proofread ega1-3.6 and ega1-3.7; proofread ega1-3

ega1/ega1-3.tex

@@ -930,12 +930,12 @@ \subsection{Fibres}
 \begin{prop}[3.6.1]
 \label{1.3.6.1}
 Let $f:X\to Y$ be a morphism, $y$ a point of $Y$, and $\fk{a}_y$ an ideal of definition for $\OO_y$ for the $\fk{m}_y$-preadic topology.
-The projection $p:X\times_Y\Spec(\OO_y/\fk{a}_y)\to X$ is a homeomorphism from the underlying space of the prescheme $X\times_Y\Spec(\OO_y/\fk{a}_y)$ to the fibre $f^{-1}(y)$ equipped with the topology induced from that of the underlying space of $X$.
+Then the projection $p:X\times_Y\Spec(\OO_y/\fk{a}_y)\to X$ is a homeomorphism from the underlying space of the prescheme $X\times_Y\Spec(\OO_y/\fk{a}_y)$ to the fibre $f^{-1}(y)$ equipped with the topology induced from that of the underlying space of $X$.
 \end{prop}
 
 \begin{proof}
 \label{proof-1.3.6.1}
-Since $\Spec(\OO_y/\fk{a}_y)\to Y$ is radicial (\sref{1.3.5.4} and \sref{1.2.4.7}), $\Spec(\OO_y/\fk{a}_y)$ is a single point, and the ideal $\fk{m}_y/\fk{a}_y$ is nilpotent by hypothesis \sref{1.1.1.12}, we already know (\sref{1.3.5.10} and \sref{1.3.3.4}) that $p$ identifies, as \emph{sets}, the underlying space of $X\times_Y\Spec(\OO_y/\fk{a}_y)$ with $f^{-1}(y)$; everything boils down to proving that $p$ is a homeomorphism.
+Since $\Spec(\OO_y/\fk{a}_y)\to Y$ is radicial (\sref{1.3.5.4} and \sref{1.2.4.7}), since $\Spec(\OO_y/\fk{a}_y)$ is a single point, and since the ideal $\fk{m}_y/\fk{a}_y$ is nilpotent by hypothesis \sref{1.1.1.12}, we already know (\sref{1.3.5.10} and \sref{1.3.3.4}) that $p$ identifies, as \emph{sets}, the underlying space of $X\times_Y\Spec(\OO_y/\fk{a}_y)$ with $f^{-1}(y)$; everything reduces to proving that $p$ is a homeomorphism.
 By \sref{1.3.2.7}, the question is local on $X$ and $Y$, and so we can suppose that $X=\Spec(B)$ and $Y=\Spec(A)$, with $B$ being an $A$-algebra.
 The morphism $p$ then corresponds to the homomorphism $1\otimes\vphi:B\to B\otimes_A A'$, where $A'=A_y/\fk{a}_y$ and $\vphi$ is the canonical map from $A$ to $A'$.
 Then every element of $B\otimes_A A'$ can be written as
@@ -965,20 +965,20 @@ \subsection{Fibres}
 \begin{prop}[3.6.4]
 \label{1.3.6.4}
 \emph{(Transitivity of fibres)}
-Let $f:X\to Y$ and $g:Y'\to Y$ be two morphisms; let $X'=X\times_Y Y'=X_{(Y')}$ and $f'=f_{(Y')}:X'\to Y'$.
+Let $f:X\to Y$ and $g:Y'\to Y$ be morphisms; let $X'=X\times_Y Y'=X_{(Y')}$ and $f'=f_{(Y')}:X'\to Y'$.
 For every $y'\in Y'$, if we let $y=g(y')$, then the prescheme $f'^{-1}(y')$ is isomorphic to $f^{-1}(y)\otimes_{\kres(y)}\kres(y')$.
 \end{prop}
 
 \begin{proof}
 \label{proof-1.3.6.4}
-Indeed, it reduces to remarking that the two preschemes $(X\otimes_Y\kres(y))\otimes_{\kres(y)}\kres(y')$ and $(X\times_Y Y')\otimes_{Y'}\kres(y')$ are both canonically isomorphic to $X\times_Y\Spec(\kres(y'))$ by (3.3.9.1).
+Indeed, it suffices to remark that the two preschemes $(X\otimes_Y\kres(y))\otimes_{\kres(y)}\kres(y')$ and $(X\times_Y Y')\otimes_{Y'}\kres(y')$ are both canonically isomorphic to $X\times_Y\Spec(\kres(y'))$ by \sref{1.3.3.9.1}.
 \end{proof}
 
