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authorGravatar Tim Hosgood <thosgood@users.noreply.github.com> 2021-07-29 00:39:17 +0100
committerGravatar GitHub <noreply@github.com> 2021-07-29 00:39:17 +0100
commit730b4d511e81de852c8d562473ad184b4991c020 (patch)
treeed9478c40bfd303115598b861d086f5b57ff58db
parent1b06dd70d1310b8bb86d72589943db258de101f3 (diff)
parent44565f097e5e12e184cd06a6a64faa2e32c5eb6a (diff)
downloadega-730b4d511e81de852c8d562473ad184b4991c020.tar.gz
ega-730b4d511e81de852c8d562473ad184b4991c020.zip
Merge pull request #193 from ryankeleti/ega2-2
finished II.2
-rw-r--r--README.md2
-rw-r--r--STYLE.md1
-rw-r--r--ega1/ega1-1.tex2
-rw-r--r--ega1/ega1-2.tex2
-rw-r--r--ega2/ega2-1.tex2
-rw-r--r--ega2/ega2-2.tex448
-rw-r--r--ega2/ega2-4.tex58
-rw-r--r--ega2/ega2-7.tex14
-rw-r--r--ega2/ega2-8.tex100
-rw-r--r--ega4/ega4-16.tex34
-rw-r--r--ega4/ega4-17.tex4
-rw-r--r--intro.tex2
-rw-r--r--preamble-base.tex3
13 files changed, 557 insertions, 115 deletions
diff --git a/README.md b/README.md
index 1f0edb5..886073f 100644
--- a/README.md
+++ b/README.md
@@ -87,7 +87,7 @@ Here is the current status of the translation, along with who is currently worki
+ [x] 0. Summary _(@ryankeleti / proofread by @thosgood)_
+ [x] 1. Affine morphisms _(@ryankeleti)_
-+ [ ] 2. Homogeneous prime spectra (~30 pages) _(@thosgood)_
++ [x] 2. Homogeneous prime spectra _(@thosgood)_
+ [ ] 3. Homogeneous prime spectrum of a sheaf of graded algebras (~20 pages)
+ [x] 4. Projective bundles; Ample sheaves _(@thosgood)_
+ [x] 5. Quasi-affine morphisms; quasi-projective morphisms; proper morphisms; projective morphisms _(@thosgood)_
diff --git a/STYLE.md b/STYLE.md
index 74cb3bf..1df9ec6 100644
--- a/STYLE.md
+++ b/STYLE.md
@@ -234,6 +234,7 @@ if in EGA II, page 41 ends with `Hi! Schemes` and page 42 begins with `are cool.
* `\red` --- reduced, i.e. `X_\red` for X<sub>red</sub>
* `\supertilde` --- for when `\widetilde{}` is used as a subscript, i.e. `\sh{F}\supertilde` instead of `\sh{F}^\sim` (note the lack of `^`)
* `\bullet` --- to be used instead of `*` when denoting a grading, e.g. `A_\bullet` instead of `A_*` for a graded module
+* `\widehat` and `\widetilde` --- to be used instead of `\hat` and `\tilde` (which have been redefined to mean this anyway)
## References
diff --git a/ega1/ega1-1.tex b/ega1/ega1-1.tex
index eab90c4..3b96f64 100644
--- a/ega1/ega1-1.tex
+++ b/ega1/ega1-1.tex
@@ -270,7 +270,7 @@ for each $f'\in A'$, we therefore have, by definition,
\begin{proof}
The relation ${}^a\vphi(x)\in V(E')$ is, by definition, equivalent to $E'\subset\vphi^{-1}(\mathfrak{j}_x)$, so $\vphi(E')\subset\mathfrak{j}_x$, and finally $x\in V(\vphi(E'))$, hence (i).
To prove (ii), we can suppose that $\mathfrak{a}$ is equal to its radical, since $V(\rad(\mathfrak{a}))=V(\mathfrak{a})$ \sref{I.1.1.2}[(v)] and $\vphi^{-1}(\rad(\mathfrak{a}))=\rad(\vphi^{-1}(\mathfrak{a}))$;
-if we set $Y=V(\mathfrak{a})$, and $\mathfrak{a}'=\mathfrak{j}({}^a\varphi(Y))$, then we have $\overline{{}^a(Y)=V(\mathfrak{a}')}$ (\sref{I.1.1.4}[(ii)])
+if we set $Y=V(\mathfrak{a})$, and $\mathfrak{a}'=\mathfrak{j}({}^a\vphi(Y))$, then we have $\overline{{}^a(Y)=V(\mathfrak{a}')}$ (\sref{I.1.1.4}[(ii)])
the relation $f'\in\mathfrak{a}'$ is, by definition, equivalent to $f'(x')=0$ for each $x\in{{}^a\vphi(Y)}$, so, by Equation~\hyperref[1.1.2.1]{(1.2.1.1)}, it is also equivalent to $\vphi(f')(x)=0$ for each $x\in Y$, or to $\vphi(f')\in\mathfrak{j}(Y)=\mathfrak{a}$, since $\mathfrak{a}$ is equal to its radical;
hence (ii).
\end{proof}
diff --git a/ega1/ega1-2.tex b/ega1/ega1-2.tex
index a94a5d9..2db8cc0 100644
--- a/ega1/ega1-2.tex
+++ b/ega1/ega1-2.tex
@@ -281,7 +281,7 @@ This establishes a canonical bijective correspondence between the set of morphis
\end{proposition}
Indeed, for all $x\in X$, we have that $a\in\overline{\{x\}}$, so $\psi(a)\in\overline{\{\psi(x)\}}$, which shows that $\psi(X)$ is contained in every affine open subset that contains $\psi(a)$.
-So it suffices to consider the case where $(Y,\sh{O}_Y)$ is an affine scheme of ring $B$, and then we have that $u=({}^a\vphi,\tilde{\vphi})$, where $\vphi\in\Hom(B,A)$ \sref{I.1.7.3}.
+So it suffices to consider the case where $(Y,\sh{O}_Y)$ is an affine scheme of ring $B$, and then we have that $u=({}^a\vphi,\widetilde{\vphi})$, where $\vphi\in\Hom(B,A)$ \sref{I.1.7.3}.
Further, we have that $\vphi^{-1}(\mathfrak{j}_a)=\mathfrak{j}_{\psi(a)}$, and hence that the image under $\vphi$ of any element of $B\setmin\mathfrak{j}_{\psi(a)}$ is invertible in the local ring $A$;
the factorization in the result follows from the universal property of the ring of fractions \sref[0]{0.1.2.4}.
Conversely, to each local homomorphism $\sh{O}_y\to A$ there is a unique corresponding morphism $(\psi,\theta):X\to\Spec(\sh{O}_y)$ such that $\psi(a)=y$ \sref{I.1.7.3}, and, by composing with the canonical morphism $\Spec(\sh{O}_y)\to Y$, we obtain a morphism $X\to Y$, which proves the proposition.
diff --git a/ega2/ega2-1.tex b/ega2/ega2-1.tex
index 10cc74f..123c4d0 100644
--- a/ega2/ega2-1.tex
+++ b/ega2/ega2-1.tex
@@ -220,7 +220,7 @@ Let $X$ be a prescheme affine over $S$; for $X$ to be reduced, it is necessary a
The question is local on $S$; by Corollary~\sref{II.1.3.2}, the corollary follows from \sref[I]{I.5.1.1} and \sref[I]{I.5.1.4}.
\end{proof}
-\subsection{Quasi-coherent sheaves over a prescheme affine over $S$}
+\subsection{Quasi-coherent sheaves over an affine prescheme over $S$}
\label{subsection:II.1.4}
\begin{proposition}[1.4.1]
diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex
index 9d782b6..147ae60 100644
--- a/ega2/ega2-2.tex
+++ b/ega2/ega2-2.tex
@@ -6,7 +6,7 @@
\begin{notation}[2.1.1]
\label{II.2.1.1}
-Given a \emph{positively-graded} ring $S$, we denote by $S_n$ the subset of $S$ consisting of homogeneous elements of degree $n$ ($n\geq 0$), by $S_+$ the (direct) sum of the $S_n$ for $n>0$;
+Given a \emph{positively graded} ring $S$, we denote by $S_n$ the subset of $S$ consisting of homogeneous elements of degree $n$ ($n\geq 0$), by $S_+$ the (direct) sum of the $S_n$ for $n>0$;
we have $1\in S_0$, $S_0$ is a subring of $S$, $S_+$ is a graded ideal of $S$, and $S$ is the direct sum of $S_0$ and $S_+$.
