Deconstructing Auslander's formulas, I. Fundamental sequences associated with additive functors
Alex Martsinkovsky
TL;DR
This work establishes a unified homological framework for additive functors between module categories by introducing right and left fundamental sequences that connect derived functors, satellites, and (co)stabilizations. It generalizes Auslander's classical formulas to arbitrary rings and modules, derives universal coefficient theorems for cohomology and homology without restrictive hypotheses, and develops the notions of injective/projective stabilization and tensor-copresented functors to handle non-finitely-presented cases. The paper achieves explicit decompositions for half-exact finitely presented functors, provides UCTs for cohomology and homology in broad contexts, and clarifies the role of the Auslander-Gruson-Jensen framework in this setting. These results yield a cohesive toolkit for analyzing homological behavior of Hom and tensor-related functors across general abelian categories with enough projectives/injectives, and they illuminate the structure of stabilizations and satellites in both covariant and contravariant settings.
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.