$p$-可除群
一个$p$-可除群是一个仿射群概形塔$$G_{1}\subset G_{2}\subset G_{3}\subset G_{4}\subset \cdot\cdot\cdot$$并满足某些额外性质。
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1. 定义
一个诺特环$k$上的高度为$h$的$p$-可除群（或Barsotti-Tate群）是一个$k$上的仿射群概形序列$G_1, G_2, G_3, . . .$，以及对每个$n \geq 1$一个态射$i_n : G_n \to G_{n+1}$，对每个$n \geq 1$满足以下特点：
(i) $G_n$是$p^{nh}$阶有限平坦的；
(ii) 态射$i_ n$是单射的，并有像$G_{n+1}[p^{n}]$。
换句话说，在将$G_n$看成与它在$G_{n+1}$中的像一样的意义下，一个$p$-可除群是一个仿射群概形塔$$G_{1}\subset G_{2}\subset G_{3}\subset G_{4}\subset \cdot\cdot\cdot$$使得$G_{n} = G_{n+1}[p^{n}]$对所有$n \geq 1$，并且$G_{n}$是$p^{nh}$阶有限平坦的。
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Some thought of p-divisible groups and p-adic Hodge theory
The name "p-divisible group" is somewhat misleading, which in fact has another name "Barsotti-Tate group". A p-divisible group is not merely just a kind of group. lt is more general. In fact, a p-divisible group can be viewed as a tower of affine group schemes with some extra conditions. Historically, p-divisible groups were the main stimulus for p-adic Hodge theory. So l think that studying p-divisible groups is the key to study p-adic Hodge theory.However, today's p-adic Hodge theory is much more complicated. l failed to understand Fontaine rings in the past. But this won't suppress my interest in p-adic Hodge theory. For now, I'm still studying Peter Scholze's Perfectoid Space, which was published nearly ten years ago and has already obtained a lot of beautiful results. lf l have time, l will still try to study p-adic Hodge theory.----------This short articles was originally published at June 24, 2021.
2024-09-30 22:44:09