If $M$ is a \emph{graded} module over $S$ (with positive or negative degrees), we similarly denote by $M_n$ the $S_0$-module consisting of homogeneous elements of $M$ of degree $n$ (with $n\in\bb{Z}$).
@@ -60,7 +60,7 @@ for two graded $S$-modules $M$ and $N$.
\oldpage[II]{21}
Let $S$ and $S'$ be two graded rings;
-a homomorphism of \emph{graded rings $\vphi:S\to S'$} is a homomorphism of rings such that $\vphi(S_n)\subset S_n'$ for all $n\in\bb{Z}$ (in other words, $\vphi$ must be a homomorphism \emph{of degree $0$} of graded $\bb{Z}$-modules).
+a homomorphism of \emph{graded rings $\varphi:S\to S'$} is a homomorphism of rings such that $\varphi(S_n)\subset S_n'$ for all $n\in\bb{Z}$ (in other words, $\varphi$ must be a homomorphism \emph{of degree $0$} of graded $\bb{Z}$-modules).
The data of such a homomorphism defines on $S'$ a \emph{graded} $S'$-module structure;
equipped with this structure and its graded ring structure, we say that $S'$ is a \emph{graded $S'$-algebra}.
@@ -576,7 +576,7 @@ we immediately see that these affine schemes are isomorphic to $K[T]$, and that
\begin{proposition}[2.4.4]
\label{II.2.4.4}
-Let $S$ be a positively-graded ring, and let $X$ be the scheme $\Proj(S)$.
+Let $S$ be a positively graded ring, and let $X$ be the scheme $\Proj(S)$.
\begin{enumerate}
\item[{\rm(i)}] If $\mathfrak{N}_+$ is the nilradical of $S_+$ \sref{II.2.1.10}, then the scheme $X_\red$ is canonically isomorphic to $\Proj(S/\mathfrak{N}_+)$;
in particular, if $S$ is essentially reduced, then $\Proj(S)$ is reduced.
@@ -639,7 +639,7 @@ is commutative, for homogeneous $f,g$ in $S_+$.
\begin{proposition}[2.4.7]
\label{II.2.4.7}
-Let $S$ be a positively-graded ring.
+Let $S$ be a positively graded ring.
\begin{enumerate}
\item[{\rm(i)}] For every integer $d>0$, there exists a canonical isomorphism from the scheme $\Proj(S)$ to the scheme $\Proj(S^{(d)})$.
\item[{\rm(ii)}] Let $S'$ be the graded ring such that $S_0=\bb{Z}$ and $S'_n=S_n$ (considered as a $\bb{Z}$-module) for $n>0$.
@@ -934,7 +934,7 @@ this follows from the commutativity of the diagram
\[
\xymatrix{
\widetilde{\mathfrak{J}}\otimes_{\sh{O}_X}\widetilde{M} \ar[rr]^\lambda \ar[dr]
- && (\mathfrak{J}\otimes_S M)\supertilde
+ && (\mathfrak{J}\otimes_S M)\supertilde \ar[dl]
\\&\widetilde{M}
}
\]
@@ -1317,3 +1317,441 @@ But, by the definition of $N$, and by \sref{II.2.5.4}, the restriction of the is
\subsection{Functorial behaviour}
\label{subsection:II.2.8}
+
+\begin{env}[2.8.1]
+\label{II.2.8.1}
+Let $S$ and $S'$ be positively graded rings, and $\varphi:S'\to S$ a homomorphism of graded rings.
+We denote by $G(\varphi)$ the open subset of $X=\Proj(S)$ given by the complement of $V_+(\varphi(S'_+))$, or, equivalently, the union of the $D_+(\varphi(f'))$ where $f'$ runs over the set of homogeneous elements of $S'_+$.
+The restriction to $G(\varphi)$ of the continuous map ${}^a\varphi$ from $\Spec(S)$ to $\Spec(S')$ \sref[I]{I.1.2.1} is thus a continuous map from $G(\varphi)$ to $\Proj(S')$, which we again denote, with an abuse of language, by ${}^a\varphi$.
+If $f'\in S'_+$ is homogeneous, then
+\[
+\label{II.2.8.1.1}
+ {}^a\varphi^{-1}(D_+(f')) = D_+(\varphi(f'))
+\tag{2.8.1.1}
+\]
+taking into account the fact that ${}^a\varphi$ sends $G(\varphi)$ to $\Proj(S')$, as well as \sref[I]{I.1.2.2.2}.
+The homomorphism $\varphi$ also canonically defines (with the same notation) a homomorphism of graded rings $S'_{f'}\to S_f$, whence, by restriction to the degree~$0$ elements,
+\oldpage[II]{42}
+a homomorphism $S'_{(f')}\to S_{(f)}$, which we denote by $\varphi_{(f)}$;
+there is a corresponding \sref[I]{I.1.6.1} morphism of affine schemes $({}^a\varphi_{(f)},\widetilde{\varphi}_{(f)}):\Spec(S_{(f)})\to\Spec(S'_{(f')})$.
+If we canonically identify $\Spec(S_{(f)})$ with the scheme induced by $\Proj(S)$ on $D_+(f)$ \sref{II.2.3.6}, then we have defined a morphism $\Phi_f:D_+(f)\to D_+(f')$, and ${}^a\varphi_{(f)}$ is identified with the restriction of ${}^a\varphi$ to $D_+(f)$.
+It is also immediate that, if $g'$ is another homogeneous element of $S'_+$, and $g=\varphi(g')$, then the diagram
+\[
+ \xymatrix{
+ D_+(f) \ar[r]^{\Phi_f}
+ & D_+(f')
+ \\D_+(fg) \ar[u] \ar[r]_{\Phi_{fg}}
+ & D_+(f'g') \ar[u]
+ }
+\]
+commutes, by the fact that the diagram
+\[
+ \xymatrix{
+ S'_{(f')} \ar[r]^{\varphi_{(f)}} \ar[d]_{\omega_{f'g',f'}}
+ & S_{(f)} \ar[d]^{\omega_{fg,f}}
+ \\S'_{(f'g')} \ar[r]_{\varphi_{(fg)}}
+ & S_{(fg)}
+ }
+\]
+commutes.
+Taking the definition of $G(\varphi)$, along with \sref{II.2.3.3.2}, we thus see that:
+\end{env}
+
+\begin{proposition}[2.8.2]
+\label{II.2.8.2}
+Given a homomorphism of graded rings $\varphi: S'\to S$, there exists exactly one morphism $({}^a\varphi,\widetilde{\varphi})$ from the induced prescheme $G(\varphi)$ to $\Proj(S')$ (said to be \emph{associated to $\varphi$}, and denoted by $\Proj(\varphi)$) such that, for every homogeneous element $f'\in S'_+$, the restriction of this morphism to $D_+(\varphi(f'))$ agrees with the morphism associated to the homomorphism $S'_{(f')}\to S_{(\varphi(f'))}$ corresponding to $\varphi$.
+\end{proposition}
+
+\begin{proof}
+With the above notation, if $f'\in S'_d$, then the diagram
+\[
+\label{II.2.8.2.1}
+ \xymatrix{
+ S'_{(f')} \ar[r]^{\varphi_{(f)}} \ar[d]_{\sim}
+ & S_{(f)} \ar[d]^{\sim}
+ \\{S'}^{(d)}/(f'-1){S'}^{(d)} \ar[r]
+ & S^{(d)}/(f-1)S^{(d)}
+ }
+\tag{2.8.2.1}
+\]
+commutes (the vertical arrows being the isomorphisms \sref{II.2.2.5}).
+\end{proof}
+
+\begin{corollary}[2.8.3]
+\label{II.2.8.3}
+\begin{enumerate}
+ \item[{\rm(i)}] The morphism $\Proj(\varphi)$ is affine.
