Abelian étale coverings currents on graphs and semistable reduction of curves
Let \(R\) be a discrete valuation ring which is strictly henselian. Let \(\Xi\) be a semi-stable curve over \(R\) with generic fiber \(\Xi_\eta\) smooth and geometrically connected. The main result is the following (theorem 4.3 (a)): let \(\overline K\), resp. \(k\), be an algebraic closure of \(K=\text{Quot}(R)\), resp. the residue field of \(R\), then, for a positive integer \(n\) prime to the characteristic of \(k\), one has the exact sequence \[ 0\to H^1_{et} (\Xi\times_Rk, \mathbb{Z}/n \mathbb{Z}) \to H^1_{et} (\Xi \times_R \overline K, \mathbb{Z}/n \mathbb{Z})\to {\mathcal C} (\Gamma, \mathbb{Z}/n \mathbb{Z}) \to 0 \] where \({\mathcal C} (\Gamma, \mathbb{Z}/n \mathbb{Z})\) are the currents (also called harmonic cochains or harmonic cocycles) defined on the intersection graph \(\Gamma\) of \(\Xi \times_Rk\) and with values in \(\mathbb{Z}/n \mathbb{Z}\). The exactness of this sequence is implies, among other properties, by the fact that every \(\mu_n\)-torsor over \(\Xi\times_R \overline K\) is defined over \(\text{Quot} (R')\), where \(R'\) is a finite extension of \(R\) (theorem 4.2 (b)). This result has been announced by \textit{M. Saïdi} in C. R. Acad. Sci., Paris, Sér. I 316, No. 12, 1299-1302 (1993; Zbl 0788.14023). As applications (of results and of arguments of the proofs), the author gives descriptions of semi-stable models for cyclic étale coverings of \(\Xi_\eta\) of degrees prime to the characteristic of \(k\).
scientific article; zbMATH DE number 866134
scientific article