In this book, the projection formula stated as follows;
Let $f:X\to Y$ a separated, quasi-compact morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on X, $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Then $(R^pf_*\mathcal{F})\otimes_{\mathcal{O}_Y}\mathcal{G}\cong R^pf_*(\mathcal{F}\otimes_{\mathcal{O}_X}f^*\mathcal{G})$ when $\mathcal{G}$ is flat over $Y$.
But in other references, like Hartshorne or Vakil, it is little different: isomorphism holds when $\mathcal{G}$ is locally free (of finite rank).
I think flat $\mathcal{O}_Y$-module is not locally free in general case. (These two are equivalent when $Y$ is locally noetherian and $\mathcal{G}$ is coherent)
Q. Can we prove the formula just with flat condition?
Actually, the proof in the book seems wrong; the author states $R^pf_*(\mathcal{F}\otimes_{\mathcal{O}_X}f^*\mathcal{G})(V)=H^p(f^{-1}(V),\mathcal{F}|_{f^{-1}(V)}\otimes_{\mathcal{O}_Y(V)}\mathcal{G}(V))$(this is the Cech cohomology) for affine open $V$. But, I think, this equality does not hold. If it is true, how can I prove?
Best Answer
As $V$ is affine, one has $$R^pf_∗(F\otimes_{O_X}f^∗G)(V)=H^p(f^{−1}(V), (F\otimes f^*G)|_{f^{-1}(V)})$$ (see e.g. Prop. 5.2.28 of the book you cite). But $(F\otimes f^*G)|_{f^{-1}(V)}=F|_{f^{-1}(V)}\otimes G(V)$. Hence the equality you are after.