+ \item[{\rm(ii)}] If $\Ker(\varphi)$ is nilpotent (and, in particular, if $\varphi$ is injective), then the morphism $\Proj(\varphi)$ is dominant.
+\end{enumerate}
+\end{corollary}
+
+\begin{proof}
+Claim~(i) is an immediate consequence of \sref{II.2.8.2} and \sref{II.2.8.1.1}.
+Claim~(ii) follows since, if $\Ker(\varphi)$ is nilpotent, then, for every homogeneous $f'$ in $S'_+$, we immediately see that $\Ker(\varphi_f)$ (with $f=\varphi(f')$) is nilpotent, and thus so too is $\Ker(\varphi_{(f)})$;
+we then apply \sref{II.2.8.2} and \sref[I]{I.1.2.7}
+\end{proof}
+
+\oldpage[II]{43}
+We note that there are, in general, morphisms from $\Proj(S)$ to $\Proj(S')$ that are not affine, and that thus do not come from homomorphisms of graded rings $S'\to S$;
+an example is the structure morphism $\Proj(S)\to\Spec(A)$ when $A$ is a field ($\Spec(A)$ thus being identified with $\Proj(A[T])$, cf.~\sref{II.3.1.7});
+indeed, this follows from \sref[I]{I.2.3.2}.
+
+\begin{env}[2.8.4]
+\label{II.2.8.4}
+Let $S''$ be another positively graded ring, and $\varphi':S''\to S'$ a homomorphism of graded rings, and set $\varphi''=\varphi\circ\varphi'$.
+By \sref{II.2.8.1.1} and the formula ${}^a\varphi''=({}^a\varphi')\circ({}^a\varphi)$, we immediately see that $G(\varphi'')\subset G(\varphi)$, and that, if $\Phi$, $\Phi'$, and $\Phi''$ are the morphisms associated to $\varphi$, $\varphi'$, and $\varphi''$ (respectively), then $\Phi''=\Phi'\circ(\Phi|G(\varphi''))$.
+\end{env}
+
+\begin{env}[2.8.5]
+\label{II.2.8.5}
+Suppose that $S$ (resp. $S'$) is a graded $A$-algebra (resp. a graded $A'$-algebra), and let $\psi:A'\to A$ be a ring homomorphism such that the diagram
+\[
+ \xymatrix{
+ A' \ar[r]^{\psi} \ar[d]
+ & A \ar[d]
+ \\S' \ar[r]_{\varphi}
+ & S
+ }
+\]
+commutes.
+We can then consider $G(\varphi)$ (resp. $\Proj(S')$) as a scheme over $\Spec(A)$ ($resp. \Spec(A')$);
+if $\Phi$ (resp. $\Psi$) is the morphism associated to $\varphi$ (resp. $\psi$), then the diagram
+\[
+ \xymatrix{
+ G(\varphi) \ar[r]^{\Phi} \ar[d]
+ & \Proj(S') \ar[d]
+ \\\Spec(A) \ar[r]_{\Psi}
+ & \Spec(A')
+ }
+\]
+commutes: it suffices to prove this for the restriction of $\Phi$ to $D_+(f)$, where $f=\varphi(f')$, with $f'$ homogeneous in $S'_+$;
+this then follows from the fact that the diagram
+\[
+ \xymatrix{
+ A' \ar[r]^{\psi} \ar[d]
+ & A \ar[d]
+ \\S'_{(f')} \ar[r]_{\varphi_{(f)}}
+ & S_{(f)}
+ }
+\]
+commutes.
+\end{env}
+
+\begin{env}[2.8.6]
+\label{II.2.8.6}
+Now let $M$ be a graded $S$-module, and consider the $S'$-module $M_{[\varphi]}$, which is evidently graded.
+Let $f'$ be homogeneous in $S'_+$, and let $f=\varphi(f')$;
+we know \sref[0]{0.1.5.2} that there is a canonical isomorphism $(M_{[\varphi]})_{f'}\xrightarrow{\sim}(M_f)_{[\varphi_f]}$, and it is immediate that this isomorphism preserves degree, whence a canonical isomorphism $(M_{[\varphi]})_{(f')}\xrightarrow{\sim}(M_{(f)})_{[\varphi_{(f)}]}$.
+To this isomorphism, there canonically corresponds an isomorphism of sheaves $(M_{[\varphi]})\supertilde|D_+(f')\xrightarrow{\sim}(\Phi_f)_*(\widetilde(M)|D_+(f))$ (\sref{II.2.5.2} and \sref[I]{I.1.6.3}).
+Furthemore,
+\oldpage{43}
+if $g'$ is another homogeneous element of $S'_+$, and $g=\varphi(g')$, then the diagram
+\[
+ \xymatrix{
+ (M_{[\varphi]})_{(f')} \ar[r]^{\sim} \ar[d]
+ & (M_{(f)})_{[\varphi_{(f)}]} \ar[d]
+ \\(M_{[\varphi]})_{(f'g')} \ar[r]^{\sim}
+ & (M_{(fg)})_{[\varphi_{(fg)}]}
+ }
+\]
+commutes, whence we immediately conclude that the isomorphism
+\[
+ (M_{[\varphi]})\supertilde|D_+(f'g') \xrightarrow{\sim} (\Phi_{fg})_*(\widetilde{M}|D_+(fg))
+\]
+is the restriction to $D_+(f'g')$ of the isomorphism $(M_{[\varphi]})\supertilde|D_+(f')\xrightarrow{\sim}(\Phi_f)_*(\widetilde{M}|D_+(f))$.
+Since $\Phi_f$ is the restriction to $D_+(f)$ of the morphism $\Phi$, we see that, taking \sref{II.2.8.1.1} into account, and setting $X'=\Proj(S)'$:
+\end{env}
+
+\begin{proposition}[2.8.7]
+\label{II.2.8.7}
+There exists a canonical functorial isomorphism from the $\sh{O}_{X'}$-module $(M_{[\varphi]})^\supertilde$ to the $\sh{O}_{X'}$-module $\Phi_*(\widetilde{M}|G(\varphi))$.
+\end{proposition}
+
+We thus immediately deduce a canonical functorial map from the set of $\varphi$-morphisms $M'\to M$ from a graded $S'$-module to the graded $S$-module $M$, to the set of $\Phi$-morphisms $\widetilde{M'}\to\widetilde{M}|G(\varphi)$.
+With the notation of \sref{II.2.8.4}, if $M''$ is a graded $S''$-module, then, to the composition of a $\varphi$-morphism $M'\to M$ and a $\varphi'$-morphism $M''\to M'$, canonically corresponds the composition of $\widetilde{M'}G(\varphi')\to\widetilde{M}|G(\varphi'')$ and $\widetilde{M''}\to\widetilde{M'}|G(\varphi')$.
+
+\begin{proposition}[2.8.8]
+\label{II.2.8.8}
+Under the hypotheses of \sref{II.2.8.1}, let $M'$ be a graded $S'$-module.
+Then there exists a canonical functorial homomorphism $\nu$ from the $(\sh{O}_X|G(\varphi))$-module $\Phi^*(\widetilde{M'})$ to the $(\sh{O}_X|G(\varphi))$-module $(M'\otimes_{S'}S)\supertilde|G(\varphi)$.
+If the ideal $S'_+$ is generated by $S'_1$, then $\nu$ is an isomorphism.
+\end{proposition}
+
+\begin{proof}
+Indeed, for $f'\in S'_d$ ($d>0$), we define a canonical functorial homomorphism of $S_{(f)}$-modules (where $f=\varphi(f')$)
+\[
+\label{II.2.8.8.1}
+ \nu_f: M'_{(f')}\otimes_{S'_{(f')}}S_{(f)} \to (M'\otimes_{S'}S)_{(f)}
+\tag{2.8.8.1}
+\]
+by composing the homomorphism $M'_{(f')}\otimes_{S'_{(f')}}S_{(f)}\to M'_{f'}\otimes_{S'_{f'}}S_f$ and the canonical isomorphism $M'_{f'}\otimes_{S'_{f'}}S_f\xrightarrow{\sim}(M'\otimes_{S'}S)_f$ \sref[0]{0.1.5.4}, and noting that the latter preserves degrees.
