Differential torsion theories on Eilenberg-Moore categories of monads

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 $EM_U$ 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 $M$.