Lemma 29.46.2. Let $A$ be a ring. Let $B = \mathop{\mathrm{colim}}\nolimits B_\lambda $ be a filtered colimit of $A$-algebras. If each $f_\lambda : \mathop{\mathrm{Spec}}(B_\lambda ) \to \mathop{\mathrm{Spec}}(A)$ is a universal homeomorphism, resp. a universal homeomorphism inducing isomorphisms on residue fields, resp. universally closed, resp. universally closed and universally injective, then the same thing is true for $f : \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$.

**Proof.**
If $f_\lambda $ is universally closed, then $B_\lambda $ is integral over $A$ by Lemma 29.44.7. Hence $B$ is integral over $A$ and $f$ is universally closed (by the same lemma). This proves the case where each $f_\lambda $ is universally closed.

For a prime $\mathfrak q \subset B$ lying over $\mathfrak p \subset A$ denote $\mathfrak q_\lambda \subset B_\lambda $ the inverse image. Then $\kappa (\mathfrak q) = \mathop{\mathrm{colim}}\nolimits \kappa (\mathfrak q_\lambda )$. Thus if $A \to B_\lambda $ induces purely inseparable extensions of residue fields, then the same is true for $A \to B$. This proves the case where $f_\lambda $ is universally closed and universally injective, see Lemma 29.10.2.

The case where $f$ is a universal homeomorphism follows from the remarks above and Lemma 29.45.5 combined with the fact that prime ideals in $B$ are the same thing as compatible sequences of prime ideals in all of the $B_\lambda $.

If $A \to B_\lambda $ induces isomorphisms on residue fields, then so does $A \to B$ (see argument in second paragraph). In this way we see that the lemma holds in the remaining case. $\square$

## 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Your email address will not be published.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)

There are also: