas are recovered for Hom and tensor on finitely presented modules.

Universal coefficient theorems are established for arbitrary complexes.

Abstract

For any additive functor from modules (or, more generally, from an abelian category with enough projectives or injectives), we construct long sequences tying up together the derived functors, the satellites, and the stabilizations of the functor. For half-exact functors, the obtained sequences are exact. For general functors, nontrivial homology may only appear at the derived functors. Specializing to the familiar Hom and tensor product functors on finitely presented modules, we recover the classical formulas of Auslander. Unlike those formulas, our results hold for arbitrary rings and arbitrary modules, finite or infinite. The same formalism leads to universal coefficient theorems for homology and cohomology of arbitrary complexes. The new results are even more explicit for the cohomology of projective complexes and the homology of flat complexes.