Bull. Soc. Math. Belg. 40 (1988), 239-253

Bernhard Keller and Dieter Vossieck

The aim of the present paper is to demonstrate the usefulness of aisles for studying the tilting theory of Db(mod A), where A is a finite-dimensional algebra. In section 1, we establish the equivalence of "aisles" with "t-structures" in the sense of Beilinson-Bernstein-Deligne and give a characterization of aisles in molecular categories. Section 2 contains an application to the generalized tilting theory of hereditary algebras. Using aisles, we then give a geometrical proof of the theorem of Happel which states that a finite-dimensional algebra which shares its derived category with a Dynkin-algebra A can be transformed into A by a finite number of reflections. The techniques developed so far naturally lead to the classification of the tilting sets in Db(mod k An) presented in section 5. Finally, we consider the classification problem for aisles in Db(mod A) where A is a Dynkin-algebra. We reduce it to the classification of the silting sets in Db(mod A), which we carry out for the quivers with underlying graph An. We thank P. Gabriel for lectures on these topics and for his helpful criticisms during the preparation of the manuscript.

Bernhard Keller, le 9 juin 2003.
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