The Stacks project

Lemma 33.24.3. Let $k$ be a field. Let $f : A \to X$, $g : B \to Y$ be quasi-compact morphisms of schemes over $k$. Let $Z \subset X$ be the scheme theoretic image of $f$, see Morphisms, Definition 29.6.2. Similarly, let $Z' \subset Y$ be the scheme theoretic image of $g$. Then $Z \times _ k Z'$ is the scheme theoretic image of $f \times g$.

Proof. Recall that $Z$ is the smallest closed subscheme of $X$ through which $f$ factors. Similarly for $Z'$. Let $W \subset X \times _ k Y$ be the scheme theoretic image of $f \times g$. As $f \times g$ factors through $Z \times _ k Z'$ we see that $W \subset Z \times _ k Z'$.

To prove the other inclusion let $U \subset X$ and $V \subset Y$ be affine opens. By Morphisms, Lemma 29.6.3 the scheme $Z \cap U$ is the scheme theoretic image of $f|_{f^{-1}(U)} : f^{-1}(U) \to U$, and similarly for $Z' \cap V$ and $W \cap U \times _ k V$. Hence we may assume $X$ and $Y$ affine. As $f$ and $g$ are quasi-compact this implies that $A = \bigcup U_ i$ is a finite union of affines and $B = \bigcup V_ j$ is a finite union of affines. Then we may replace $A$ by $\coprod U_ i$ and $B$ by $\coprod V_ j$, i.e., we may assume that $A$ and $B$ are affine as well. In this case $Z$ is cut out by $\mathop{\mathrm{Ker}}(\Gamma (X, \mathcal{O}_ X) \to \Gamma (A, \mathcal{O}_ A))$ and similarly for $Z'$ and $W$. Hence the result follows from the equality

\[ \Gamma (A \times _ k B, \mathcal{O}_{A \times _ k B}) = \Gamma (A, \mathcal{O}_ A) \otimes _ k \Gamma (B, \mathcal{O}_ B) \]

which holds as $A$ and $B$ are affine. Details omitted. $\square$

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