D\'{e}vissage for generation in derived categories

TL;DR

This paper extends the understanding of how derived categories of Noetherian schemes can be generated by specific complexes, providing explicit descriptions of strong generators especially for schemes with singularities.

Contribution

It generalizes Takahashi's affine result to global schemes and analyzes generation behavior under derived pushforward, enabling explicit identification of strong generators.

Findings

Bounded derived category generated by perfect complexes and structure sheaves.

Generation behavior under proper surjective morphisms analyzed.

Explicit strong generators identified for schemes with isolated singularities.

Abstract

We study a form of d\'{e}vissage for generation in derived categories of Noetherian schemes. First, we extend a result of Takahashi from the affine context to the global setting, showing that the bounded derived category is classically generated by a perfect complex together with structure sheaves of closed subschemes supported on the singular locus. Second, we make an observation for how generation behaves under the derived pushforward of a proper surjective morphism between Noetherian schemes. These results enable us to explicitly identify strong generators for projective schemes with isolated singularities and for singular varieties over a perfect field.