文章

Geometric quotient

作者：Lunifans

发布于 2026-07-09T13:02:58Z · 公开内容更新于 2026-08-17T13:52:31Z

主要讲讲简单的几何商知识。

Geometric quotient

Abstract

主要讲讲简单的几何商知识。

Definition 1.

这里的 pre-scheme 应该就是现代的 scheme.

Definition 2.

An algebraic group $𝐺$ over a field $𝑘$ , is a group scheme $𝐺 / 𝑘$ which is an algebraic scheme, smooth over $𝑘$ .

Definition 3.

A group pre-scheme $𝐺 / 𝑆$ acts or operates on a pre-scheme $𝑋 / 𝑆$ if an $𝑆$ -morphism $𝜎 : 𝐺 \times 𝑆 𝑋 \to 𝑋$ is given, such that:

[(a)]
1.

commutes.
2.

The composition:
$𝑋 ≅ 𝑆 \times 𝑆 𝑋 𝑒 \times 1 \leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow \to 𝐺 \times 𝑆 𝑋 𝜎 \to 𝑋$
equals $1 𝑋$ .

Definition 4.

Let $𝑓 : 𝑇 \to 𝑋$ be a $𝑇$ -valued point of $𝑋$ . Then $𝜎 \circ (1 𝐺 \times 𝑓)$ is a morphism from $𝐺 \times 𝑆 𝑇$ to $𝑋$ .

𝐺 \times 𝑇 1 𝐺 \times 𝑓 \leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow \to 𝐺 \times 𝑆 𝑋 𝜎 \to 𝑋 .

Define the morphism

𝜓 𝐺 𝑓 : 𝐺 \times 𝑆 𝑇 \to 𝑋 \times 𝑆 𝑇

as $(𝜎 \circ (1 𝐺 \times 𝑓), 𝑝 2)$ .

𝐺 \times 𝑆 𝑇 1 𝐺 \times 𝑓 \leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow\leftarrow \to 𝐺 \times 𝑆 𝑋 𝜎 \to 𝑋

𝐺 \times 𝑆 𝑇 𝑝 2 ⟶ 𝑇

If no confusion arises, $𝜓 𝐺 𝑓$ will be shortened to $𝜓 𝑓$ . If $𝑓 = 1 𝑋$ , $𝜓 𝑓$ will be denoted $Ψ$ . $Ψ$ is simply:

(𝜎, 𝑝 2) : 𝐺 \times 𝑆 𝑋 \to 𝑋 \times 𝑆 𝑋 .

The image of $𝜓 𝑓$ will be denoted $𝑂 (𝑓)$ and called the orbit of $𝑓$ . Now $𝑋 \times 𝑆 𝑇$ , as a scheme over $𝑇$ , has a canonical section, namely $(𝑓, 1 𝑇)$ . Via this, we set up a fibre product defining $𝑆 (𝑓)$ :

Definition 5.

Given an action $𝜎$ of $𝐺 / 𝑆$ on $𝑋 / 𝑆$ , a pair $(𝑌, 𝜙)$ consisting of a pre-scheme $𝑌 / 𝑆$ and an $𝑆$ -morphism $𝜙 : 𝑋 \to 𝑌$ will be called a categorical quotient (of $𝑋$ by $𝐺$ ) if

[(i)]
1.

the diagram:

commutes.
2.

(universal property) given any pair $(𝑍, 𝜓)$ consisting of a pre-scheme $𝑍$ over $𝑆$ , and an $𝑆$ -morphism $𝜓 : 𝑋 \to 𝑍$ sucht that $𝜓 \circ 𝜎 = 𝜓 \circ 𝑝 2$ ,i.e. (i) holds for $𝑍$ and $𝜓$ , then there is a unique $𝑆$ -morphism $X : 𝑌 \to 𝑍$ such that $𝜓 = X \circ 𝜙$ .

Definition 6.

Given an action $𝜎$ of $𝐺 / 𝑆$ on $𝑋 / 𝑆$ , a pair $(𝑌, 𝜙)$ consisting of a pre-scheme $𝑌$ over $𝑆$ and an $𝑆$ -morphism $𝜙 : 𝑋 \to 𝑌$ will be called a geometric quotient (of $𝑋$ by $𝐺$ ) if

[(i)]
1.

$𝜙 \circ 𝜎 = 𝜙 \circ 𝑝 2$ .

commutes.
2.

$𝜙$ is surjective, and the image of $Ψ$ is $𝑋 \times 𝑌 𝑋$ .
3.

$𝜙$ is submersive, i.e. a subset $𝑈 \subset 𝑌$ is open $⟺ 𝜙 - 1 (𝑈)$ is open in $𝑋 .$
4.

the fundamental sheaf $O 𝑌$ is the subsheaf of $𝜙 * (O 𝑌)$ consisting of invariant functions, i.e. if $𝑓 \in Γ (𝑈, 𝜙 * (O 𝑋)) = Γ (𝜙 - 1 (𝑈), O 𝑋)$ , then $𝑓 \in Γ (𝑈, O 𝑌)$ if and only if

commutes.

Definition 7.

Given an action $𝜎$ of $𝐺 / 𝑆$ on $𝑋 / 𝑆$ , a pair $(𝑌, 𝜙)$ as above will be called a universal categorical quotient(universal geometric quotient) if, for all morphisms $𝑌' \to 𝑌$ , we put $𝑋' = 𝑋 \times 𝑌 𝑌'$ and let $𝜙' : 𝑋' \to 𝑌'$ denotes $𝑝 2$ , then $(𝑌', 𝜙')$ is a categorical quotient(geometric quotient) of $𝑋'$ by $𝐺$ .

If this holds only for flat morphism $𝑌' \to 𝑌$ , then $(𝑌, 𝜙)$ will be called a uniform categorical quotient(uniform geometric quotient).

Geometric quotient

Abstract

Definition 1.

Definition 2.

Definition 3.

Definition 4.

Definition 5.

Definition 6.

Definition 7.

References