Bounding regularity of FIm-modules

Abstract

Let FI be a skeleton of the category of finite sets and injective maps, and ${\mathrm{FI}}^m$ the product of m copies of FI. We prove that if an ${\mathrm{FI}}^m$-module is generated in degree ⩽d and related in degree ⩽r, then its regularity is bounded above by a function of m, d, and r.