Castelnuovo-Mumford regularity of $\mathrm{FI}^m$-modules presented in finite degrees

Liping Li

Published 2019-03-20Version 1

Let $V$ be a representation of the category $\mathrm{FI}^m$, a product of $m$ copies of the category of finite sets and injections, over an arbitrary commutative coefficient ring. We show in this paper that $V$ has finite Castelnuovo-Mumford regularity if and only if it is presented in finite degrees. In particular, the category of $\mathrm{FI}^m$-modules presented in finite degrees is abelian.