If f is flat and locally of finite presentation, then f is universally open .< ref > EGA IV < sub > 2 </ sub >, Théorème 2. 4. 6 .</ ref > However, if f is faithfully flat and quasi-compact, it is not in general true that f is open, even if X and Y are noetherian .< ref > EGA IV < sub > 2 </ sub >, Remarques 2. 4. 8 ( i ).</ ref > Furthermore, no converse to this statement holds: If f is the canonical map from the reduced scheme X < sub > red </ sub > to X, then f is a universal homeomorphism, but for X noetherian, f is never flat .< ref > EGA IV < sub > 2 </ sub >, Remarques 2. 4. 8 ( ii ).</ ref >

If f is flat and locally of finite presentation, then for each of the following properties P, the set of points where f has P is open :< ref > EGA IV < sub > 3 </ sub >, Théorème 12. 1. 6 .</ ref >

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