Explicit HRS-Tilting

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.

💡 Research Summary

The paper gives a concrete description of the Happel‑Reiten‑Smalo (HRS) tilted heart (\mathcal{B}) associated with a torsion pair ((\mathcal{T},\mathcal{F})) in an abelian category (\mathcal{A}). Traditionally, (\mathcal{B}) is defined abstractly as the heart of a t‑structure on the bounded derived category (D^{b}(\mathcal{A})) generated by (\mathcal{T}