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.Comment: 31 pages, introduction expanded, minor errors correcte

Lectures on derived and triangulated categories

These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.Comment: Lecture notes. To appear in the proceedings volume of the "International Workshop on Noncommutative Geometry", IPM, Tehran 200