Rapoport-Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let $\mathscr{M}_{\infty}$ be an infinite-level Rapoport-Zink space of EL type, and let $\mathscr{M}_{\infty}^\circ$ be one geometrically connected component of it. We show that $\mathscr{M}_{\infty}^{\circ}$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^\infty)^{\circ}$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

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无限级 Rapoport-Zink 空间中的光滑轨迹

Rapoport-Zink 空间是具有附加结构的 $p$-可分群的变形空间。在无限层次上，它们成为完美的空间。令 $\mathscr{M}_{\infty}$ 是一个无限级的 EL 类型的 Rapoport-Zink 空间，让 $\mathscr{M}_{\infty}^\circ$ 是它的一个几何连通分量. 我们证明 $\mathscr{M}_{\infty}^{\circ}$ 包含一个密集的开放子集，它在 Scholze 的意义上是上同调平滑的。这是没有任何额外自同态的 $p$-可分群的轨迹。作为推论，我们发现无限级模曲线 $X(p^\infty)^{\circ}$ 中的上同调光滑轨迹正是超奇异归约椭圆曲线 $E$ 的轨迹，使得形式化$E$ 组没有额外的内同态。