Projective Dimensions in Cluster-Tilted Categories

TL;DR

This paper investigates the projective dimensions of functors restricted to rigid subcategories in triangulated categories, providing characterizations of objects with infinite projective dimension in cluster-tilted contexts.

Contribution

It offers a new criterion linking morphisms in triangulated categories to the projective dimension of associated functors, advancing understanding of cluster-tilted categories.

Findings

Projective dimension at most one corresponds to absence of certain morphisms.

Characterization of infinite projective dimension objects in cluster-tilted categories.

Provides conditions for when functors have finite or infinite projective dimension.

Abstract

We study the projective dimensions of the restriction of functors Hom(-,X) to a contravariantly finite rigid subcategory T of a triangulated category C. We show that the projective dimension of Hom(-,X)|T is at most one if and only if there are no non-zero morphisms between objects in T[1] factoring through X, when the object X belongs to a suitable subcategory of C. As a consequence, we obtain a characterisation of the objects of infinite projective dimension in the category of finitely presented contravariant functors on a cluster-tilting subcategory of C.