# Explicit HRS-Tilting

## TL;DR

This paper provides explicit descriptions of the abelian category associated with a torsion pair, including chain complexes and derived categories, along with DG structures, offering new proofs of existing results.

## Contribution

It introduces explicit descriptions of categories related to torsion pairs and decorated complexes, enhancing understanding of their structure and DG properties.

## Key findings

- Explicit descriptions of $B$, $Ch(B)$, and $D(B)$ for a torsion pair
- New proofs of results by Happel-Reiten-Smalo
- Description of DG structure on $Ch(B)$

## Abstract

For an abelian category $A$ equipped with a torsion pair, we give an explicit description for the abelian category $B$ introduced by Happel-Reiten-Smalo, and also for the category of chain complexes $Ch(B)$ and the derived category $D(B)$ of $B$. We also describe the DG structure on $Ch(B)$. As a consequence, we find new proofs of certain results of Happel-Reiten-Smalo. The main ingredient is the category of {\em decorated} complexes.