# Vanishing of higher direct image of finite morphisms relative to the fppf topology

Let $$f:X \to Y$$ be a finite morphism of schemes. Let $$\mathcal{F}$$ be a sheaf of abelian groups on the the etale site of $$X$$ then we know that $$R^{i}f_{*} \mathcal{F} = 0$$. Is this statement also true when $$\mathcal{F}$$ is sheaf of abelian groups on the fppf site?

## 1 Answer

No, it is not true. Let $$k$$ be an algebraically closed field of characteristic $$p > 0$$ and set $$k' := k[x]/(x^2)$$. Let $$f \colon \mathrm{Spec}(k') \rightarrow \mathrm{Spec}(k)$$ be the corresponding map. Then the sequence $$0 \rightarrow \mu_p \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 0$$ will show you that $$(R^1f_*(\mu_p))(k) \cong k'^{\times}/k'^{\times p} \neq 0$$.

As far as I know, it is an open question whether such vanishing is true when $$f$$ is a closed immersion.