Let $X$ be a projective scheme and $X\_0 \subset X$ a subscheme defined by a nilpotent ideal. Denote by $i:X\_0 \to X$ the closed immersion. Let $\mathcal{F}$ be a locally free sheaf sheaf on $X\_{\mathrm{et}}$ (the etale site associated to $X$). Denote by $i^\*\mathcal{F}$ the pullback (not just inverse image) of $\mathcal{F}$ to $X\_{0,\mathrm{et}}$. Is it true that the induced morphism $H^0(X\_{\mathrm{et}}, \mathcal{F}) \to H^0(X\_{0,\mathrm{et}}, i^\*\mathcal{F})$ surjective?



Etale cohomology and topological invariance
Since global sections of a coherent sheaf in Zariski or étale cohomology coincide, your question is really about Zariski cohomology of coherent (locally free) sheaves.


It has a negative answer, although it is not too easy to find a counterexample. I explain below one counter-example. The variety involved is a particular case of the ribbons studied here : <http://www.math.columbia.edu/~bayer/papers/Ribbons_BE95.pdf> . Using the material of this paper, you should be able to produce many more examples.


Let $k$ be a field, and $X\subset\mathbb{P}^2\_k$ be the double conic defined by $(xy+z^2)^2=0$. Let $X\_0$ be the reduction of $X$: it is the smooth conic defined by $xy+z^2=0$. It is easy to compute $H^0(X\_0, \mathcal{O})=H^0(X, \mathcal{O})=k$, $H^1(X\_0, \mathcal{O})=0$ and $H^1(X,\mathcal{O})\neq 0$.


Fix a nontrivial element $e\in H^1(X,\mathcal{O})$. Let $0\to\mathcal{O}\_X\to\mathcal{F}\to\mathcal{O}\_X\to 0$ be the associated extension: it is a rank $2$ vector bundle on $X$. In the associated long exact sequence $0\to H^0(X,\mathcal{O})\to H^0(X,\mathcal{F})\to H^0(X,\mathcal{O})\to H^1(X,\mathcal{O})$, the image of $1$ in $H^1(X,\mathcal{O})$ is the extension class $e$, so that $H^0(X,\mathcal{F})$ has dimension $1$.


On the other hand the restriction $0\to\mathcal{O}\_{X\_0}\to i^\*\mathcal{F}\to\mathcal{O}\_{X\_0}\to 0$ has to split because $H^1(X\_0, \mathcal{O})=0$. As a consequence, $H^0(X\_0,i^\*\mathcal{F})$ has dimension $2$.


For dimension reasons, $H^0(X,\mathcal{F})\to H^0(X\_0,i^\*\mathcal{F})$ cannot be surjective.


