Use lemma from commit 964258e

This actually shortens the exposition for once.

more-morphisms.tex

@@ -3964,11 +3964,11 @@ \section{Pushouts in the category of schemes, I}
 \\
 Y \ar[r]
 &
-Y \amalg_X X'
+Y'
 }
 $$
-in the category of schemes. Moreover $Y' = Y \amalg_X X'$ is a
-thickening of $Y$ and
+in the category of schemes. Moreover, $Y \subset Y'$ is a
+thickening, $X = Y \times_{Y'} X'$, and
 $$
 \mathcal{O}_{Y'} = \mathcal{O}_Y \times_{f_*\mathcal{O}_X} f'_*\mathcal{O}_{X'}
 $$
@@ -4003,36 +4003,18 @@ \section{Pushouts in the category of schemes, I}
 claim follows.
 
 \medskip\noindent
-Finally, we prove the universal property of the pushout holds for
-$Y'$ and the morphisms $Y \to Y'$ and $X' \to Y'$. Namely, let $S$ be
-a scheme and let $b : Y \to S$ and $a' : X' \to S$ be morphisms such that
-$$
-\xymatrix{
-X \ar[r] \ar[d]
-&
-X' \ar[d]^{a'}
-\\
-Y \ar[r]^b
-&
-S
-}
-$$
-commutes. Note that $a' = b \circ f'$ on underlying topological spaces.
-Denote also $(a')^\sharp : b^{-1}\mathcal{O}_S \to f'_*\mathcal{O}_{X'}$
-the map which is adjoint to $(a')^\sharp :
-(a')^{-1}\mathcal{O}_S = (f')^{-1} b^{-1}\mathcal{O}_S \to \mathcal{O}_{X'}$.
-Then we get a map
-$$
-b^{-1}\mathcal{O}_S
-\xrightarrow{(b^\sharp, (a')^\sharp)}
-\mathcal{O}_Y \times_{f_*\mathcal{O}_X} f'_*\mathcal{O}_{X'}
-=
-\mathcal{O}_{Y'}
-$$
-which defines a morphism of ringed spaces $b' : Y' \to S$ compatible
-with $a'$ and $b$. Since $Y \subset Y'$ is a thickening it follows that
-$b'$ is a morphism of locally ringed spaces, i.e., a morphism of schemes.
-This finishes the proof.
+It remains to show that $Y'$ is the pushout.
+The discussion above shows the scheme $Y'$
+has an affine open covering $Y' = \bigcup W'_i$
+such that the corresponding opens
+$U'_i \subset X'$, $W_i \subset Y$, and
+$U_i \subset X$ are affine open.
+Moreover, if $A'_i$, $B_i$, $A_i$ are the rings corresponding to
+$U'_i$, $W_i$, $U_i$, then
+$W'_i$ corresponds to $B_i \times_{A_i} A'_i$.
+Thus we can apply Lemmas \ref{lemma-basic-example-pushout} and
+\ref{lemma-pushout-fpqc-local} to conclude our construction is a pushout
+in the category of schemes.
 \end{proof}
 
 \noindent
@@ -16824,12 +16806,12 @@ \section{Pushouts in the category of schemes, II}
 }
 $$
 The diagram is a fibre square, the morphism $a$ is integral,
-the morphism $b$ is a closed immersion, and there is a short exact sequence
+the morphism $b$ is a closed immersion, and
 $$
-0 \to \mathcal{O}_W \to a_*\mathcal{O}_X \oplus b_*\mathcal{O}_Y \to
-c_*\mathcal{O}_Z \to 0
+\mathcal{O}_{Y \amalg_Z X} =
+b_*\mathcal{O}_Y \times_{c_*\mathcal{O}_Z} a_*\mathcal{O}_X
 $$
-where $c = a \circ i = b \circ j$.
+as sheaves of rings where $c = a \circ i = b \circ j$.
 \end{proposition}
 
 \begin{proof}
@@ -16886,56 +16868,17 @@ \section{Pushouts in the category of schemes, II}
 $B_i$. Hence $b$ is a closed immersion.
 The ring map $A_i \times_{C_i} B_i \to A_i$ is integral by
 More on Algebra, Lemma \ref{more-algebra-lemma-fibre-product-integral}
-hence $a$ is integral.
-The short exact sequence comes from the short exact sequence
-$$
-0 \to A_i \times_{C_i} B_i \to A_i \times B_i \to C_i \to 0
-$$
-The diagram is cartesian because
+hence $a$ is integral. The diagram is cartesian because
 $$
 C_i \cong B_i \otimes_{B_i \times_{C_i} A_i} A_i
 $$
 This follows as $B_i \times_{C_i} A_i \to B_i$
 and $A_i \to C_i$ are surjective maps whose kernels are the same.
 
 \medskip\noindent
-We finish the proof by showing our construction gives a pushout
+Finally, we can apply Lemmas \ref{lemma-basic-example-pushout} and
+\ref{lemma-pushout-fpqc-local} to conclude our construction is a pushout
 in the category of schemes.
-Let $f : X \to T$ and $g : Y \to T$ be morphisms of schemes
-with $f \circ i = g \circ j$. Then we obtain a map of topological
-spaces $(g, f) : Y \amalg_Z X \to T$. We can use the maps
-$f^\sharp : f^{-1}\mathcal{O}_T \to \mathcal{O}_X$ and
-$g^\sharp : g^{-1}\mathcal{O}_T \to \mathcal{O}_Y$
-the equalities $f = (g, f) \circ a$, $g = (g, f) \circ b$
-and adjunction of the pairs $(a^{-1}, a_*)$ and $(b^{-1}, b_*)$
-to get maps
-$$
-(g, f)^{-1}\mathcal{O}_T \to a_*\mathcal{O}_X
-\quad\text{and}\quad
-(g, f)^{-1}\mathcal{O}_T \to b_*\mathcal{O}_Y
-$$
-A computation (omitted) shows that the compositions
-into $c_*\mathcal{O}_Z$ are equal. Thus we get a morphism
-of ringed spaces
-$$
-h : Y \amalg_Z X \longrightarrow T
-$$
-by our choice of the structure sheaf of $Y \amalg_Z X$
-as the fibre product of $a_*\mathcal{O}_X$ and $b_*\mathcal{O}_Y$
-over $c_*\mathcal{O}_Z$. To show that $h$ is a morphism of
-locally ringed spaces (and hence a morphism of schemes),
-let $s \in Y \amalg_Z X$ be a point mapping to $t \in T$.
-Then either $s$ is the image of a $y \in Y$ or the image of
-an $x \in X$. Consider
-$$
-\mathcal{O}_{T, t} \to \mathcal{O}_{Y \amalg_Z X, s}
-\to \mathcal{O}_{Y, y}
-\quad\text{or}\quad
-\mathcal{O}_{T, t} \to \mathcal{O}_{Y \amalg_Z X, s}
-\to \mathcal{O}_{X, x}
-$$
-Since the composition and the second map are local ring homomorphisms,
-we conclude.
 \end{proof}
 
 \begin{lemma}