Luna's fundamental lemma for diagonalizable groups

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.