Paper detail

Ladders and simplicity of derived module categories

Recollements of derived module categories are investigated, using a new technique, ladders of recollements, which are mutation sequences. The position in the ladder is shown to control whether a recollement restricts from unbounded to another level of derived category. Ladders also turn out to control derived simplicity on all levels. An algebra is derived simple if its derived category cannot be deconstructed, that is, if it is not the middle term of a non-trivial recollement whose outer terms are again derived categories of algebras. Derived simplicity on each level is characterised in terms of heights of ladders. These results are complemented by providing new classes of examples of derived simple algebras, in particular indecomposable commutative rings, as well as by a finite-dimensional counterexample to the Jordan--Hölder property for derived module categories. Moreover, recollements are used to compute homological and K-theoretic invariants.