The Stacks project

Lemma 26.17.2. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. If $X, Y, S$ are all affine then $X \times _ S Y$ is affine.

Proof. Suppose that $X = \mathop{\mathrm{Spec}}(A)$, $Y = \mathop{\mathrm{Spec}}(B)$ and $S = \mathop{\mathrm{Spec}}(R)$. By Lemma 26.6.7 the affine scheme $\mathop{\mathrm{Spec}}(A \otimes _ R B)$ is the fibre product $X \times _ S Y$ in the category of locally ringed spaces. Hence it is a fortiori the fibre product in the category of schemes. $\square$

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