Lemma 15.6.2. In Situation 15.6.1 we have

$\mathop{\mathrm{Spec}}(B') = \mathop{\mathrm{Spec}}(B) \amalg _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A')$

as topological spaces.

Proof. Since $B' = B \times _ A A'$ we obtain a commutative square of spectra, which induces a continuous map

$can : \mathop{\mathrm{Spec}}(B) \amalg _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A') \longrightarrow \mathop{\mathrm{Spec}}(B')$

as the source is a pushout in the category of topological spaces (which exists by Topology, Section 5.29).

To show the map $can$ is surjective, let $\mathfrak q' \subset B'$ be a prime ideal. If $I \subset \mathfrak q'$ (here and below we take the liberty of considering $I$ as an ideal of $B'$ as well as an ideal of $A'$), then $\mathfrak q'$ corresponds to a prime ideal of $B$ and is in the image. If not, then pick $h \in I$, $h \not\in \mathfrak q'$. In this case $B_ h = A_ h = 0$ and the ring map $B'_ h \to A'_ h$ is an isomorphism, see Lemma 15.5.3. Thus we see that $\mathfrak q'$ corresponds to a unique prime ideal $\mathfrak p' \subset A'$ which does not contain $I$.

Since $B' \to B$ is surjective, we see that $can$ is injective on the summand $\mathop{\mathrm{Spec}}(B)$. We have seen above that $\mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(B')$ is injective on the complement of $V(I) \subset \mathop{\mathrm{Spec}}(A')$. Since $V(I) \subset \mathop{\mathrm{Spec}}(A')$ is exactly the image of $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(A')$ a trivial set theoretic argument shows that $can$ is injective.

To finish the proof we have to show that $can$ is open. To do this, observe that an open of the pushout is of the form $V \amalg U'$ where $V \subset \mathop{\mathrm{Spec}}(B)$ and $U' \subset \mathop{\mathrm{Spec}}(A')$ are opens whose inverse images in $\mathop{\mathrm{Spec}}(A)$ agree. Let $v \in V$. We can find a $g \in B$ such that $v \in D(g) \subset V$. Let $f \in A$ be the image. Pick $f' \in A'$ mapping to $f$. Then $D(f') \cap U' \cap V(I) = D(f') \cap V(I)$. Hence $V(I) \cap D(f')$ and $D(f') \cap (U')^ c$ are disjoint closed subsets of $D(f') = \mathop{\mathrm{Spec}}(A'_{f'})$. Write $(U')^ c = V(J)$ for some ideal $J \subset A'$. Since $A'_{f'} \to A'_{f'}/IA'_{f'} \times A'_{f'}/JA'_{f'}$ is surjective by the disjointness just shown, we can find an $a'' \in A'_{f'}$ mapping to $1$ in $A'_{f'}/IA'_{f'}$ and mapping to zero in $A'_{f'}/JA'_{f'}$. Clearing denominators, we find an element $a' \in J$ mapping to $f^ n$ in $A$. Then $D(a'f') \subset U'$. Let $h' = (g^{n + 1}, a'f') \in B'$. Since $B'_{h'} = B_{g^{n + 1}} \times _{A_{f^{n + 1}}} A'_{a'f'}$ by a previously cited lemma, we see that $D(h')$ pulls back to an open neighbourhood of $v$ in the pushout, i.e., the image of $V \amalg U'$ contains an open neighbourhood of the image of $v$. We omit the (easier) proof that the same thing is true for $u' \in U'$ with $u' \not\in V(I)$. $\square$

Comment #5899 by Thibaud van den Hove on

In the second paragraph, I think you want to distinguish between whether or not $I\subseteq\mathfrak{q}'$ instead of whether or not $I\cap \mathfrak{q}'=0$: the inverse image in $B\times_A A'$ of a prime ideal in $B$ will always contain $(0,x)$ for all $x\in I$. Similarly, you also want $h\in I\setminus \mathfrak{q}'$ instead of $h\in I\cap \mathfrak{q}'$.

I also found a few typo's:

In the second paragraph, you want to consider $I$ as an ideal of $B'$ and $A'$ instead of $B'$ and $A$. The following are all in the last paragraph: you wrote $\to '_f/IA'_f$, and forgot the $A$ (the $f$ should be $f'$, but I had some trouble formatting, sorry about that), you want $h'=(g^{n+1},a'f')$ instead of $(g^n,a'f')$, you wrote $D(h)$ instead of $D(h')$, you wrote $V\coprod U$ instead of $V \coprod U'$, * you use both $J$ and $J'$, but they should mean the same thing.

Comment #5900 by Thibaud van den Hove on

I tried putting the typo's in a list, but it looks like I has some more formatting problems, sorry about this.

Comment #6101 by on

Thanks very much. This can still be cleaned up further but I am going to take your comments as essentially telling me the proof isn't wrong. Fixes are here.

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