A fine moduli space is constructed, for cyclic-by-$\mathsf{ p}$ covers of an affine curve over an algebraically closed field $k$ of characteristic $\mathsf{ p}>0$. An intersection of finitely many fine moduli spaces for cyclic-by-$\mathsf{ p}$ covers of affine curves gives a moduli space for $\mathsf{ p}'$-by-$\mathsf{ p}$ covers of an affine curve. A local moduli space is also constructed, for cyclic-by-$\mathsf{ p}$ covers of $Spec(k((x)))$, which is the same as the global moduli space for cyclic-by-$\mathsf{ p}$ covers of $\mathbb{ P}^1-\{ 0\}$ tamely ramified over $\infty$ with the same Galois group. Then it is shown that a restriction morphism is finite with degrees on connected components $\mathsf{ p}$ powers: There are finitely many deleted points of an affine curve from its smooth completion. A cyclic-by-$\mathsf{ p}$ cover of an affine curve gives a product of local covers with the same Galois group of the punctured infinitesimal neighbourhoods of the deleted points. So there is a restriction morphism from the global moduli space to a product of local moduli spaces.
Categories: math.AG