Relative dimension of morphisms and dimension for algebraic stacks

TL;DR

This paper develops a versatile framework for analyzing relative dimensions and codimension in algebraic stacks, facilitating deformation arguments and advancing the understanding of moduli spaces.

Contribution

It introduces a new language for lower bounds on relative dimension of morphisms of stacks and explores properties of codimension, enhancing tools for moduli theory.

Findings

Robust theory applicable to various algebraic stacks

Simplified dimension-based deformation techniques

Foundational properties of codimension for stacks

Abstract

Motivated by applications in moduli theory, we introduce a flexible and powerful language for expressing lower bounds on relative dimension of morphisms of schemes, and more generally of algebraic stacks. We show that the theory is robust and applies to a wide range of situations. Consequently, we obtain simple tools for making dimension-based deformation arguments on moduli spaces. Additionally, in a complementary direction we develop the basic properties of codimension for algebraic stacks. One of our goals is to provide a comprehensive toolkit for working transparently with dimension statements in the context of algebraic stacks.