On the relation between cluster and classical tilting

TL;DR

This paper explores the relationship between cluster tilting subcategories in a triangulated category and support tilting subcategories in its abelian quotient, establishing a correspondence and lifting properties.

Contribution

It demonstrates how cluster tilting subcategories project to support tilting subcategories and how these can be uniquely lifted back, linking triangulated and abelian categories.

Findings

Projection from D to D/U maps cluster tilting to support tilting subcategories.

Support tilting subcategories in D/U can be uniquely lifted to maximal 1-orthogonal subcategories in D.

The quotient category D/U has finite global dimension, enabling this correspondence.

Abstract

Let D be a triangulated category with a cluster tilting subcategory U. The quotient category D/U is abelian; suppose that it has finite global dimension. We show that projection from D to D/U sends cluster tilting subcategories of D to support tilting subcategories of D/U, and that, in turn, support tilting subcategories of D/U can be lifted uniquely to maximal 1-orthogonal subcategories of D.