Remarks on diagonal dimension for algebraic stacks

TL;DR

This paper investigates the diagonal dimension of algebraic stacks and morphisms to establish bounds on Rouquier dimension, with results for smooth morphisms, fiber products, and varieties with mild singularities.

Contribution

It provides explicit upper bounds for diagonal dimension in various contexts, extending understanding of derived categories on algebraic stacks.

Findings

Upper bound for smooth morphisms with regular targets

Identification of strong generators for fiber products

Diagonal dimension of certain varieties is at most twice their Krull dimension

Abstract

This note is concerned with the Rouquier dimension of the bounded derived category of coherent complexes on a Noetherian algebraic stack. Specifically, we study the diagonal dimension of a morphism, which can be used to produce upper bounds on Rouquier dimension. First, we obtain an explicit upper bound for smooth morphisms with a regular target. Second, we identify strong generators of a fiber product, recovering a result of Elagin--Lunts--Schn\"{u}rer. Finally, we show that the diagonal dimension of a variety in arbitrary characteristic with mild singularities is at most twice its Krull dimension.