摘要翻译:
设$X$是一个简化的连通$k$-Scheme指向X(k)$中的一个rational点$X\。利用tannakian技巧构造了满足条件$h^0(Y,\Mathcal{O}_Y)=k$的本质有限的$k$-态射$f:Y\到x$的Galois闭包;这个Galois闭包是一个torsor$P:\hat{X}_y\to X$通过一个$X$-morphism$\lambda:\hat{X}_y\to y$控制$f$并且对于这个性质是通用的。此外,我们还证明了$\lambda:\hat{X}_y\to y$在我们描述的有限群格式下是一个torsor。进一步证明了在$y$上本质有限向量丛的直像仍然是在$x$上本质有限向量丛。对于torsors和本质有限态射,我们发展了一个类似于通常的Galois对应。作为应用,我们证明了在满足条件$h^0(Y,\mathcal{O}_Y)=k$的有限群格式下,对于任意尖torsor$f:Y\to X$,$Y$有一个基本群格式$\pi_1(Y,Y)$与$\pi_1(X,X)$以短精确序列拟合。
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英文标题:
《Galois Closure of Essentially Finite Morphisms》
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作者:
Marco Antei, Michel Emsalem
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最新提交年份:
2010
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分类信息:
一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
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一级分类:Mathematics 数学
二级分类:Number Theory 数论
分类描述:Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数,丢番图方程,解析数论,代数数论,算术几何,伽罗瓦理论
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英文摘要:
Let $X$ be a reduced connected $k$-scheme pointed at a rational point $x \in X(k)$. By using tannakian techniques we construct the Galois closure of an essentially finite $k$-morphism $f:Y\to X$ satisfying the condition $H^0(Y,\mathcal{O}_Y)=k$; this Galois closure is a torsor $p:\hat{X}_Y\to X$ dominating $f$ by an $X$-morphism $\lambda:\hat{X}_Y\to Y$ and universal for this property. Moreover we show that $\lambda:\hat{X}_Y\to Y$ 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 $f:Y \to X$ under a finite group scheme satisfying the condition $H^0(Y,\mathcal{O}_Y)=k$, $Y$ has a fundamental group scheme $\pi_1 (Y,y)$ fitting in a short exact sequence with $\pi_1 (X,x)$.
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PDF链接:
https://arxiv.org/pdf/0901.1551