The Stacks project

Lemma 29.4.6. Let $S$ be a scheme. Let $X, Y \subset S$ be closed subschemes. Let $X \cup Y$ be the scheme theoretic union of $X$ and $Y$. Let $X \cap Y$ be the scheme theoretic intersection of $X$ and $Y$. Then $X \to X \cup Y$ and $Y \to X \cup Y$ are closed immersions, there is a short exact sequence

\[ 0 \to \mathcal{O}_{X \cup Y} \to \mathcal{O}_ X \times \mathcal{O}_ Y \to \mathcal{O}_{X \cap Y} \to 0 \]

of $\mathcal{O}_ S$-modules, and the diagram

\[ \xymatrix{ X \cap Y \ar[r] \ar[d] & X \ar[d] \\ Y \ar[r] & X \cup Y } \]

is cocartesian in the category of schemes, i.e., $X \cup Y = X \amalg _{X \cap Y} Y$.

Proof. The morphisms $X \to X \cup Y$ and $Y \to X \cup Y$ are closed immersions by Lemma 29.2.2. In the short exact sequence we use the equivalence of Lemma 29.4.1 to think of quasi-coherent modules on closed subschemes of $S$ as quasi-coherent modules on $S$. For the first map in the sequence we use the canonical maps $\mathcal{O}_{X \cup Y} \to \mathcal{O}_ X$ and $\mathcal{O}_{X \cup Y} \to \mathcal{O}_ Y$ and for the second map we use the canonical map $\mathcal{O}_ X \to \mathcal{O}_{X \cap Y}$ and the negative of the canonical map $\mathcal{O}_ Y \to \mathcal{O}_{X \cap Y}$. Then to check exactness we may work affine locally. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open of $S$ and let $X \cap U$ and $Y \cap U$ correspond to the ideals $I \subset A$ and $J \subset A$. Then $(X \cup Y) \cap U$ corresponds to $I \cap J \subset A$ and $X \cap Y \cap U$ corresponds to $I + J \subset A$. Thus exactness follows from the exactness of

\[ 0 \to A/I \cap J \to A/I \times A/J \to A/(I + J) \to 0 \]

To show the diagram is cocartesian, suppose we are given a scheme $T$ and morphisms of schemes $f : X \to T$, $g : Y \to T$ agreeing as morphisms $X \cap Y \to T$. Goal: Show there exists a unique morphism $h : X \cup Y \to T$ agreeing with $f$ and $g$. To construct $h$ we may work affine locally on $X \cup Y$, see Schemes, Section 26.14. If $s \in X$, $s \not\in Y$, then $X \to X \cup Y$ is an isomorphism in a neighbourhood of $s$ and it is clear how to construct $h$. Similarly for $s \in Y$, $s \not\in X$. For $s \in X \cap Y$ we can pick an affine open $V = \mathop{\mathrm{Spec}}(B) \subset T$ containing $f(s) = g(s)$. Then we can choose an affine open $U = \mathop{\mathrm{Spec}}(A) \subset S$ containing $s$ such that $f(X \cap U)$ and $g(Y \cap U)$ are contained in $V$. The morphisms $f|_{X \cap U}$ and $g|_{Y \cap V}$ into $V$ correspond to ring maps

\[ B \to A/I \quad \text{and}\quad B \to A/J \]

which agree as maps into $A/(I + J)$. By the short exact sequence displayed above there is a unique lift of these ring homomorphism to a ring map $B \to A/I \cap J$ as desired. $\square$

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