×

Aisles in derived categories. (English) Zbl 0671.18003

Let \({\mathcal T}\) be a triangulated category with suspension functor \({\mathcal S}\). An aisle in \({\mathcal T}\) is a full additive coreflective subcategory \({\mathcal U}\) of \({\mathcal T}\) which is stable under extensions and satisfies \({\mathcal S}{\mathcal U}\subset {\mathcal U}\). The article aims to demonstrate the usefulness of aisles in the tilting theory of the bounded derived category \({\mathcal D}^ b(A)\) of mod A, where A is a finite dimensional algebra over an algebraically closed field k. A geometric proof of a result of D. Happel concerning finite dimensional k-algebras B such that \({\mathcal D}^ b(B)\) is \({\mathcal S}\)-equivalent to \({\mathcal D}^ b(A)\) is given; and the problem of classifying aisles in \({\mathcal D}^ b(k\Delta)\) is considered, where \(k\Delta\) is the path algebra of a Dynkin quiver \(\Delta\). This classification problem is shown to reduce to the problem of classifying a generalization, termed silting sets by the authors, of tilting sets in \({\mathcal D}^ b(k\Delta)\). Indeed, the authors obtain a classification of silting sets in the case when \(\Delta =A_ n\), having already gotten a classification of tilting sets in \({\mathcal D}^ b(kA_ n)\) earlier in the paper.
Reviewer: R.Gordon

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
PDFBibTeX XMLCite