Report on locally finite triangulated categories (Q2905332)

The main aim of this very useful report is, as the author says, ``to provide the foundation for studying the locally finite triangulated categories''. In order to do this the author presents characterizations for some finiteness conditions on essentially small triangulated categories. For instance locally noetherian categories are characterized in Theorem 2.1, and locally finite idemsplit categories are characterized in Proposition 2.3. The proofs of these results are also useful since they are good examples about applications of the abelianisation of a triangulated categories.NEWLINENEWLINEThe paper also contains information about Auslander-Reiten triangles (Section 3), the lattice of thick subcategories (Section 4) with an example of an application to the Auslander-Reiten theory (Theorem 4.6), and applications to simply connected triangulated categories (Theorem 5.1), respectively to the classification of some connected hereditary Artin algebras (Theorem 6.10).