 In particular, if $V$ is an open neighborhood of $y$ in $Y$, and we denote by $f_V$ the restriction of $f$ to the induced prescheme on $f^{-1}(V)$, then the preschemes $f^{-1}(y)$ and $f^{-1}_V(y)$ are canonically identified.
 
 \begin{prop}[3.6.5]
 \label{1.3.6.5}
-Let $f:X\to Y$ be a morphism, $y$ a point of $Y$, $Z$ the local prescheme $\Spec(\OO_y)$, and $p=(\psi,\theta)$ the projection $X\times_Y Z\to X$; then $p$ is a homeomorphism from the underlying space of $X\times_Y Z$ to the subspace $f^{-1}(Z)$ of $X$ (\emph{when the underlying space of $Z$ is identified with a subspace of $Y$, cf.~\sref{1.2.4.2}}), and, for all $t\in X\times_Y Z$, letting $z=\psi(t)$, $\theta_t^\sharp$ is an isomorphism from $\OO_x$ to $\OO_t$.
+Let $f:X\to Y$ be a morphism, $y$ a point of $Y$, $Z$ the local prescheme $\Spec(\OO_y)$, and $p=(\psi,\theta)$ the projection $X\times_Y Z\to X$; then $p$ is a homeomorphism from the underlying space of $X\times_Y Z$ to the subspace $f^{-1}(Z)$ of $X$ (\emph{when the underlying space of $Z$ is identified with a subspace of $Y$, cf.~\sref{1.2.4.2}}), and, for all $t\in X\times_Y Z$, letting $z=\psi(t)$, we have that $\theta_t^\sharp$ is an isomorphism from $\OO_x$ to $\OO_t$.
 \end{prop}
 
 \begin{proof}
@@ -990,7 +990,7 @@ \subsection{Fibres}
 \subsection{Application: reduction of a prescheme mod.~$\mathfrak{J}$}
 \label{subsection:reduction-of-a-prescheme}
 
-\emph{This section, which makes use of notions and results from Chapter~I and Chapter~II, will not be used in the following, and is only intended for readers familiar with classical algebraic geometry}.
+\emph{This section, which makes use of notions and results from Chapter~I and Chapter~II, will not be used in what follows in this treatise, and is only intended for readers familiar with classical algebraic geometry}.
 
