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

Luna's fundamental lemma for diagonalizable groups