A note on homological systems

TL;DR

This paper provides an elementary proof demonstrating that the category of Δ-filtered modules, linked to a finite homological system, remains closed under direct summands, clarifying a known mathematical fact.

Contribution

It offers a simplified proof of the closure property of Δ-filtered modules under direct summands in the context of finite homological systems.

Findings

Proves closure of Δ-filtered modules under direct summands.

Simplifies understanding of homological systems.

Clarifies properties of module categories.

Abstract

I give an elementary proof of the known fact that the category $\mathfrak{F} \left( \Delta \right)$ of $\Delta -$filtered modules, associated to a given finite homological system $\left( \Delta ; \Omega , \leq \right) ,$ is closed under direct summands.