# Could we prove the flat base change theorem for cohomology via injective resolution?

Let $X$ be a quasi-separated scheme over a base ring $A$ and $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $A\to B$ be a flat morphism and $X^{\prime}:=X\times_{\text{spec}(A)}\text{spec}(B)$ be the base change scheme. Let $p: X^{\prime}\to X$ be the projection and $\mathcal{F}^{\prime}:=p^*\mathcal{F}$. The flat base change theorem claims that the canonical map $$H^i(X,\mathcal{F})\otimes_A B\to H^i(X^{\prime},\mathcal{F}^{\prime})$$ is an isomorphism. See Stack Project Lemma 29.5.2.

The proof given in Stack Project Lemma 29.5.2 is via Cech cohomology. I wonder if we could prove this result through injective resolution. I know one problem is that the pullback of an injective sheaf is not necessarily injective, but do we have a way to overcome this problem?

• Have you looked at the proof in Kempf's paper Some elemetary proofs of basic theorems in cohomology of quasicoherent sheaves .He uses flabby resolutions and Lazard's theorem – Mohan Ramachandran May 24 '17 at 20:28