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

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

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)