+We can immediately verify the compatibility of $\nu_f$ with the restriction operators from $D_+(f)$ to $D_+(fg)$ (for any $g'\in S'_+$ and $g=\varphi(g')$), whence the definition of the homomorphism
+\[
+ \nu: \Phi^*(\widetilde{M'}) \to (M'\otimes_{S'}S)\supertilde|G(\varphi)
+\]
+taking \sref[I]{I.1.6.5} into account.
+To prove the second claim, it suffices to show that $\nu_f$ is an isomorphism for all $f'\in S_1$, since $G(\varphi)$ is then a union of the $D_+(\varphi(f'))$.
+We first define a $\bb{Z}$-bilinear $M'_m\times S_n\to M'_{(f')}\otimes_{S'_{(f')}}S_{(f)}$ by sending $(x',s)$ to the element $(x'/{f'}^m)\otimes(s/f^n)$ (with the convention that $x'/{f'}^m$ is ${f'}^{-m}x'/1$ when $m<0$).
+\oldpage[II]{45}
+We claim that, in the proof of \sref{II.2.5.13}, this map gives rise to a di-homomorphism of modules
+\[
+ \eta_f: M'\otimes_{S'}S \to M'_{(f')}\otimes_{S_{(f')}}S_{(f)}.
+\]
+Furthermore, if, for $r>0$, we have $f^r\sum_i(x'_i\otimes s_i)=0$, then this can also be written as $\sum_i({f'}^rx'_i\otimes s_i)=0$, whence, by \sref[0]{0.1.5.4}, $\sum_i({f'}^rx_i/{f'}^{m_i+r})\otimes(s_i/f^{n_i})=0$, i.e. $\eta_f(\sum_i x_i\otimes y_i)0=$, which proves that $\eta_f$ factors as $M'\otimes_{S'}S\to(M'\otimes_{S'}S)_f\xrightarrow{\eta'_f}M'_{(f')}\otimes_{S'_{(f')}}S_{(f)}$;
+we finally can prove that $\eta'_f$ and $\nu_f$ are inverse isomorphisms to one another.
+
+In particular, it follows from \sref{II.2.1.2.1} that we have a canonical homomorphism
+\[
+\label{II.2.8.8.2}
+ \Phi^*(\sh{O}_{X'}(n)) \xrightarrow{\sim} \sh{O}_X(n)|G(\varphi)
+\tag{2.8.8.2}
+\]
+for all $n\in\bb{Z}$.
+\end{proof}
+
+\begin{env}[2.8.9]
+\label{II.2.8.9}
+Let $A$ and $A'$ be rings, and $\psi:A'\to A$ a ring homomorphism, defining a morphism $\Psi:\Spec(A)\to\Spec(A')$.
+Let $S'$ be a positively graded $A'$-algebra, and set $S=S'\otimes_{A'}A$, which is evidently an $A$-algebra graded by the $S'_n\otimes_{A'}A$;
+the map $\varphi:s'\to s'\otimes1$ is then a graded ring homomorphism that makes the diagram \sref{II.2.8.5.1} commute.
+Since $S_+$ is here the $A$-module generated by $\varphi(S'_+)$, we have $G(\varphi)=\Proj(S)=X$;
+whence, setting $X'=\Proj(S')$, we have the commutative diagram
+\[
+\label{II.2.8.9.1}
+ \xymatrix{
+ X \ar[r]^{\Phi} \ar[d]_p
+ & X' \ar[d]
+ \\Y \ar[r]_{\Psi}
+ & Y'
+ }
+\tag{2.8.9.1}
+\]
+
+Now let $M'$ be a graded $S'$-module, and set $M=M'\otimes_{A'}A=M'\otimes_{S'}S$.
+Under these conditions:
+\end{env}
+
+\begin{proposition}[2.8.10]
+\label{II.2.8.10}
+The diagram \sref{II.2.8.9.1} identifies the scheme $X$ with the product $X'\times_{Y'}Y$;
+furthermore, the canonical homomorphism $\nu:\Phi^*(\widetilde{M'})\to\widetilde{M}$ \sref{II.2.8.8} is an isomorphism.
+\end{proposition}
+
+\begin{proof}
+The first claim will be proven if we show that, for every homogenous $f'$ in $S'_+$, setting $f=\varphi(f')$, the restrictions of $\Phi$ and $p$ to $D_+(f)$ identify this scheme with the product $D_+(f')\times_{Y'}Y$ \sref[I]{I.3.2.6.2};
+in other words, it suffices \sref[I]{I.3.2.2} to prove that $S_{(f)}$ is canonically identified with $S_f\xrightarrow{\sim}S'_{f'}\otimes_{A'}A$, which is immediate by the existence of the canonical isomorphism $S_f\xrightarrow{\sim}S'_{f'}\otimes_{A'}A$ that preserves degrees \sref[0]{0.1.5.4}.
+The second claim then follows from the fact that $M'_{(f')}\otimes_{S'_{(f')}}S_{(f)}$ can be identified, by the above, with $M'_{(f')}\otimes_{A'}A$, and this can be identified with $M_{(f)}$, since $M_f$ is canonically identified with $M'_{f'}\otimes_{A'}A$ by an isomorphism that preserves degrees.
+\end{proof}
+
+\begin{corollary}[2.8.11]
+\label{II.2.8.11}
+For every integer $n\in\bb{Z}$, $\widetilde{M}(n)$ can be identified with $\Phi^*(\widetilde{M'}(n))=\widetilde{M'}(n)\otimes_{Y'}\sh{O}_Y$;
+in particular, $\sh{O}_X(n)=\Phi^*(\sh{O}_{X'}(n))=\sh{O}_{X'}(n)\otimes_{Y'}\sh{O}_Y$.
+\end{corollary}
+
+\begin{proof}
+This follows from \sref{II.2.8.10} and \sref{II.2.5.15}.
+\end{proof}
+
+\oldpage[II]{46}
+
+\begin{env}[2.8.12]
+\label{II.2.8.12}
+Under the hypotheses of \sref{II.2.8.9}, for $f'\in S'_d$ ($d>0$) and $f=\varphi(f')$, the diagram
+\[
+ \xymatrix{
+ M'_{(f')} \ar[r]^-{\sim} \ar[d]
+ & {M'}^{(d)}/(f'-1){M'}^{(d)} \ar[d]
+ \\M_{(f)} \ar[r]^-{\sim}
+ & M^{(d)}/(f-1)M^{(d)}
+ }
+\]
+(cf. \sref{II.2.2.5}) commutes.
+\end{env}
+
+\begin{env}[2.8.13]
+\label{II.2.8.13}
+Keep the notation and hypotheses of \sref{II.2.8.9}, and let $\sh{F}'$ be an $\sh{O}_{X'}$-module;
+if we set $\sh{F}=\Phi^*(\sh{F}')$, then, for all $n\in\bb{Z}$, we have $\sh{F}(n)=\Phi^*(\sh{F}'(n))$, by \sref{II.2.8.11} and \sref[0]{0.4.3.3}.
+Then \sref[0]{0.3.7.1} we have a canonical homomorphism
+\[
+ \Gamma(\rho): \Gamma(X',\sh{F}'(n)) \to \Gamma(X,\sh{F}(n))
+\]
+which gives a canonical di-homomorphism of graded modules
+\[
+ \Gamma_\bullet(\sh{F}') \to \Gamma_\bullet(\sh{F}).
+\]
+
+Suppose that the ideal $S_+$ is generated by $S_1$, and that $\sh{F}'=\widetilde{M'}$, thus $\sh{F}=\widetilde{M}$ with $M=M'\otimes_{A'}A$.
+If $f'$ is homogeneous in $S'_+$, and $f=\varphi(f')$, then we have seen that $M_{(f)}=M'_{(f')}\otimes_{A'}A$, and the diagram
+\[
+ \xymatrix{
+ M'_0 \ar[r] \ar[d]
+ & M'_{(f')} \ar[d]
+ & =\Gamma(D_+(f'),\widetilde{M'})
+ \\M_0 \ar[r]
+ & M_{(f)}
+ & =\Gamma(D_+(f),\widetilde{M})
+ }
+\]
+thus commutes;
+we immediately conclude from this remark, and from the definition of the homomorphism $\alpha$ \sref{II.2.6.2}, that the diagram
+\[
+\label{II.2.8.13.1}
+ \xymatrix{
+ M' \ar[r]^-{\alpha_{M'}} \ar[d]
+ & \Gamma_\bullet(\widetilde{M'}) \ar[d]
+ \\M \ar[r]_-{\alpha_M}
+ & \Gamma_\bullet(\widetilde{M})
+ }
+\tag{2.8.13.1}
+\]
+commutes.
