## AISLES IN DERIVED CATEGORIES

## 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 D^{b}(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 D^{b}(mod k
A_{n}) presented in section 5. Finally, we consider the
classification problem for aisles in D^{b}(mod A) where A is a
Dynkin-algebra. We reduce it to the classification of the silting sets
in D^{b}(mod A), which we carry out for the quivers with
underlying graph A_{n}.
We thank P. Gabriel for lectures on these topics and for his
helpful criticisms during the preparation of the manuscript.

http://www.math.jussieu.fr/~keller/publ/smashabs.html

Bernhard Keller, le 9 juin 2003.

keller@math.jussieu.fr

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