Families of Berkovich spaces
Families of Berkovich spaces
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伯科维奇空间族
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发表时间：
2011
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通讯作者：
Antoine Ducros
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作者：
Antoine Ducros
本文致力于系统地研究相对性质的背景下，伯科维奇解析空间。我们首先发展了一个理论的平坦度在这种设置。在通过反例表明，天真的平坦性不能是正确的概念，因为它是不稳定的基础变化，我们定义平坦性的基础变化下的{em要求}不变性，我们研究了第一类重要的平坦态射，即准光滑态射。然后，我们表明存在的地方{em d 'evissages}（在雷诺和格鲁森的精神）相干层，我们使用，连同研究的地方环的“通用纤维”的态射，证明了一个{em boundaryless}，天真平坦的态射是平坦的。然后证明了紧致解析空间的平坦态射的象可以被紧致的、相对Cohen-Macaulay的和零维的多剖复盖，并且证明了后者的象是目标的紧致解析域;从而恢复了Raynaud关于紧致解析空间的平坦态射的象是目标的紧致解析域的结果。在本文的最后一部分，我们研究了一些效度位点。我们首先证明了解析空间的一个给定态射的平坦轨迹是源的一个Zakirki-开子集。然后，我们看看在{em纤维}的有效性轨迹通常的交换代数性质。我们证明了我们所期望的结果实际上是成立的：这样一个属性的纤维有效性轨迹是Zerkiki-可构造的，并且在适当的额外假设（平坦性，有时也是等维性）下是Zerkiki-开放的。
This text is devoted to the systematic study of relative properties in the context of Berkovich analytic spaces. We first develop a theory of flatness in this setting. After having shown through a counter-example that naive flatness cannot be the right notion because it is not stable under base change, we define flatness by {em requiring} invariance under base change, and we study a first important class of flat morphisms, that of quasi-smooth ones. We then show the existence of local {em d'evissages} (in the spirit of Raynaud and Gruson) for coherent sheaves, which we use, together with a study of the local rings of "generic fibers" of morphisms, to prove that a {em boundaryless}, naively flat morphism is flat. After that we prove that the image of a compact analytic space by a flat morphism can be covered by a compact, relatively Cohen-Macaulay and zero-dimensional multisection, and the image of the latter is shown to be a compact analytic domain of the target; we thus recover the result by Raynaud telling that the image of a compact analytic space under a flat morphism is a compact analytic domain of the target. In the last part of this work we study some validity loci. We first prove that the flatness locus of a given morphism of analytic spaces is a Zariski-open subset of the source. We then look at the {em fiberwise} validity loci of the usual commutative algebra properties. We prove that the results we could expect actually hold: the fiberwise validity locus of such a property is Zariski-constructible, and Zariski-open under suitable extra assumptions (flatness, and also sometimes equidimentionality).