论文标题
非库赛田地的代数品种的共同体学
Cohomology of algebraic varieties over non-archimedean fields
论文作者
论文摘要
我们基于Hrushovski-Loeser的稳定完成，开发了一个代数封闭的非估值的非Archimedean Field $ k $的代数封闭的代数品种的融资协会理论。同时，我们在热带半集团$γ_\ infty $的O最低扩展中开发了可确定子集的捆绑共同体，其中$γ$表示$ k $的价值组。对于准标准品种，这两个共同体都通过稳定完成同质形态的变形缩回到$γ_\ infty $的可确定子集的变形缩回。在这两种情况下，我们都表明，相应的辅助理论满足了Eilenberg-Steenrod公理，有限和不变性，并且我们在每种情况下都提供了共同体学维度的自然界限。作为一种应用，我们表明可确定家庭中有限的同构类型的同构类型。此外，由于代数品种的稳定完成与V. Berkovich意义上的分析之间的牢固关系，我们恢复并扩展了有关代数品种有关有限性和不变性的拓扑结构的结果。
We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $Γ_\infty$, where $Γ$ denotes the value group of $K$. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $Γ_\infty$. In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the topological cohomology of the analytification of algebraic varieties concerning finiteness and invariance.
