For F-finite animated rings in characteristic p, every map factors as a free finite-type map, a relatively perfect map, and a surjective map, and relatively perfect maps coincide with formally etale maps in the Noetherian case.
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Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de sché- mas. (Première partie). Rédigé avec la colloboration de J. Dieudonné
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Relative Inverse Limit Perfection of Derived Commutative Rings
For F-finite animated rings in characteristic p, every map factors as a free finite-type map, a relatively perfect map, and a surjective map, and relatively perfect maps coincide with formally etale maps in the Noetherian case.