Separation and excision in functor homology

TL;DR

This paper develops separation and excision tools in functor homology for $k[\mathcal{A}]\text{-Mod}$, clarifying how antipolynomial and polynomial data decouple in Ext and Tor computations via AP-type bifunctors. It proves an excision theorem characterizing when restriction along an additive functor $\phi$ preserves Ext and Tor up to a degree $e$, and derives polynomial analogues that apply to reduced and polynomial functors through cross-effects and Pirashvili’s vanishing lemma. A central separation theorem shows that Ext and Tor between AP-type bifunctors behave multiplicatively after restricting to the diagonal, enabling Kunneth-type computations in a diagonal setting. The results rely on simplicial resolutions and local Hurewicz theorems to connect homotopical and homological data, with applications to vanishing results and to simplifying computations by passing to additive quotients with finite Hom-sets. Overall, the work provides a robust framework to transfer homological information across decompositions of functors, mirroring classical excision phenomena in algebraic K-theory and yielding practical vanishing and reduction results for functor categories.

Abstract

We prove separation and excision results in functor homology. These results explain how the global Steinberg decomposition of functors proved by Djament, Touz{é} and Vespa behaves in Ext and Tor computations.

Theorems & Definitions (61)

• Theorem 1: separation
• Corollary 2: Separation for tensor products
• proof
• Remark 1.1: The interest of tensor products
• Theorem 3: excision
• Proposition 5
• Theorem 6: Polynomial excision
• Remark 2.1
• Proposition 2.2
• Remark 2.3