The dimension of the product is the sum of the dimensions.

Lemma 33.20.5. Let $k$ be a field. Let $X$, $Y$ be locally algebraic $k$-schemes.

1. For $z \in X \times Y$ lying over $(x, y)$ we have $\dim _ z(X \times Y) = \dim _ x(X) + \dim _ y(Y)$.

2. We have $\dim (X \times Y) = \dim (X) + \dim (Y)$.

Proof. Proof of (1). Consider the factorization

$X \times Y \longrightarrow Y \longrightarrow \mathop{\mathrm{Spec}}(k)$

of the structure morphism. The first morphism $p : X \times Y \to Y$ is flat as a base change of the flat morphism $X \to \mathop{\mathrm{Spec}}(k)$ by Morphisms, Lemma 29.25.8. Moreover, we have $\dim _ z(p^{-1}(y)) = \dim _ x(X)$ by Morphisms, Lemma 29.28.3. Hence $\dim _ z(X \times Y) = \dim _ x(X) + \dim _ y(Y)$ by Morphisms, Lemma 29.28.2. Part (2) is a direct consequence of (1). $\square$

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