Luna's fundamental lemma for diagonalizable groups

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 → 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,étale, smooth, regular, syntomic, or lci.