Luna's fundamental lemma for diagonalizable groups

TL;DR

This paper extends Luna's fundamental lemma to diagonalizable group actions on schemes, providing criteria for equivariance, base change, and descent properties such as étale and smooth morphisms.

Contribution

It generalizes Luna's fundamental lemma specifically for diagonalizable groups, establishing new criteria for strong equivariance and descent of various morphism properties.

Findings

Criteria for a $G$-equivariant morphism to be strongly equivariant.

Conditions under which the quotient morphism inherits properties like étale, smooth, or regular.

Extension of Luna's fundamental lemma to a broader class of group actions.

Abstract

We study relatively affine actions of a diagonalizable group $G$ on locally noetherian schemes. In particular, we generalize Luna's fundamental lemma when applied to a diagonalizable group: we obtain criteria for a $G$-equivariant morphism $f: X'\to X$ to be $strongly\ equivariant$, namely the base change of the morphism $f/\!/G$ of quotient schemes, and establish descent criteria for $f/\!/G$ to be an open embedding, \'etale, smooth, regular, syntomic, or lci.