Lemma 59.96.2. Let $K$ be a field.
If $f : X \to Y$ is a morphism of finite type schemes over $K$, then $\text{cd}(f) < \infty $.
If $\text{cd}(K) < \infty $, then $\text{cd}(X) < \infty $ for any finite type scheme $X$ over $K$.
Proof. Proof of (1). We may assume $Y$ is affine. We will use the induction principle of Cohomology of Schemes, Lemma 30.4.1 to prove this. If $X$ is affine too, then the result holds by Lemma 59.95.8. Thus it suffices to show that if $X = U \cup V$ and the result is true for $U \to Y$, $V \to Y$, and $U \cap V \to Y$, then it is true for $f$. This follows from the relative Mayer-Vietoris sequence, see Lemma 59.50.2.
Proof of (2). We will use the induction principle of Cohomology of Schemes, Lemma 30.4.1 to prove this. If $X$ is affine, then the result holds by Proposition 59.95.6. Thus it suffices to show that if $X = U \cup V$ and the result is true for $U$, $V$, and $U \cap V $, then it is true for $X$. This follows from the Mayer-Vietoris sequence, see Lemma 59.50.1. $\square$
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