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 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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