+Similarly, the diagram
+\[
+\label{II.2.8.13.2}
+ \xymatrix{
+ (\Gamma_\bullet(\sh{F}'))\supertilde \ar[r]^-{\beta_{\sh{F}'}} \ar[d]
+ & \sh{F}' \ar[d]
+ \\(\Gamma_\bullet(\sh{F}))\supertilde \ar[r]_-{\beta_{\sh{F}}}
+ & \sh{F}
+ }
+\tag{2.8.13.2}
+\]
+commutes (the vertical arrow on the right being the canonical $\Phi$-morphism $\sh{F}'\to\Phi^*(\sh{F}')=\sh{F}$).
+\end{env}
+
+\oldpage[II]{47}
+
+\begin{env}[2.8.14]
+\label{II.2.8.14}
+Still keeping the notation and hypotheses of \sref{II.2.8.9}, let $N'$ be another graded $S'$-module, and let $N=N'\otimes_{A'}A$.
+It is immediate that the canonical di-homomorphisms $M'\to M$ and $N'\to N$ give a di-homomorphism $M'\otimes_{S'}N'\to M\otimes_S N$ (with respect to the canonical ring homomorphism $S'\to S$), and thus also an $S$-homomorphism $(M'\otimes_{S'}N')\otimes_{A'}A\to M\otimes_S N$ of degree~$0$, to which corresponds (taking \sref{II.2.8.10} into account) a homomorphism of $\sh{O}_X$-modules
+\[
+\label{II.2.8.14.1}
+ \Phi^*((M'\otimes_{S'}N')\supertilde) \to (M\otimes_S N)\supertilde.
+\tag{2.8.14.1}
+\]
+
+Furthermore, we can immediately verify that the diagram
+\[
+\label{II.2.8.14.2}
+ \xymatrix{
+ \Phi^*(\widetilde{M'}\otimes_{\sh{O}_{X'}}\widetilde{N'}) \ar[r]^\sim \ar[d]_{\Phi^*(\lambda)}
+ & \widetilde{M}\otimes_{\sh{O}_X}\widetilde{N} \ar[d]^\lambda
+ & =\Phi^*(\widetilde{M'})\otimes_{\sh{O}_X}\Phi^*(\widetilde{N'})
+ \\\Phi^*((M'\otimes_{S'}N')\supertilde) \ar[r]
+ & (M\otimes_S N)\supertilde
+ }
+\tag{2.8.14.2}
+\]
+commutes, with the first line being the canonical isomorphism \sref[0]{0.4.3.3}.
+If the ideal $S'_+$ is generated by $S'_1$, then it is clear that $S_+$ is generated by $S_1$, and the two vertical arrows of \sref{II.2.8.14.2} are then isomorphisms \sref{II.2.5.13};
+it is thus also the case for \sref{II.2.8.14.1}.
+
+We similarly have a canonical di-homomorphism $\Hom_{S'}(M',N')\to\Hom_S(M,N)$ by sending a homomorphism $u'$ of degree~$k$ to the homomorphism $u'\otimes1$, which is also of degree~$k$;
+from this, we again deduce a $S$-homomorphism of degree~$0$
+\[
+ (\Hom_{S'}(M',N'))\otimes_{A'}A \to \Hom_S(M,N)
+\]
+whence a homomorphism of $\sh{O}_X$-modules
+\[
+\label{II.2.8.14.3}
+ \Phi^*((\Hom_{S'}(M',N'))\supertilde) \to (\Hom_S(M,N))^\supertilde.
+\tag{2.8.14.3}
+\]
+
+Furthermore, the diagram
+\[
+ \xymatrix{
+ \Phi^*((\Hom_{S'}(M',N'))\supertilde) \ar[r] \ar[d]_{\Phi^*(\mu)}
+ & (\Hom_S(M,N))^\supertilde \ar[d]^\mu
+ \\\Phi^*(\shHom_{\sh{O}_{X'}}(\widetilde{M'},\widetilde{N'})) \ar[r]
+ & \shHom_{\sh{O}_X}(\widetilde{M},\widetilde{N})
+ }
+\]
+commutes (the second horizontal line being the canonical homomorphism \sref[0]{0.4.4.6}).
+\end{env}
+
+\oldpage[II]{48}
+
+\begin{env}[2.8.15]
+\label{II.2.8.15}
+With the notation and hypotheses of \sref{II.2.8.1}, it follows from \sref{II.2.4.7} that we do not change the morphism $\Phi$, up to isomorphism, when we replace $S_0$ and $S'_0$ by $\bb{Z}$, and $\varphi_0$ by the identity map, and thus when we replace $S$ and $S'$ by $S^{(d)}$ and ${S'}^{(d)}$ (respectively) ($d>0$), and $\varphi$ by its restriction $\varphi^{(d)}$ to $S^{(d)}$.
+\end{env}
+
+
+\subsection{Closed subschemes of a scheme $\operatorname{Proj}(S)$}
+\label{subsection:II.2.9}
+
+\begin{env}[2.9.1]
+\label{II.2.9.1}
+If $\varphi:S\to S'$ is a homomorphism of graded rings, then we say that $\varphi$ is (TN)-\emph{surjective} (resp. (TN)-\emph{injective}, (TN)-\emph{bijective}) if there exists an integer $n$ such that, for $k\geq n$, $\varphi_k:S_k\to S'_k$ is \emph{surjective} (resp. \emph{injective}, \emph{bijective}).
+Instead of saying that $\varphi$ is (TN)-bijective, we sometimes say that it is a (TN)-\emph{isomorphism}.
+\end{env}
+
+\begin{proposition}[2.9.2]
+\label{II.2.9.2}
+Let $S$ be a positively graded ring, and let $X=\Proj(S)$.
+\begin{enumerate}
+ \item[\rm{(i)}] If $\varphi:S\to S'$ is a (TN)-surjective homomorphism of graded rings, then the corresponding morphism $\Phi$ \sref{II.2.8.1} is defined on the whole of $\Proj(S')$, and is a closed immersion of $\Proj(S')$ into $X$.
+ If $\mathfrak{J}$ is the kernel of $\varphi$, then the closed subscheme of $X$ associated to $\Phi$ is defined by the quasi-coherent sheaf of ideals $\widetilde{\mathfrak{J}}$ of $\sh{O}_X$.
+ \item[\rm{(ii)}] Suppose further that the ideal $S_+$ is generated by a finite number of homogeneous elements of degree~$1$.
+ Let $X'$ be a closed subscheme of $X$ defined by a quasi-coherent sheaf of ideals $\sh{J}$ of $\sh{O}_X$.
+ Let $\mathfrak{J}$ be the graded ideal of $S$ given by the inverse image of $\Gamma_\bullet(\sh{J})$ under the canonical homomorphism $\alpha:S\to\Gamma_\bullet(\sh{O}_X)$ \sref{II.2.6.2}, and set $S'=S/\mathfrak{J}$.
+ Then $X'$ is the subscheme associated to the closed immersion $\Proj(S')\to X$ corresponding to the canonical homomorphism of graded rings $S\to S'$.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+\begin{enumerate}
+ \item[\rm{(i)}] We can suppose that $\varphi$ is surjective \sref{II.2.9.1}.
+ Since, by hypothesis, $\varphi(S_+)$ generates $S'_+$, we have $G(\varphi)=\Proj(S')$.
+ Now, the second claim can be checked locally on $X$;
+ so let $f$ be a homogeneous element of $S_+$, and set $f'=\varphi(f)$.
+ Since $\varphi$ is a surjective homomorphism of graded rings, we immediately see that $\varphi_{(f')}:S_{(f)}\to S'_{(f')}$ is surjective, and that its kernel is $\mathfrak{J}_{(f)}$, which proves (i) \sref[I]{I.4.2.3}.
