文章

Local System

发布于 2026-06-17T05:48:24Z · 公开内容更新于 2026-08-17T13:52:21Z

An A-local system on the topological space X is a sheaf mod(A X) which is locally constant, i.e. there is an open covering U i of X and a family of A -modules M

Local System

1 第一节

1.1 小节

Definition 1.1.

An A-local system on the topological space $𝑋$ is a sheaf $L \in 𝑚 𝑜 𝑑 (𝐴 𝑋)$ which is locally constant, i.e. there is an open covering ${𝑈 𝑖}$ of $𝑋$ and a family of $𝐴$ -modules ${𝑀 𝑖}$ such that

L | 𝑈 𝑖 ≃ 𝑀 𝑖,

the constant sheaf on $𝑈 𝑖$ associated to the module $𝑀 𝑖$ .

Proposition 1.1.

The following categories are equivalent

1.

A - local systems on $𝑋$ .
2.

covariant functors $𝐿$ from the fundamental groupoid of $𝑋$ to the category of $𝐴$ -modules.
3.

representations $𝜌 : 𝜋 1 (𝑋, 𝑥 0) \to 𝐴 𝑢 𝑡 (𝑀)$ , where $𝜋 1 (𝑋, 𝑥 0)$ is the fundamental group of $𝑋$ based at a given point $𝑥 0$ and $𝑀$ is an $𝐴$ -module. 其中若 $𝑥 0 \in 𝑈 𝑖$ , 则 $𝑀 : = 𝑀 𝑖$ .

设 $ℍ$ 是连通概型 $𝑆$ 上的一个 local system, 其对应一个 $𝜌 : 𝜋 (𝑆, 𝑥 0) \to 𝐴 𝑢 𝑡 (ℍ 𝑥 0)$ . 我们取 $𝑚 \in ℍ 𝑥 0$ , 一条从 $𝑥 0$ 到 $𝑥$ 的路径，将会给出 $𝑠 𝑥 : ℍ 𝑥 0 \to ℍ 𝑥$ 的映射，如果这个映射在 $𝑚$ 点的取值 $𝑠 𝑥 (𝑚)$ 与路径无关，则称 $𝑠 (𝑚) : 𝑥 \mapsto 𝑠 𝑥 (𝑚)$ 为 $𝑚$ 对应的 flat section(horizontal section).

Proposition 1.2.

Let $𝑓 : 𝑋 \to 𝑌$ be a continuous map between topological spaces. If $L$ is a local system on $𝑌$ given by a representation $(𝑀, 𝜌)$ , then the inverse image sheaf $𝑓 - 1 L$ is a local system on $𝑋$ corresponding to the representation $(𝑀, 𝜌 \cdot 𝑓 *)$ with $𝑓 * : 𝜋 1 (𝑋, 𝑥 0) \to 𝜋 1 (𝑌, 𝑦 0), 𝑦 0 = 𝑓 (𝑥 0)$ the homomorphism induced by $𝑓$ at the level of fundamental groups.