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Effective models and extension of torsors over a discrete valuation ring of unequal characteristic. (English) Zbl 1194.14069

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)

MSC:

14L30 Group actions on varieties or schemes (quotients)
14L15 Group schemes
13A50 Actions of groups on commutative rings; invariant theory

Citations:

Zbl 0234.14002
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