The Stacks project

78.14 The finite part of a groupoid

In this section we are going to use the idea explained in Section 78.13 to take the finite part of a groupoid in algebraic spaces.

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$. Assumption: The morphisms $s, t$ are separated and locally of finite type. This notation and assumption will we be fixed throughout this section.

Denote $R_ s$ the algebraic space $R$ seen as an algebraic space over $U$ via $s$. Let $U' = (R_ s/U, e)_{fin}$. Since $s$ is separated and locally of finite type, by Proposition 78.12.11 and Lemma 78.12.15, we see that $U'$ is an algebraic space endowed with an ├ętale morphism $g : U' \to U$. Moreover, by Lemma 78.12.1 there exists a universal open subspace $Z_{univ} \subset R \times _{s, U, g} U'$ which is finite over $U'$ and such that $(1_{U'}, e \circ g) : U' \to R \times _{s, U, g} U'$ factors through $Z_{univ}$. Moreover, by Lemma 78.12.4 the open subspace $Z_{univ}$ is also closed in $R \times _{s, U', g} U$. Picture so far:

\[ \xymatrix{ Z_{univ} \ar[d] \ar[rd] & \\ R \times _{s, U, g} U' \ar[d] \ar[r] & U' \ar[d]^ g \\ R \ar[r]^ s & U } \]

Let $T$ be a scheme over $B$. We see that a $T$-valued point of $Z_{univ}$ may be viewed as a triple $(u, Z, z)$ where

  1. $u : T \to U$ is a $T$-valued point of $U$,

  2. $Z \subset R \times _{s, U, u} T$ is an open and closed subspace finite over $T$ such that $(e \circ u, 1_ T)$ factors through it, and

  3. $z : T \to R$ is a $T$-valued point of $R$ with $s \circ z = u$ and such that $(z, 1_ T)$ factors through $Z$.

Having said this, it is morally clear from the discussion in Section 78.13 that we can turn $(Z_{univ}, U')$ into a groupoid in algebraic spaces over $B$. To make sure will define the morphisms $s', t', c', e', i'$ one by one using the functorial point of view. (Please don't read this before reading and understanding the simple construction in Section 78.13.)

The morphism $s' : Z_{univ} \to U'$ corresponds to the rule

\[ s' : (u, Z, z) \mapsto (u, Z). \]

The morphism $t' : Z_{univ} \to U'$ is given by the rule

\[ t' : (u, Z, z) \mapsto (t \circ z, c(Z, i \circ z)). \]

The entry $c(Z, i \circ z)$ makes sense as the map $c(-, i \circ z) : R \times _{s, U, u} T \to R \times _{s, U, t \circ z} T$ is an isomorphism with inverse $c(-, z)$. The morphism $e' : U' \to Z_{univ}$ is given by the rule

\[ e' : (u, Z) \mapsto (u, Z, (e \circ u, 1_ T)). \]

Note that this makes sense by the requirement that $(e \circ u, 1_ T)$ factors through $Z$. The morphism $i' : Z_{univ} \to Z_{univ}$ is given by the rule

\[ i' : (u, Z, z) \mapsto (t \circ z, c(Z, i \circ z), i \circ z). \]

Finally, composition is defined by the rule

\[ c' : ((u_1, Z_1, z_1), (u_2, Z_2, z_2)) \mapsto (u_2, Z_2, z_1 \circ z_2). \]

We omit the verification that the axioms of a groupoid in algebraic spaces hold for $(U', Z_{univ}, s', t', c', e', i')$.

A final piece of information is that there is a canonical morphism of groupoids

\[ (U', Z_{univ}, s', t', c', e', i') \longrightarrow (U, R, s, t, c, e, i) \]

Namely, the morphism $U' \to U$ is the morphism $g : U' \to U$ which is defined by the rule $(u, Z) \mapsto u$. The morphism $Z_{univ} \to R$ is defined by the rule $(u, Z, z) \mapsto z$. This finishes the construction. Let us summarize our findings as follows.

Lemma 78.14.1. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$. Assume the morphisms $s, t$ are separated and locally of finite type. There exists a canonical morphism

\[ (U', Z_{univ}, s', t', c', e', i') \longrightarrow (U, R, s, t, c, e, i) \]

of groupoids in algebraic spaces over $B$ where

  1. $g : U' \to U$ is identified with $(R_ s/U, e)_{fin} \to U$, and

  2. $Z_{univ} \subset R \times _{s, U, g} U'$ is the universal open (and closed) subspace finite over $U'$ which contains the base change of the unit $e$.

Proof. See discussion above. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04RT. Beware of the difference between the letter 'O' and the digit '0'.