Remark 30.22.2. A consequence of Lemma 30.22.1 is that there exists a finite complex of finite projective $A$-modules $M^\bullet $ such that we have

\[ H^ i(X_{A'}, \mathcal{F}_{A'}) = H^ i(M^\bullet \otimes _ A A') \]

functorially in $A'$. The condition that $\mathcal{F}$ is flat over $A$ is essential, see [Hartshorne].

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