Descent conditions for generation in derived categories

TL;DR

This paper identifies conditions under which strong generation in derived categories is preserved under proper morphisms, providing bounds on Rouquier dimension and exploring tensor functors in algebraic geometry.

Contribution

It introduces a new criterion for preservation of strong generation in derived categories and derives bounds on Rouquier dimension, with applications to affine varieties.

Findings

Established a condition for preservation of strong generation under proper morphisms.

Provided upper bounds on Rouquier dimension of derived categories.

Identified a necessary and sufficient condition for tensor-exact functors to preserve strong generators.

Abstract

This work establishes a condition that determines when strong generation in the bounded derived category of a Noetherian $J\textrm{-}2$ scheme is preserved by the derived pushforward of a proper morphism. Consequently, we can produce upper bounds on the Rouquier dimension of the bounded derived category, and applications concerning affine varieties are studied. In the process, a necessary and sufficient constraint is observed for when a tensor-exact functor between rigidly compactly generated tensor triangulated categories preserves strong $\oplus$-generators.