The homology of additive functors in prime characteristic
Aurélien Djament, Antoine Touzé
TL;DR
This work studies Ext and Tor in the category of all functors from a small ${\mathbb{F}_p}$-linear additive category ${\mathcal{A}}$ to ${\mathcal V}_{\Bbbk}$, connecting it to Ext and Tor in the full subcategory of additive functors. The authors construct and exploit the ${\aleph}$-additive envelope and Frobenius-twist machinery to establish a precise isomorphism: for a perfect field ${\Bbbk}$ of characteristic $p>0$ and ${\mathcal{A}}$ ${\mathbb{F}_p}$-linear, the additive Ext data tensor an explicit Frobenius-twist Ext algebra $E^*_{\infty}$ recovers the full Ext in ${\mathcal{F}}({\mathcal{A}},\Bbbk)$ via an isomorphism $\mathrm{Ext}^*_{\mathbf{Add}}(\pi,\rho) \otimes E^*_{\infty} \cong \mathrm{Ext}^*_{{\mathcal{F}}({\mathcal{A}},\Bbbk)}(\pi,\rho)$. A key step is the identification $E^*_{\infty} \cong E^*$ (where $E^* = \mathrm{Ext}^*_{{\mathcal{F}}({\mathbf P}_{\Bbbk},\Bbbk)}(I,I)$) via Frobenius twists, together with a natural formula $\Psi(e\otimes e') = \Phi(e) \circ \pi^*_{\infty}(e')$. The paper also derives dual Tor-Ext statements, provides a framework for computing Tor between non-additive functors from additive data, and applies these results to the homology of general linear groups, yielding concrete product decompositions and new computable cases. Overall, it offers a robust, characteristic-driven bridge between additive and non-additive functor homology with direct implications for representation stability and GL$_\infty$-homology computations.
Abstract
We compute certain Ext and Tor groups in the category of all functors from an Z/p-linear additive category A to vector spaces in terms of Ext and Tor computed in the full subcategory of additive functors from A to vector spaces. We thus obtain group homology computations for general linear groups.