## Local-to-global extensions of representations of fundamental groups.(English)Zbl 0564.14013

Let $$K$$ be a field of characteristic $$p>0$$, $$C$$ a proper, smooth, geometrically connected curve over $$K$$, and $$0$$ and $$\infty$$ two $$K$$-rational points on $$C$$. We show that any representation of the local Galois group at $$\infty$$ extends to a representation of the fundamental group of $$C\setminus\{0,\infty\}$$ which is tamely ramified at $$0$$, provided either that $$K$$ is separately closed or that $$C$$ is $$\mathbb P^1$$. In the latter case, we show there exists a unique such extension, called “canonical”, with the property that the image of the geometric fundamental group has a unique $$p$$-Sylow subgroup. As an application, we give a global cohomological construction of the Swan representation in equal characteristic.

### MSC:

 14H30 Coverings of curves, fundamental group 14H25 Arithmetic ground fields for curves 11S20 Galois theory 14G20 Local ground fields in algebraic geometry
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### References:

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