Differential torsion theories on Eilenberg-Moore categories of monads

TL;DR

This paper proves that all hereditary torsion theories on Eilenberg-Moore categories of certain monads are differential, and shows derivations extend uniquely to modules of quotients, enriching the structure theory of these categories.

Contribution

It establishes the differential property of hereditary torsion theories on Eilenberg-Moore categories and demonstrates the unique extension of derivations to modules of quotients.

Findings

Hereditary torsion theories are differential on these categories.

Derivations on modules extend uniquely to modules of quotients.

Results apply to monads that are exact and preserve colimits.

Abstract

Let $\mathcal C$ be a Grothendieck category and $U$ be a monad on $\mathcal C$ that is exact and preserves colimits. In this article, we prove that every hereditary torsion theory on the Eilenberg-Moore category of modules over a monad $U$ is differential. Further, if $\delta:U\longrightarrow U$ denotes a derivation on a monad $U$, then we show that every $\delta$-derivation on a $U$-module $M$ extends uniquely to a $\delta$-derivation on the module of quotients of $M$.