Properties of Flat Morphisms
Let f : X → Y be a morphism of schemes. For a morphism g : Y′ → Y, let X′ = X ×Y Y′ and f′ = f × g : X′ → Y′. f is flat if and only if for every g, the pullback f′* is an exact functor from the category of quasi-coherent -modules to the category of quasi-coherent -modules.
Assume that f : X → Y and g : Y → Z are morphisms of schemes. Assume furthermore that f is flat at x in X. Then g is flat at f(x) if and only if gf is flat at x. In particular, if f is faithfully flat, then g is flat or faithfully flat if and only if gf is flat or faithfully flat, respectively.
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