(Co)homological vanishing for non-additive representations of a semi-additive category

TL;DR

This work demonstrates Ext-vanishing for linearizations of additive functors from a semi-additive category $\mathcal{E}$ to $k$-modules, under $k$-triviality and finiteness conditions. The authors develop a model category framework on $s\mathrm{Rep}(\mathcal{E})$ to build simplicial projective resolutions and reduce Ext computations to the homology of simplicial monoids, enabling explicit vanishing results in key cases, including semilattices and $k$-trivial categories. They establish dualities $k[A] \cong k^{A^{\#}}$ (when $k$ contains all roots of unity) and derive Ext-vanishing via Tor-Ext spectral sequences, with applications to generalized correspondence functors and Bouc–Thévenaz's category of correspondences. The paper further extends these vanishing results to generalized correspondence functors defined by finite distributive lattices, thereby broadening the known Ext-vanishing phenomena in this area and providing tools for analyzing simple functors and polynomial behavior in $\mathcal{F}(\mathcal{E};k)$.

Abstract

We show the vanishing of higher extension groups and torsion groups between linearisation of additive functors from a semi-additive category satisfying some conditions to a category of vector spaces. In particular, we apply our results to the category of correspondences functors of Bouc-Thévenaz.

Theorems & Definitions (60)

• Definition 2.1
• Definition 2.2
• Definition 3.1
• Lemma 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Lemma 3.5
• proof