+ \item[\rm{(ii)}] By (i), we are led to proving that the homomorphism $\widetilde{j}:\widetilde{\mathfrak{J}}\to\sh{O}_X$ induced by the canonical injection $j:\mathfrak{J}\to S$ is an isomorphism from $\widetilde{\mathfrak{J}}$ to $\sh{J}$, which follows from \sref{II.2.7.11}.
+\end{enumerate}
+\end{proof}
+
+We note that $\mathfrak{J}$ is the \emph{largest} of the graded ideals $\mathfrak{J}'$ of $S$ such that $\widetilde{j}(\widetilde{\mathfrak{J'}})=\sh{J}$, since we can immediately show, using the definitions \sref{II.2.6.2}, that this equation implies that $\alpha(\mathfrak{J}')\subset\Gamma_\bullet(\sh{J})$.
+
+\begin{corollary}[2.9.3]
+\label{II.2.9.3}
+Suppose that the hypotheses of \sref{II.2.9.2}[(i)] are satisfied, and further that the ideal $S_+$ is generated by $S_1$;
+then $\Phi^*((S(n))\supertilde)$ is canonically isomorphic to $(S'(n))\supertilde$ for all $n\in\bb{Z}$, and so $\Phi^*(\sh{F}(n))$ is canonically isomorphic to $\Phi^*(\sh{F})(n)$ for every $\sh{O}_X$-module $\sh{F}$.
+\end{corollary}
+
+\begin{proof}
+This is a particular case of \sref{II.2.8.8}, taking \sref{II.2.5.10.2} into account.
+\end{proof}
+
+\begin{corollary}[2.9.4]
+\label{II.2.9.4}
+Suppose that the hypotheses of \sref{II.2.9.2}[(ii)] are satisfied.
+For the closed sub-prescheme $X'$ of $X$ to be integral, it is necessary and sufficient for the graded ideal $\mathfrak{J}$ to be prime in $S$.
+\end{corollary}
+
+\oldpage[II]{49}
+
+\begin{proof}
+Since $X'$ is isomorphic to $\Proj(S/\mathfrak{J})$, the condition is sufficient by \sref{II.2.4.4}.
+To see that it is necessary, consider the exact sequence $0\to\sh{J}\to\sh{O}_X\to\sh{O}_X/\sh{J}$, which gives the exact sequence
+\[
+ 0 \to \Gamma_\bullet(\sh{J}) \to \Gamma_\bullet(\sh{O}_X) \to \Gamma_\bullet(\sh{O}_X/\sh{J}).
+\]
+
+It suffices to prove that, if $f\in S_m$ and $g\in S_n$ are such that the image in $\Gamma_\bullet(\sh{O}_X/\sh{J})$ of $\alpha_{n+m}(fg)$ is zero, then the image of either $\alpha_m(f)$ or $\alpha_n(g)$ is zero.
+But, by definition, these images are sections of invertible $(\sh{O}_X/\sh{J})$-modules $\sh{L}=(\sh{O}_X/\sh{J})(m)$ and $\sh{L}'=(\sh{O}_X/\sh{J})(n)$ over the integral scheme $X'$;
+the hypothesis implies that the product of these two sections is zero in $\sh{L}\otimes\sh{L}'$ (\sref{II.2.9.3} and \sref{II.2.5.14.1}), and so one of them is zero by \sref[I]{I.7.4.4}.
+\end{proof}
+
+\begin{corollary}[2.9.5]
+\label{II.2.9.5}
+Let $A$ be a ring, $M$ an $A$-module, $S$ a graded $A$-algebra generated by the set $S_1$ of homogeneous elements of degree~$1$, $u:M\to S_1$ a surjective homomorphism of $A$-modules, and $\overline{u}:\bb{S}(M)\to S$ the homomorphism (of $A$-algebras) from the symmetric algebra $\bb{S}(M)$ of $M$ to $S$ that extends $u$.
+Then the morphism corresponding to $\overline{u}$ is a closed immersion of $\Proj(S)$ into $\Proj(\bb{S}(M))$.
+\end{corollary}
+
+\begin{proof}
+Indeed, $\overline{u}$ is surjective by hypothesis, and so it suffices to apply \sref{II.2.9.2}
+\end{proof}
diff --git a/ega2/ega2-4.tex b/ega2/ega2-4.tex
index 251e4d4..f3ee19a 100644
--- a/ega2/ega2-4.tex
+++ b/ega2/ega2-4.tex
@@ -135,7 +135,7 @@ Since the functor $r^*$ is right exact \sref[0]{0.4.3.1}, we obtain, from the su
But $r^*(p^*(\sh{E}))=q^*(\sh{E})$, and $r^*(\sh{O}_P(1))$ is locally isomorphic to $r^*(\sh{O}_P)=\sh{O}_X$, or, in other words, the latter is an \emph{invertible} sheaf $\sh{L}_r$ on $\sh{O}_X$, and so we have defined, given $r$, a canonical surjective $\sh{O}_X$-homomorphism
\[
\label{II.4.2.1.1}
- \varphi_r:q^*(\sh{E}) \to \sh{L}_r.
+ \vphi_r:q^*(\sh{E}) \to \sh{L}_r.
\tag{4.2.1.1}
\]
@@ -143,15 +143,15 @@ When $Y=\Spec(A)$ is affine, and $\mathscr{E}=\widetilde{E}$, we can further cla
given $f\in E$, it follows from \sref{II.2.6.3} that
\[
\label{II.4.2.1.2}
- r^{-1}(D_+(f)) = X_{\varphi_r^\flat(f)}.
+ r^{-1}(D_+(f)) = X_{\vphi_r^\flat(f)}.
\tag{4.2.1.2}
\]
Now let $V$ be an affine open subset of $X$ that is contained inside $r^{-1}(D_+(f))$, and let $B$ be its ring, which is an $A$-algebra;
let $S=\bb{S}_A(E)$;
the restriction of $r$ to $V$ corresponds to an $A$-homomorphism $\omega:\bb{S}_f\to B$, and we have that $q^*(\sh{E})|V = (E\otimes_A B)\supertilde$ and $\sh{L}_r|V = \widetilde{L_r}$, whence $L_r = (S(1))_{(f)}\otimes_{S_{(f)}}B_{[\omega]}$ \sref[I]{I.1.6.5}.
-The restriction of $\varphi_r$ to $q^*(\sh{E})|V$ corresponds to the $B$-homomorphism $u:E\otimes_A B\to L_r$, which sends $x\otimes1$ to $(x/1)\otimes f = (f/1)\otimes\omega(x/f)$.
-The canonical extension of $\varphi_r$ to a homomorphism of $\sh{O}_X$-algebras
+The restriction of $\vphi_r$ to $q^*(\sh{E})|V$ corresponds to the $B$-homomorphism $u:E\otimes_A B\to L_r$, which sends $x\otimes1$ to $(x/1)\otimes f = (f/1)\otimes\omega(x/f)$.
+The canonical extension of $\vphi_r$ to a homomorphism of $\sh{O}_X$-algebras
\[
\psi_r: q^*(\bb{S}(\sh{E})) = \bb{S}(q^*(\sh{E})) \to \bb{S}(\sh{L}_r) = \bigoplus_{n\geq0}\sh{L}_r^{\otimes n}
\]
@@ -161,32 +161,32 @@ is thus such that the restriction of $\psi_r$ to $q^*(\bb{S}_n(\sh{E}))|V$ corre
\begin{env}[4.2.2]
\label{II.4.2.2}
Conversely, suppose that we are given a morphism $q:X\to Y$, an invertible $\sh{O}_X$-module $\sh{L}$, and a quasi-coherent $\sh{O}_Y$-module $\sh{E}$;
-to each homomorphism $\varphi:q^*(\sh{E})\to\sh{L}$ there canonically corresponding homomorphism of quasi-coherent $\sh{O}_X$-algebras
+to each homomorphism $\vphi:q^*(\sh{E})\to\sh{L}$ there canonically corresponding homomorphism of quasi-coherent $\sh{O}_X$-algebras
\[
\psi: \bb{S}(q^*(\sh{E})) = q^*(\bb{S}(\sh{E})) \to \bigoplus_{n\geq0}\sh{L}^{\otimes n}
\]
-and thus \sref{II.3.7.1} a $Y$-morphism $r_{\sh{L},\psi}:G(\psi)\to\Proj(\bb{S}(\sh{E}))=\bb{P}(\sh{E})$, which we denote by $r_{\sh{L},\varphi}$.
