Coequalizer or pushout fpqc local on it

descent.tex

@@ -3127,6 +3127,39 @@ \section{Fpqc coverings are universal effective epimorphisms}
 morphism $\Spec(R) \to S$.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-coequalizer-fpqc-local}
+Consider schemes $X, Y, Z$ and morphisms $a, b : X \to Y$ and
+a morphism $c : Y \to Z$ with $c \circ a = c \circ b$. Set
+$d = c \circ a = c \circ b$. If there exists an
+fpqc covering $\{Z_i \to Z\}$ such that
+\begin{enumerate}
+\item for all $i$ the morphism $Y \times_{c, Z} Z_i \to Z_i$
+is the coequalizer of $(a, 1) : X \times_{d, Z} Z_i \to Y \times_{c, Z} Z_i$
+and $(b, 1) : X \times_{d, Z} Z_i \to Y \times_{c, Z} Z_i$, and
+\item for all $i$ and $i'$ the morphism
+$Y \times_{c, Z} (Z_i \times_Z Z_{i'}) \to (Z_i \times_Z Z_{i'})$
+is the coequalizer of
+$(a, 1) : X \times_{d, Z} (Z_i \times_Z Z_{i'}) \to
+Y \times_{c, Z} (Z_i \times_Z Z_{i'})$ and
+$(b, 1) : X \times_{d, Z} (Z_i \times_Z Z_{i'}) \to
+Y \times_{c, Z} (Z_i \times_Z Z_{i'})$
+\end{enumerate}
+then $c$ is the coequalizer of $a$ and $b$.
+\end{lemma}
+
+\begin{proof}
+Namely, for a scheme $T$ a morphism $Z \to T$ is the same thing as
+a collection of morphism $Z_i \to T$ which agree on overlaps by
+Lemma \ref{lemma-fpqc-universal-effective-epimorphisms}.
+\end{proof}
+
+
+
+
+
+
+
 
 
 

more-morphisms.tex

@@ -2686,7 +2686,43 @@ \section{Pushouts in the category of schemes}
 \noindent
 In this section we collect some results on pushouts in the category of schemes.
 See Categories, Section \ref{categories-section-pushouts} for a general
-discussion of pushouts in any category. We first discuss the case where
+discussion of pushouts in any category.
+
+\begin{lemma}
+\label{lemma-pushout-fpqc-local}
+Consider a commutative diagram of schemes
+$$
+\xymatrix{
+Z \ar[r]_i \ar[d]_j & X \ar[d]^a \\
+Y \ar[r]^-b & W
+}
+$$
+and set $c = a \circ i = b \circ j$. If there exists an fpqc covering
+$\{W_i \to W\}$ such that for all $i$ and $i'$ the diagrams
+$$
+\xymatrix{
+Z \times_{c, W} W_i \ar[r] \ar[d] & X \times_{a, W} W_i \ar[d] \\
+Y \times_{b, W} W_i \ar[r] & W_i
+}
+\quad
+\xymatrix{
+Z \times_{c, W} (W_i \times_W W_{i'}) \ar[r] \ar[d] &
+X \times_{a, W} (W_i \times_W W_{i'}) \ar[d] \\
+Y \times_{b, W} (W_i \times_W W_{i'}) \ar[r] &
+(W_i \times_W W_{i'})
+}
+$$
+are cocartesian, then so is the original diagram.
+\end{lemma}
+
+\begin{proof}
+Namely, for a scheme $T$ a morphism $W \to T$ is the same thing as
+a collection of morphism $W_i \to T$ which agree on overlaps, see
+Descent, Lemma \ref{descent-lemma-fpqc-universal-effective-epimorphisms}.
+\end{proof}
+
+\noindent
+Next, we discuss existence in the case where
 both morphisms are closed immersions.
 
 \begin{lemma}