The Stacks project

77.4 Equivalence relations

Please refer to Groupoids, Section 39.3 for notation.

Definition 77.4.1. Let $B \to S$ as in Section 77.3. Let $U$ be an algebraic space over $B$.

  1. A pre-relation on $U$ over $B$ is any morphism $j : R \to U \times _ B U$ of algebraic spaces over $B$. In this case we set $t = \text{pr}_0 \circ j$ and $s = \text{pr}_1 \circ j$, so that $j = (t, s)$.

  2. A relation on $U$ over $B$ is a monomorphism $j : R \to U \times _ B U$ of algebraic spaces over $B$.

  3. A pre-equivalence relation is a pre-relation $j : R \to U \times _ B U$ such that the image of $j : R(T) \to U(T) \times U(T)$ is an equivalence relation for all schemes $T$ over $B$.

  4. We say a morphism $R \to U \times _ B U$ of algebraic spaces over $B$ is an equivalence relation on $U$ over $B$ if and only if for every $T$ over $B$ the $T$-valued points of $R$ define an equivalence relation on the set of $T$-valued points of $U$.

In other words, an equivalence relation is a pre-equivalence relation such that $j$ is a relation.

Lemma 77.4.2. Let $B \to S$ as in Section 77.3. Let $U$ be an algebraic space over $B$. Let $j : R \to U \times _ B U$ be a pre-relation. Let $g : U' \to U$ be a morphism of algebraic spaces over $B$. Finally, set

\[ R' = (U' \times _ B U')\times _{U \times _ B U} R \xrightarrow {j'} U' \times _ B U' \]

Then $j'$ is a pre-relation on $U'$ over $B$. If $j$ is a relation, then $j'$ is a relation. If $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation. If $j$ is an equivalence relation, then $j'$ is an equivalence relation.

Proof. Omitted. $\square$

Definition 77.4.3. Let $B \to S$ as in Section 77.3. Let $U$ be an algebraic space over $B$. Let $j : R \to U \times _ B U$ be a pre-relation. Let $g : U' \to U$ be a morphism of algebraic spaces over $B$. The pre-relation $j' : R' \to U' \times _ B U'$ of Lemma 77.4.2 is called the restriction, or pullback of the pre-relation $j$ to $U'$. In this situation we sometimes write $R' = R|_{U'}$.

Lemma 77.4.4. Let $B \to S$ as in Section 77.3. Let $j : R \to U \times _ B U$ be a pre-relation of algebraic spaces over $B$. Consider the relation on $|U|$ defined by the rule

\[ x \sim y \Leftrightarrow \exists \ r \in |R| : t(r) = x, s(r) = y. \]

If $j$ is a pre-equivalence relation then this is an equivalence relation.

Proof. Suppose that $x \sim y$ and $y \sim z$. Pick $r \in |R|$ with $t(r) = x$, $s(r) = y$ and pick $r' \in |R|$ with $t(r') = y$, $s(r') = z$. We may pick a field $K$ such that $r$ and $r'$ can be represented by morphisms $r, r' : \mathop{\mathrm{Spec}}(K) \to R$ with $s \circ r = t \circ r'$. Denote $x = t \circ r$, $y = s \circ r = t \circ r'$, and $z = s \circ r'$, so $x, y, z : \mathop{\mathrm{Spec}}(K) \to U$. By construction $(x, y) \in j(R(K))$ and $(y, z) \in j(R(K))$. Since $j$ is a pre-equivalence relation we see that also $(x, z) \in j(R(K))$. This clearly implies that $x \sim z$.

The proof that $\sim $ is reflexive and symmetric is omitted. $\square$


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 043B. Beware of the difference between the letter 'O' and the digit '0'.