-If $\varphi$ is \emph{surjective}, then so too is $\psi$, and thus \sref{II.3.7.4} $r_{\sh{L},\varphi}$ is \emph{everywhere defined}.
+and thus \sref{II.3.7.1} a $Y$-morphism $r_{\sh{L},\psi}:G(\psi)\to\Proj(\bb{S}(\sh{E}))=\bb{P}(\sh{E})$, which we denote by $r_{\sh{L},\vphi}$.
+If $\vphi$ is \emph{surjective}, then so too is $\psi$, and thus \sref{II.3.7.4} $r_{\sh{L},\vphi}$ is \emph{everywhere defined}.
Furthermore, with the notation of \sref{II.4.2.1} and \sref{II.4.2.2}:
\end{env}
\begin{proposition}[4.2.3]
\label{II.4.2.3}
-Given a morphism $q:X\to Y$ and a quasi-coherent $\sh{O}_Y$-module $\sh{E}$, maps $r\to(\sh{L}_r,\varphi_r)$ and $(\sh{L},\varphi)\to r_{\sh{L},\varphi}$ give a bijective correspondence between the set of $Y$-morphisms $r:X\to\bb{P}(\sh{E})$ and the set of equivalence classes of pairs $(\sh{L},\varphi)$ of an invertible $\sh{O}_X$-module $\sh{L}$ and a surjective homomorphism $\varphi:q^*(\sh{E})\to\sh{L}$, where such pairs $(\sh{L},\varphi)$ and $(\sh{L}',\varphi')$ are defined to be equivalent if there exists an $\sh{O}_X$-isomorphism $\tau:\sh{L}\xrightarrow{\sim}\sh{L}'$ such that $\varphi'=\tau\circ\varphi$.
+Given a morphism $q:X\to Y$ and a quasi-coherent $\sh{O}_Y$-module $\sh{E}$, maps $r\to(\sh{L}_r,\vphi_r)$ and $(\sh{L},\vphi)\to r_{\sh{L},\vphi}$ give a bijective correspondence between the set of $Y$-morphisms $r:X\to\bb{P}(\sh{E})$ and the set of equivalence classes of pairs $(\sh{L},\vphi)$ of an invertible $\sh{O}_X$-module $\sh{L}$ and a surjective homomorphism $\vphi:q^*(\sh{E})\to\sh{L}$, where such pairs $(\sh{L},\vphi)$ and $(\sh{L}',\vphi')$ are defined to be equivalent if there exists an $\sh{O}_X$-isomorphism $\tau:\sh{L}\xrightarrow{\sim}\sh{L}'$ such that $\vphi'=\tau\circ\vphi$.
\end{proposition}
\begin{proof}
-Start first with a $Y$-morphism $r:X\to\bb{P}(\sh{E})$, and construct $\sh{L}_r$ and $\varphi_r$ \sref{II.4.2.1}, and let $r'=r_{\sh{L}_r,\varphi_r}$;
+Start first with a $Y$-morphism $r:X\to\bb{P}(\sh{E})$, and construct $\sh{L}_r$ and $\vphi_r$ \sref{II.4.2.1}, and let $r'=r_{\sh{L}_r,\vphi_r}$;
it follows immediately from \sref{II.4.2.1} and \sref{II.3.7.2} that the morphisms $r$ and $r'$ are identical (by taking the generator of $\sh{L}_r$ to be the element $(f/1)\otimes1$ to define the homomorphisms $v_n$ of \sref{II.3.7.2}).
-Conversely, take a pair $(\sh{L},\varphi)$ and construct
+Conversely, take a pair $(\sh{L},\vphi)$ and construct
\oldpage[II]{74}
-$r''=r_{\sh{L},\varphi}$, and then $\sh{L}_{r''}$ and $\varphi_{r''}$;
-we will show that there exists a canonical isomorphism $\tau:\sh{L}_{r''}\xrightarrow{\sim}\sh{L}$ such that $\varphi=\tau\circ\varphi_{r''}$;
+$r''=r_{\sh{L},\vphi}$, and then $\sh{L}_{r''}$ and $\vphi_{r''}$;
+we will show that there exists a canonical isomorphism $\tau:\sh{L}_{r''}\xrightarrow{\sim}\sh{L}$ such that $\vphi=\tau\circ\vphi_{r''}$;
to define it, we can restrict to the case where $Y=\Spec(A)$ and $X=\Spec(B)$ are affine, and (with the notation of \sref{II.4.2.1} and \sref{II.3.7.2}) associate to each element $(x/1)\otimes1$ of $L_{r''}$ (where $x\in E$) the element $v_1(x)c$ of $L$.
We immediately see that $\tau$ does not depend on the chosen generator $c$ of $L$;
since $v_1$ is surjective by hypothesis, to prove that $\tau$ is an isomorphism it suffices to to show that, if $x/1=0$ in $(S(1))_{(f)}$, then $v_1(x)/1=0$ in $B_g$;
but the first equality implies that $f^nx=0$ in $\bb{S}_{n+1}(E)$ for some $n$, and this implies that $v_{n+1}(f^nx) = g^nv_1(x) = 0$ in $B$, whence the conclusion.
-Finally, it is immediate that, for any two equivalent pairs $(\sh{L},\varphi)$ and $(\sh{L}',\varphi')$, we have $r_{\sh{L},\varphi}=r_{\sh{L}',\varphi'}$.
+Finally, it is immediate that, for any two equivalent pairs $(\sh{L},\vphi)$ and $(\sh{L}',\vphi')$, we have $r_{\sh{L},\vphi}=r_{\sh{L}',\vphi'}$.
\end{proof}
In particular, for $X=Y$:
@@ -247,26 +247,26 @@ This will allow us to determine the sheaf of germs of automorphisms of $\bb{P}(\
\begin{env}[4.2.8]
\label{II.4.2.8}
Keeping the notation of \sref{II.4.2.1}, let $u:X'\to X$ be a morphism;
-if the $Y$-morphism $r:X\to P$ corresponds to the homomorphism $\varphi:q^*(\sh{E})\to\sh{L}$, then the $Y$-morphism $r\circ u$ corresponds to $u^*(\varphi):u^*(q^*(\sh{E}))\to u^*(\sh{L})$, as follows immediately from the definitions.
+if the $Y$-morphism $r:X\to P$ corresponds to the homomorphism $\vphi:q^*(\sh{E})\to\sh{L}$, then the $Y$-morphism $r\circ u$ corresponds to $u^*(\vphi):u^*(q^*(\sh{E}))\to u^*(\sh{L})$, as follows immediately from the definitions.
\end{env}
\begin{env}[4.2.9]
\label{II.4.2.9}
Let $\sh{E}$ and $\sh{F}$ be quasi-coherent $\sh{O}_Y$-modules, $v:\sh{E}\to\sh{F}$ a surjective homomorphism, and $j=\bb{P}(v)$ the corresponding closed immersion $\bb{P}(\sh{F})\to\bb{P}(\sh{E})$ \sref{II.4.1.2}.
-If the $Y$-morphism $r:X\to\bb{P}(\sh{F})$ corresponds to the homomorphism $\varphi:q^*(\sh{F})\to\sh{L}$, then the
+If the $Y$-morphism $r:X\to\bb{P}(\sh{F})$ corresponds to the homomorphism $\vphi:q^*(\sh{F})\to\sh{L}$, then the
\oldpage[II]{76}
-$Y$-morphism $j\circ r$ corresponds to $q^*(\sh{E})\xrightarrow{q^*(v)}q^*(\sh{F})\xrightarrow{\varphi}\sh{L}$;
+$Y$-morphism $j\circ r$ corresponds to $q^*(\sh{E})\xrightarrow{q^*(v)}q^*(\sh{F})\xrightarrow{\vphi}\sh{L}$;
this again follows from the definition given in \sref{II.4.2.1}.
\end{env}
\begin{env}[4.2.10]
\label{II.4.2.10}
Let $\psi:Y'\to Y$ be a morphism, and let $\sh{E}'=\psi^*(\sh{E})$.
