Locally Self-injective Property of FI$^m$

TL;DR

This paper investigates the locally self-injective property of the product category FI$^m$, demonstrating that certain functors preserve injectivity and establishing key categorical equivalences for FI$^m$-modules.

Contribution

It proves that the external tensor product commutes with coinduction in FI$^m$, leading to new insights on injective and projective modules and categorical structures.

Findings

External tensor product commutes with coinduction in FI$^m$

Every projective FI$^m$-module over a field of characteristic 0 is injective

Serre quotient of finitely generated FI$^m$-modules relates to finite dimensional modules

Abstract

In this paper we consider the locally self-injective property of the product FI$^m$ of the category FI of finite sets and injections. Explicitly, we prove that the external tensor product commutes with the coinduction functor, and hence preserves injective modules. As corollaries, every projective FI$^m$-modules over a field of characteristic 0 is injective, and the Serre quotient of the category of finitely generated FI$^m$-modules by the category of finitely generated torsion FI$^m$-modules is equivalent to the category of finite dimensional FI$^m$-modules.