Koszul resolution for linear monoidal functors
Serge Bouc, Nadia Romero
TL;DR
This work extends Hochschild cohomology to the setting of functor categories by developing Koszul-type resolutions for monoids in $\mathcal{F}$, the abelian category of $R$-linear functors on a small symmetric monoidal category. It defines the Koszul complex $K_A(\boldsymbol{\alpha})$ via regular sequences $\boldsymbol{\alpha}$ and proves it yields a resolution of $A/A\langle \boldsymbol{\alpha}\rangle$, with the regularity notion ensuring exactness. It then introduces polynomial functors $F_n$ and analyzes Hochschild cohomology for polynomial monoids $A_n$ in the tensor-idempotent commutative setting, obtaining $\mathcal{HH}^p(A_n,M)=0$ for $p>n$ and $\mathcal{HH}^p(A_n,A_n)\cong K_p^n(A_n)$ for $p\le n$, and proves a relative Hilbert syzygy theorem that guarantees finite $\mathcal{F}$-split resolutions of $A[t_1,\dots,t_n]$-modules. The framework is applicable to Green biset functors (e.g., Burnside), enabling polynomial-like analyses in this homological setting and providing tools for structural results in that domain.
Abstract
We introduce regular sequences and associated Koszul resolutions for monoids in the category of functors over an essentially small linear symmetric monoidal category. Next we define polynomials over such monoids. We compute the Hochschild cohomology functors and prove a relative analogue of Hilbert's syzygy theorem for polynomials over tensor idempotent commutative monoids.