Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber

Antei, Marco

doi:10.5802/jtnb.731

Cited by 6 publications

(8 citation statements)

References 12 publications

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“…By a quotient pointed G-torsor q : Y → X η we mean a pointed torsor whose associated morphism π 1 (X η , x η ) → G is faithfully flat. Using standard techniques (cf., for instance, [2], lemma 2.7) we are able to prove the following Theorem 3.10: Let G be a finite and commutative K-group scheme. Then every quotient pointed G-torsor q : Y → X η can be extended to a pointed Gtorsor Z → X for some (necessarily commutative) model G of G finite and flat over S.…”

Section: 3mentioning

confidence: 96%

“…We already know that if X is an abelian scheme, every (necessarily) commutative quotient (i.e., the group scheme acting on it is a quotient of the fundamental group scheme of X η ) pointed torsor over X η can be extended to X (cf. [2], §3.2). Tossici recently proved in [27], Corollary 4.9 that if S = Spec(R), where R is a d.v.r.…”

Section: Introductionmentioning

confidence: 95%

“…As already discussed in [2], the study of the fundamental group scheme is tightly related to the problem of extending finite and pointed torsors over the generic fiber X η of X to torsors over X. Here we concentrate our attention on commutative torsors.…”

Section: Introductionmentioning

confidence: 99%

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On the abelian fundamental group scheme of a family of varieties

Antei¹

2011

Isr. J. Math.

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Let S be a connected Dedekind scheme and X an S-scheme provided with a section x. We prove that the morphism between fundamental group schemes π 1 (X, x) ab → π 1 (Alb X/S , 0 Alb X/S ) induced by the canonical morphism from X to its Albanese scheme Alb X/S (when the latter exists) fits in an exact sequence of group schemes 0 → (NS τ X/S ) ∨ → π 1 (X, x) ab → π 1 (Alb X/S , 0 Alb X/S ) → 0, where the kernel is a finite and flat S-group scheme. Furthermore, we prove that any finite and commutative quotient pointed torsor over the generic fiber Xη of X can be extended to a finite and commutative pointed torsor over X.

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Section: 3mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 95%

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On the abelian fundamental group scheme of a family of varieties

Antei¹

2011

Isr. J. Math.

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“…Now it is easy to find an R-model H of G (commutative, finite and flat) and a pointed (over 0 NJ ) H-torsor Y → N J whose generic fiber is isomorphic to T → J (cf. for instance [2], §2.2). Then finally Y ′ := Y × NJ X, the pull back over u ′ , is a finite, commutative H-torsor over X (pointed over x) extending T ′ → X η .…”

Section: Commutative Torsorsmentioning

confidence: 99%

Extension of finite solvable torsors over a curve

Antei¹

2012

manuscripta math.

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Let R be a discrete valuation ring with fraction field K and with algebraically closed residue field of positive characteristic p. Let X be a smooth fibered surface over R with geometrically connected fibers endowed with a section x ∈ X(R). Let G be a finite solvable K-group scheme and assume that either |G| = p n or G has a normal series of length 2. We prove that every quotient pointed G-torsor over the generic fiber X η of X can be extended to a torsor over X after eventually extending scalars and after eventually blowing up X at a closed subscheme of its special fiber X s .Mathematics Subject Classification: 14H30, 14L15.

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“…A pointed torsor (Y , G, y) is a quotient torsor if and only if G is a quotient of the fundamental group scheme π 1 (X, x), that is the canonical morphism π 1 (X, x) → G is faithfully flat (cf. for instance [1], Corollary 2.8).…”

Section: The Fundamental Group Schemementioning

confidence: 99%

Galois closure of essentially finite morphisms

Antei

Emsalem

2011

Journal of Pure and Applied Algebra

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a b s t r a c tLet X be a reduced connected k-scheme pointed at a rational point x ∈ X (k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphismthis Galois closure is a torsor p :X Y → X dominating f by an X -morphism λ :X Y → Y and universal for this property. Moreover, we show that λ :X Y → Y is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X . We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f : Y → X under a finite group scheme satisfying the condition H 0 (Y , O Y ) = k, Y has a fundamental group scheme π 1 (Y , y) fitting in a short exact sequence with π 1 (X, x).

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Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber

Cited by 6 publications

References 12 publications

On the abelian fundamental group scheme of a family of varieties

On the abelian fundamental group scheme of a family of varieties

Extension of finite solvable torsors over a curve

Galois closure of essentially finite morphisms

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