Lemma 36.32.4. Let f : X \to S be a morphism of finite presentation. Let \mathcal{F} be an \mathcal{O}_ X-module of finite presentation, flat over S with support proper over S. If R^ if_*\mathcal{F} = 0 for i > 0, then f_*\mathcal{F} is locally free and its formation commutes with arbitrary base change (see proof for explanation).
Proof. By Lemma 36.30.1 the object E = Rf_*\mathcal{F} of D(\mathcal{O}_ S) is perfect and its formation commutes with arbitrary base change, in the sense that Rf'_*(g')^*\mathcal{F} = Lg^*E for any cartesian diagram
\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S }
of schemes. Since there is never any cohomology in degrees < 0, we see that E (locally) has tor-amplitude in [0, b] for some b. If H^ i(E) = R^ if_*\mathcal{F} = 0 for i > 0, then E has tor amplitude in [0, 0]. Whence E = H^0(E)[0]. We conclude H^0(E) = f_*\mathcal{F} is finite locally free by More on Algebra, Lemma 15.74.2 (and the characterization of finite projective modules in Algebra, Lemma 10.78.2). Commutation with base change means that g^*f_*\mathcal{F} = f'_*(g')^*\mathcal{F} for a diagram as above and it follows from the already established commutation of base change for E. \square
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