# Existence of finite etale covering and cohomology of the profinite completion of the fundamental group

Let $$X$$ be a connected complex-analytic space. , $$G = \pi_1(X)$$ the fundamental group of $$X$$ , $$\hat{G}=\varprojlim G/N$$ its profinite completion.

Let $$\beta\in H^2(X,\mathcal{O}_X^\times).$$, say with $$n\beta=0$$, and choose a class $$\alpha\in H^2(X,\mu_n)$$ mapping to $$\beta$$.

suppose that : $$H^2(G,\mu_n(\mathbb{C}) ) \simeq H^2(\hat{G},\mu_n(\mathbb{C}))\simeq \varinjlim H^2(G/N,\mu_n(\mathbb{C})).$$ and $$H^2(G,\mu_n(\mathbb{C}) )\simeq H^2(X,\mu_n)$$ Then there is an finite étale covering $$g:X'\rightarrow X$$ with $$g^*(\beta)=0$$, and in particular $$g^*(\alpha)=0$$.

I couldn't realize how can we get this finite étale covering $$g :X'\rightarrow X$$ from the previous assertions and why $$g^*(\beta)=0$$?.

• There has to be some $N$ such that $\beta$ is in the H2 of G/N. Pick the corresponding cover and by inflation restriction or something like that this element maps to zero in H2(N,..). I think this works... – Asvin Aug 11 at 4:47
• Thank dear Asvin, may this work for the vanishing of β, but still don't understand why the corresponding cover is finite and etale – Moutand Mohammed yesterday
• The cover is automatically finite etale, by definition. – Asvin yesterday