On splitting of morphisms induced by unit map of adjoint functors

TL;DR

This paper characterizes when the unit map of an adjunction between triangulated categories splits, linking it to the additive closure of shifts of objects in the functor's image, with applications in geometry.

Contribution

It provides a new criterion for split monomorphisms induced by the unit map in triangulated categories, connecting it to the additive closure of shifted objects.

Findings

Unit map is a split monomorphism iff the object is in the additive closure of shifts of the functor's image.

Applications to geometric contexts like derived splinters and rational singularities.

Establishes a link between categorical properties and geometric singularity conditions.

Abstract

Given a right adjoint functor between triangulated categories and an object in the target category, we show that the unit map of adjunction on that object is a split monomorphism if and only if the object belongs to the additive closure of (all possible) shifts of an object in the image of the functor. Applications to geometric context related to (derived) splinters and rational singularities are given.