# Silting and cosilting classes in derived categories

## TL;DR

This paper extends tilting theory to derived categories by characterizing silting and cosilting classes as certain orthogonal and definable subcategories, revealing their structure within t-structures and co-t-structures.

## Contribution

It provides a derived category analogue of classical tilting and cotilting class characterizations, introducing silting and cosilting classes and their properties.

## Key findings

- Silting classes are intermediate and Ext-orthogonal to a set of compact objects.
- Cosilting classes are cosuspended, definable, and co-intermediate.
- Characterization of silting and cosilting classes within derived categories.

## Abstract

An important result in tilting theory states that a class of modules over a ring is a tilting class if and only if it is the Ext-orthogonal class to a set of compact modules of bounded projective dimension. Moreover, cotilting classes are precisely the resolving and definable subcategories of the module category whose Ext-orthogonal class has bounded injective dimension. In this article, we prove a derived counterpart of the statements above in the context of silting theory. Silting and cosilting complexes in the derived category of a ring generalise tilting and cotilting modules. They give rise to subcategories of the derived category, called silting and cosilting classes, which are part of both a t-structure and a co-t-structure. We characterise these subcategories: silting classes are precisely those which are intermediate and Ext-orthogonal classes to a set of compact objects, and cosilting classes are precisely the cosuspended, definable and co-intermediate subcategories of the derived category.