Tossici, Dajano Effective models and extension of torsors over a discrete valuation ring of unequal characteristic. (English) Zbl 1194.14069 Int. Math. Res. Not. 2008, Article ID rnn111, 68 p. (2008). This paper is mainly devoted to the problem of extending torsors: let \(R\) be a discrete valuation ring of mixed characteristic \((0,p)\), i.e. with function field \(K\) and residual field \(k\) respectively of characteristic \(0\) and \(p>0\), \(X\) a faithfully flat \(R\)-scheme with generic and special fibers denoted (resp.) \(X_K\) and \(X_k\). We are given a finite \(K\)-group scheme \(G_K\) and a \(G_K\)-torsor \(f_K:Y_K\to X_K\); moreover let \(Y\) be the normalization of \(X\) in \(Y_K\) and \(f:Y\to X\) the natural morphism, then the author gives an answer, in some interesting cases that we will describe, to the following two problems:1. “Weak extension”: is it possible to find a finite and flat \(R\)-group scheme \(G'\) whose generic fiber is isomorphic to \(G_K\) and a \(G'\)-torsor \(f':Y'\to X\) extending \(f_K\), i.e. such that the generic fiber of \(f'\) is isomorphic, as a \(G_K\)-torsor, to \(f_K\)?2. “Strong extension”: is it possible to find a finite and flat \(R\)-group scheme \(G\) whose generic fiber is isomorphic to \(G_K\) such that \(f\) is a \(G\)-torsor extending \(f_K\)?For any \(G_K\) such that \((|G_K|,p)=1\) both problems are classical and a positive answer has already been given [cf. for instance A. Grothendieck, Revêtements étales et groupe fondamental. Lecture Notes in Mathematics. 224. Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0234.14002)]. A solution to the weak extension problem when \(G_K\) is commutative, constant and of order \(m\) is given by Corollary 4.2.8 for a normal \(X\) with integral fibers such that \({}_m\mathrm{Pic}(X)={}_m\mathrm{Pic}(X_K)\) and assuming that \(R\) contains a primitive \(m\)-th root of unity and that \(Y_k\), special fiber of \(Y\), is reduced. If \(X\) is affine (with some further assumptions), \(G_K=(\mathbb{Z}/p\mathbb{Z})_K\) and \(Y_K\) is connected then Theorem 5.1 gives a positive answer to the strong extension problem assuming that \(R\) contains a primitive \(p\)-th root of unity and that \(Y_k\) is reduced. With similar assumptions, but with \(G_K=(\mathbb{Z}/p^2\mathbb{Z})_K\), then Corollary 6.2.7 gives a criterion for extending (strongly) \(Y_K\). Reviewer: Marco Antei (Bonn) Cited in 11 Documents MSC: 14L30 Group actions on varieties or schemes (quotients) 14L15 Group schemes 13A50 Actions of groups on commutative rings; invariant theory Keywords:torsors; group actions; effective models Citations:Zbl 0234.14002 PDFBibTeX XMLCite \textit{D. Tossici}, Int. Math. Res. Not. 2008, Article ID rnn111, 68 p. (2008; Zbl 1194.14069) Full Text: DOI arXiv