Lemma 39.7.12. Let $k$ be a field. Let $T = \mathop{\mathrm{Spec}}(A)$ where $A$ is a directed colimit of algebras which are finite products of copies of $k$. For any scheme $X$ over $k$ we have $|T \times _ k X| = |T| \times |X|$ as topological spaces.

Proof. By taking an affine open covering we reduce to the case of an affine $X$. Say $X = \mathop{\mathrm{Spec}}(B)$. Write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ with $A_ i = \prod _{t \in T_ i} k$ and $T_ i$ finite. Then $T_ i = |\mathop{\mathrm{Spec}}(A_ i)|$ with the discrete topology and the transition morphisms $A_ i \to A_{i'}$ are given by set maps $T_{i'} \to T_ i$. Thus $|T| = \mathop{\mathrm{lim}}\nolimits T_ i$ as a topological space, see Limits, Lemma 32.4.6. Similarly we have

\begin{align*} |T \times _ k X| & = |\mathop{\mathrm{Spec}}(A \otimes _ k B)| \\ & = |\mathop{\mathrm{Spec}}(\mathop{\mathrm{colim}}\nolimits A_ i \otimes _ k B)| \\ & = \mathop{\mathrm{lim}}\nolimits |\mathop{\mathrm{Spec}}(A_ i \otimes _ k B)| \\ & = \mathop{\mathrm{lim}}\nolimits |\mathop{\mathrm{Spec}}(\prod \nolimits _{t \in T_ i} B)| \\ & = \mathop{\mathrm{lim}}\nolimits T_ i \times |X| \\ & = (\mathop{\mathrm{lim}}\nolimits T_ i) \times |X| \\ & = |T| \times |X| \end{align*}

by the lemma above and the fact that limits commute with limits. $\square$

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