Lemma 43.13.1. Let $X$ and $Y$ be varieties. Then $X \times Y$ is a variety and $\dim (X \times Y) = \dim (X) + \dim (Y)$.

**Proof.**
The scheme $X \times Y = X \times _{\mathop{\mathrm{Spec}}(\mathbf{C})} Y$ is a variety by Varieties, Lemma 33.3.3. The statement on dimension is Varieties, Lemma 33.20.5.
$\square$

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