Characterizations of higher derivations and higher differential torsion theories in Eilenberg-Moore categories of monads

TL;DR

The paper develops a generalized theory of higher derivations in the setting of monads and their Eilenberg-Moore categories, showing that higher derivations on a monad are governed by a recursive relation to a sequence of ordinary derivations on $T$ and extending this to modules over the monad. It then constructs a higher differential torsion theory framework: a hereditary torsion theory on $EM_T$ is higher differential of order $n$ iff its Gabriel filters are $\Delta$-invariant of order $n$, with several equivalent formulations and a focus on extensions of higher derivations to modules of quotients. A key result is the existence and uniqueness of extending a higher $\Delta$-derivation from a torsion-free module to its module of quotients $Q_\tau(M)$ precisely when the torsion theory is higher differential of order $n$. Overall, the work generalizes differential torsion theory and higher derivations from rings to monads, providing a robust monadic foundation for differential structures in EM_T and their quotients.

Abstract

Let $T$ be a monad on a category $\mathscr{C}$. In this paper, we introduce the notion of higher derivations on the monad $T$ and characterize them in terms of ordinary derivations on $T$. We also define higher derivations on modules over the monad $T$ in the Eilenberg-Moore category $EM_T$ and establish their characterization in a similar manner. We provide several examples that illustrate and support our results. Furthermore, we examine the conditions under which a torsion theory on $EM_T$ is higher differential, and show that this holds if and only if every higher derivation on a module $M \in EM_T$ extends uniquely to its module of quotients $Q_τ(M)$.

Theorems & Definitions (44)

• Theorem A
• Theorem B
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Proposition 2.7
• proof