Obstructions to compatible splittings

TL;DR

This paper investigates when compatible splittings of a map of short exact sequences exist in abelian categories, introducing an obstruction group that characterizes the conditions for such splittings.

Contribution

It defines and analyzes an obstruction group that determines the existence of compatible splittings in maps of short exact sequences.

Findings

Obstruction group characterizes compatible splitting existence

Conditions for compatible splittings depend on the obstruction group

Basic properties of the obstruction group are established

Abstract

Suppose one has a map of split short exact sequences in a category of modules, or more generally, in any abelian category. Do the short exact sequences split compatibly, i.e., does there exist a splitting of each short exact sequence which commutes with the map of short exact sequences? The answer is sometimes yes and sometimes no. We define and prove basic properties of an obstruction group to the existence of compatible splittings.