 \begin{env}[3.7.1]
 \label{1.3.7.1}
@@ -1009,19 +1009,19 @@ \subsection{Application: reduction of a prescheme mod.~$\mathfrak{J}$}
 We consider the affine scheme $Y=\Spec(A)$, formed of two points, the unique closed point
 \oldpage[I]{119}
 $y=\fk{J}$ and the generic point $(0)$, the singleton set $U$ of the generic point being thus an open $U=\Spec(K)$ in $Y$.
-If $X$ is an $A$-prescheme (said otherwise, a $Y$-prescheme), then $X\otimes_A K=X'$ is nothing but the prescheme induced by $X$ on $\psi^{-1}(U)$, denoting by $\psi$ the structure morphism $X\to Y$.
-In particular, if $\vphi$ is the structure morphism $P\to Y$, a closed subprescheme $X'$ of $P'=\vphi^{-1}(U)$ is then a (locally closed) subprescheme of $P$.
-If $P$ is Noetherian (for example, if $A$ is Noetherian and $P$ is of finite type over $A$), then there exists a smaller closed subprescheme $X=\overline{X'}$ of $G$ that majorizes $X'$ \sref{1.9.5.10}, and $X'$ is the prescheme induced by $X$ on the open $\vphi^{-1}(U)\cap X$, and so is isomorphic to $X\otimes_A K$ \sref{1.9.5.10}.
+If $X$ is an $A$-prescheme (or, in other words, a $Y$-prescheme), then $X\otimes_A K=X'$ is exactly the prescheme induced by $X$ on $\psi^{-1}(U)$, denoting by $\psi$ the structure morphism $X\to Y$.
+In particular, if $\vphi$ is the structure morphism $P\to Y$, then a closed subprescheme $X'$ of $P'=\vphi^{-1}(U)$ is a (locally closed) subprescheme of $P$.
+If $P$ is Noetherian (for example, if $A$ is Noetherian and $P$ is of finite type over $A$), then there exists a smaller closed subprescheme $X=\overline{X'}$ of $G$ that through which $X'$ factors \sref{1.9.5.10}, and $X'$ is the prescheme induced by $X$ on the open $\vphi^{-1}(U)\cap X$, and so is isomorphic to $X\otimes_A K$ \sref{1.9.5.10}.
 \emph{The immersion of $X'$ into $P'=P\otimes_A K$ thus lets us canonically consider $X'$ as being of the form $X'=X\otimes_A K$, where $X$ is an $A$-prescheme.}
-We can then consider the reduced mod.~$\fk{J}$ prescheme $X_0=X\otimes_A k$, which is nothing but the fibre $\psi^{-1}(y)$ of the closed point $y$.
+We can then consider the reduced mod.~$\fk{J}$ prescheme $X_0=X\otimes_A k$, which is exactly the fibre $\psi^{-1}(y)$ of the closed point $y$.
 Up until now, lacking the adequate terminology, we had avoided explicitly introducing the $A$-prescheme $X$.
-One ought to, however, note that all the assertions normally made about the ``reduced mod.~$\fk{J}$'' prescheme  $X_0$ should be seen as consequences of more complicated assertions concerning $X$ itself, and cannot be satisfactorily formulated or understood except by interpreting them as such.
-It seems also that the hypotheses made reduce always to hypotheses on $X$ itself (independent of the prior data of an immersion of $X'$ in $\bb{P}_K^r$), which lets us give more intrinsic statements.
+One ought to, however, note that all the claims normally made about the ``reduced mod.~$\fk{J}$'' prescheme  $X_0$ should be seen as consequences of more complicated claims about $X$ itself, and cannot be satisfactorily formulated or understood except by interpreting them as such.
+It seems also that any hypotheses made on $X_0$ always reduce to hypotheses on $X$ itself (independent of the prior data of an immersion of $X'$ in $\bb{P}_K^r$), which lets us give more intrinsic statements.
 \end{env}
 
 \begin{env}[3.7.3]
 \label{1.3.7.3}
 Lastly, we draw attention to a very particular fact, which has undoubtedly contributed to slowing the conceptual clarification of the situation envisaged here: if $A$ is a discrete valuation ring, and if $X$ is \emph{proper} over $A$ (which is indeed the case if $X$ is a closed subprescheme of some $\bb{P}_A^r$, cf. \sref[II]{2.5.5.4}), then the points of $X$ with values in $A$ and the points of $X'$ with values in $k$ are in bijective correspondence \sref[II]{2.7.3.8}.
-It is because we often believe that facts about $X'$ have been proved, when in reality we have proved facts about $X$, and these remain valuable (in this form) whenever we no longer suppose that the base local ring is of dimension~$1$.
+This is why we often believe that facts about $X'$ have been proved, when in reality we have proved facts about $X$, and these remain valuable (in this form) whenever we no longer suppose that the base local ring is of dimension~$1$.
 \end{env}
 

what.tex

@@ -64,6 +64,10 @@ \section*{Notes from the translators}
 
 \sectionbreak
 
+\unsure{Any translations about which we are not entirely sure will be marked with a}.
+
+\sectionbreak
+
 Whenever a note is made by the translators, it will be prefaced by ``[Trans.]''.
 
 \sectionbreak