The Stacks project

Lemma 55.8.1. Let $V_1 \to V_2$ be a closed immersion of algebraic schemes over $K$. If $X_2$ is a model for $V_2$, then the scheme theoretic image of $V_1 \to X_2$ is a model for $V_1$.

Proof. Using Morphisms, Lemma 29.6.3 and Example 29.6.4 this boils down to the following algebra statement. Let $A_1$ be a finite type $R$-algebra flat over $R$. Let $A_1 \otimes _ R K \to B_2$ be a surjection. Then $A_2 = A_1 / \mathop{\mathrm{Ker}}(A_1 \to B_2)$ is a finite type $R$-algebra flat over $R$ such that $B_2 = A_2 \otimes _ R K$. We omit the detailed proof; use More on Algebra, Lemma 15.22.11 to prove that $A_2$ is flat. $\square$

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