论文标题
算术喷气空间内核
Kernel of Arithmetic Jet Spaces
论文作者
论文摘要
由于此处的结果已被作者所面包的另一篇论文所取代，因此本文仅用于参考目的。 修复Dedekind域$ \ MATHCAL {O} $和INC中的非零Prime $ \ Mathfrak {p} $以及均匀器$π$。在本文的第一部分中，我们构建$ m $ chhifted $π$ typical witt vectors $ w_ {mn}（b）$，用于任何$ \ mathcal {o} $ algebra $ b $ buggy $ m+m+n+n+1 $。它们是对通常的$π$ typical witt向量的概括。随之而来的是，我们构造了Frobenius的升降机，称为frobenius $ \ tilde {f}：w_ {mn}（b）（b）\ rightArrow w_ {m（n-1）}（b）$，并表明它可以与通常的Frobenius Map满足自然身份。现在给定了一个组方案$ g $定义在$ \ mathrm {spec} 〜r $上的定义，其中$ r $是$ \ mathcal {o} $ - 带有固定的$π$衍生$δ$的代数，它自然而然地考虑了$ n $ n $ n $ n $ n $ n $ thth arithmetic quet quient quient quin $ j^ng $ j^ng $的点是$ g $。这导致了组方案的自然投影图$ u：j^{m+n} g \ rightarrow j^mg $。令$ n^{mn} g $表示$ u $的内核。 我们的主要结果之一暗示，对于任何$π$ - 形式组方案，$ \ hat {g} $上的$ \ mathrm {spf} 〜r $，$ n^{mn} \ hat {g} $ is omorphic to $ j^{n-1}（n^{m1} g）$。作为一个应用程序，如果$ \ hat {g} $是一种平稳的交换性$π$ - 尺寸$ d $和$ r $的形式组方案，其特征性0是其特征性的，以上的后，我们的结果在上面的限制为$ p-2 $，那么我们的结果意味着$ j^ng $是$ \ hat {g} $ by $ by $（n n n n n w w wer） $ \ mathbb {w} _ {n-1} $是$π$ - 形式组方案$ \ hat {\ mathbb {a}}}^n $赋予了添加witt vectors的组定律。我们的结果还给出了$ g（π^{n+1} r）$的几何表征，该$是$ g（r）$的点的子组，可在modulo $π^{n+1} $ map下降低到身份。
Since the results here have been superseded by another paper cowritten by the author, this article is available for reference purposes only. Fix a Dedekind domain $\mathcal{O}$ and a non-zero prime $\mathfrak{p}$ in it along with a uniformizer $π$. In the first part of the paper, we construct $m$-shifted $π$-typical Witt vectors $W_{mn}(B)$ for any $\mathcal{O}$ algebra $B$ of length $m+n+1$. They are a generalization of the usual $π$-typical Witt vectors. Along with it we construct a lift of Frobenius, called the lateral Frobenius $\tilde{F}: W_{mn}(B) \rightarrow W_{m(n-1)}(B)$ and show that it satisfies a natural identity with the usual Frobenius map. Now given a group scheme $G$ defined over $\mathrm{Spec}~ R$, where $R$ is an $\mathcal{O}$-algebra with a fixed $π$-derivation $δ$ on it, one naturally considers the $n$-th arithmetic jet space $J^nG$ whose points are the Witt ring valued points of $G$. This leads to a natural projection map of group schemes $u: J^{m+n}G \rightarrow J^mG$. Let $N^{mn}G$ denote the kernel of $u$. One of our main results imply that for any $π$-formal group scheme $\hat{G}$ over $\mathrm{Spf}~ R$, $N^{mn}\hat{G}$ is isomorphic to $J^{n-1}(N^{m1}G)$. As an application, if $\hat{G}$ is a smooth commutative $π$-formal group scheme of dimension $d$ and $R$ is of characteristic 0 whose ramification is bounded above by $p-2$, then our result implies that $J^nG$ is a canonical extension of $\hat{G}$ by $(\mathbb{W}_{n-1})^d$ where $\mathbb{W}_{n-1}$ is the $π$-formal group scheme $\hat{\mathbb{A}}^n$ endowed with the group law of addition of Witt vectors. Our results also give a geometric characterization of $G(π^{n+1}R)$ which is the subgroup of points of $G(R)$ that reduces to identity under the modulo $π^{n+1}$ map.
