p进数域Q_p同伦型及其在拓扑学与代数几何中的应用
$\mathbb{Q}_p$-Homotopy Types and Applications to Topology and Algebraic Geometry
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中文总结 AI 辅助
本文发展Q_p同伦理论，关联p完全空间与Q_p上的交换微分分次代数，证明其极小模型可还原同伦群等结构，并将该理论应用于拓扑与代数几何的多类问题。
中文摘要 AI 辅助
我们针对p完全空间发展了Q_p同伦理论。对任意p完全空间X，我们通过对奇异上链构成的E_∞代数S^*(X;Z_p帽)⊗_{Z_p帽}Q_p进行修正，将其与Q_p上的一个交换微分分次代数相关联。对于幂零且有限型的p完全空间，我们证明该代数的极小模型可还原出Q_p同伦群与Whitehead乘积，这与Sullivan的有理同伦理论直接类似。我们还证明，对于非单连通的p完全空间，其1-极小模型的李代数对偶是基本群的连续Mal'cev Q_p完备化对应的李代数。我们将Q_p同伦理论应用于拓扑学与代数几何中的多个问题，包括p完全空间的有限实现问题、平展同伦型的有限性性质、光滑射影簇的形式性、平展同伦群上的伽罗瓦表示，以及平展基本群的约束条件。
英文摘要
We develop a $\mathbb{Q}_p$-homotopy theory for $p$-complete spaces. To a $p$-complete space $X$, we associate a commutative differential graded algebra over $\mathbb{Q}_p$ by rectifying the $E_\infty$-algebra $S^*(X;\widehat{\mathbb{Z}}_p)\otimes_{\widehat{\mathbb{Z}}_p} \mathbb{Q}_p$ of singular cochains. For nilpotent $p$-complete finite type spaces, we prove that the minimal model of this algebra recovers the $\mathbb{Q}_p$-homotopy groups and Whitehead products, in direct analogy with Sullivan's rational homotopy theory. We also prove that, for a non-simply-connected $p$-complete space, the Lie algebra dual to its $1$-minimal model is the Lie algebra of the continuous Mal'cev $\mathbb{Q}_p$-completion of the fundamental group. We apply the $\mathbb{Q}_p$-homotopy theory to several questions in topology and algebraic geometry, including finite realization problems for $p$-complete spaces, finiteness properties of étale homotopy types, formality of smooth proper varieties, Galois representations on étale homotopy groups, and constraints on étale fundamental groups.