-If the $Y$-morphism $r:X\to P$ corresponds to the homomorphism $\varphi:q^*(\sh{E})\to\sh{L}$, then the $Y'$-morphism
+If the $Y$-morphism $r:X\to P$ corresponds to the homomorphism $\vphi:q^*(\sh{E})\to\sh{L}$, then the $Y'$-morphism
\[
r_{(Y')}: X_{(Y')} \to P' = \bb{P}(\sh{E}')
\]
-corresponds to $\varphi_{(Y')}:q_{(Y')}^*(\sh{E}') = q^*(\sh{E})\otimes_Y\sh{O}_{Y'} \to \sh{L}\otimes_Y\sh{O}_{Y'}$.
+corresponds to $\vphi_{(Y')}:q_{(Y')}^*(\sh{E}') = q^*(\sh{E})\otimes_Y\sh{O}_{Y'} \to \sh{L}\otimes_Y\sh{O}_{Y'}$.
Indeed, by \sref{II.4.1.3.1}, we have the commutative diagram
\[
\xymatrix{
@@ -450,21 +450,21 @@ Let $Y$ be a quasi-compact scheme, or a prescheme whose underlying space is Noet
\item[\rm{(i)}] Let $\sh{S}$ be a positively-graded quasi-coherent $\sh{O}_Y$-algebra, and $\psi:q^*(\sh{S})\to\bigoplus_{n\geq0}\sh{L}^{\otimes n}$ a homomorphism of graded algebras.
For $r_{\sh{L},\psi}$ to be everywhere defined and an immersion, it is necessary and
\oldpage[II]{79}
- sufficient for there to exist an integer $n\geq0$ and a quasi-coherent sub-$\sh{O}_Y$-module \emph{of finite type} $\sh{E}$ of $\sh{S}_n$ such that the homomorphism $\psi'=\psi_n\circ q^*(j):q^*(\sh{E})\to\sh{L}^{\otimes n}=\sh{L}'$ (where $j$ is the injection $\sh{E}\to\sh{S}_n$) is surjective and such that the morphism $r_{\sh{L}',\varphi'}:X\to\bb{P}(\sh{E})$ is an immersion.
- \item[\rm{(ii)}] Let $\sh{F}$ be a quasi-coherent $\sh{O}_Y$-module, and $\varphi:q^*(\sh{F})\to\sh{L}$ a surjective homomorphism.
- For the morphism $r_{\sh{L},\varphi}$ to be an immersion $X\to\bb{P}(\sh{F})$, it is necessary and sufficient for there to exist a quasi-coherent sub-$\sh{O}_Y$-module \emph{of finite type} $\sh{E}$ of $\sh{F}$ such that the homomorphism $\varphi'=\varphi\circ q(j):q^*(\sh{E})\to\sh{L}$ (where $j$ is the canonical injection $\sh{E}\to\sh{F}$) is surjective and such that the morphism $r_{\sh{L},\varphi'}:X\to\bb{P}(\sh{E})$ is an immersion.
+ sufficient for there to exist an integer $n\geq0$ and a quasi-coherent sub-$\sh{O}_Y$-module \emph{of finite type} $\sh{E}$ of $\sh{S}_n$ such that the homomorphism $\psi'=\psi_n\circ q^*(j):q^*(\sh{E})\to\sh{L}^{\otimes n}=\sh{L}'$ (where $j$ is the injection $\sh{E}\to\sh{S}_n$) is surjective and such that the morphism $r_{\sh{L}',\vphi'}:X\to\bb{P}(\sh{E})$ is an immersion.
+ \item[\rm{(ii)}] Let $\sh{F}$ be a quasi-coherent $\sh{O}_Y$-module, and $\vphi:q^*(\sh{F})\to\sh{L}$ a surjective homomorphism.
+ For the morphism $r_{\sh{L},\vphi}$ to be an immersion $X\to\bb{P}(\sh{F})$, it is necessary and sufficient for there to exist a quasi-coherent sub-$\sh{O}_Y$-module \emph{of finite type} $\sh{E}$ of $\sh{F}$ such that the homomorphism $\vphi'=\vphi\circ q(j):q^*(\sh{E})\to\sh{L}$ (where $j$ is the canonical injection $\sh{E}\to\sh{F}$) is surjective and such that the morphism $r_{\sh{L},\vphi'}:X\to\bb{P}(\sh{E})$ is an immersion.
\end{enumerate}
\end{proposition}
\begin{proof}
\medskip\noindent
\begin{enumerate}
- \item[\rm{(i)}] The fact that $r_{\sh{L},\varphi}$ is everywhere defined and is an immersion is equivalent, by \sref{II.3.8.5}, to the existence of some $n\geq0$ and $\sh{E}$ such that, if $\sh{S}'$ is the subalgebra of $\sh{S}$ generated by $\sh{E}$, the homomorphism $q^*(\sh{E})\to\sh{L}^{\otimes n}$ is surjective and the morphism $r_{\sh{L},\psi'}:X\to\Proj(\sh{S}')$ is everywhere defined and is an immersion.
+ \item[\rm{(i)}] The fact that $r_{\sh{L},\vphi}$ is everywhere defined and is an immersion is equivalent, by \sref{II.3.8.5}, to the existence of some $n\geq0$ and $\sh{E}$ such that, if $\sh{S}'$ is the subalgebra of $\sh{S}$ generated by $\sh{E}$, the homomorphism $q^*(\sh{E})\to\sh{L}^{\otimes n}$ is surjective and the morphism $r_{\sh{L},\psi'}:X\to\Proj(\sh{S}')$ is everywhere defined and is an immersion.
We already have a canonical surjective homomorphism $\bb{S}(\sh{E})\to\sh{S}'$ to which there exists a corresponding closed immersion $\Proj(\sh{S}')\to\bb{P}(\sh{E})$ \sref{II.3.6.2};
whence the conclusion.
\item[\rm{(ii)}] Since $\sh{F}$ is the inductive limit of its quasi-coherent submodules of finite type $\sh{E}_\lambda$ \sref[I]{I.9.4.9}, $\bb{S}(\sh{F})$ is the inductive limit of the $\bb{S}(\sh{E}_\lambda)$;
the conclusion then follows from \sref{II.3.8.4}, by observing that we can take all the $n_i$ in the proof of \sref{II.3.8.4} to be equal to $1$:
- indeed, supposing that $Y$ is affine, if $r=r_{\sh{L},\varphi}$ is an immersion, then $r(X)$ is a quasi-compact subspace of $\bb{P}(\sh{F})$ that we can cover by finitely many open subsets of $\bb{P}(\sh{F})$ of the form $D_+(f)$, with $f\in F$, such that $D_+(f)\cap r(X)$ is closed.
+ indeed, supposing that $Y$ is affine, if $r=r_{\sh{L},\vphi}$ is an immersion, then $r(X)$ is a quasi-compact subspace of $\bb{P}(\sh{F})$ that we can cover by finitely many open subsets of $\bb{P}(\sh{F})$ of the form $D_+(f)$, with $f\in F$, such that $D_+(f)\cap r(X)$ is closed.
\end{enumerate}
\end{proof}
@@ -474,7 +474,7 @@ Let $Y$ be a prescheme, and $q:X\to Y$ a morphism.
We say that an invertible $\sh{O}_X$-module $\sh{L}$ is \emph{very ample for $q$}, or \emph{relative to $q$} (or \emph{very ample for} (or \emph{relative to}) \emph{$Y$}, or simply \emph{very ample}, if $q$ is clear from the context) if there exists a quasi-coherent $\sh{O}_Y$-module $\sh{E}$ and a $Y$-immersion $i$ from $X$ to $P=\bb{P}(\sh{E})$ such that $\sh{L}$ is isomorphic to $i^*(\sh{O}_P(1))$.
\end{definition}
-It is equivalent \sref{II.4.2.3} to say that there exists a quasi-coherent $\sh{O}_Y$-module $\sh{E}$ and a \emph{surjective} homomorphism $\varphi:q^*(\sh{E})\to\sh{L}$ such that $r_{\sh{L},\varphi}:X\to\bb{P}(\sh{E})$ is an \emph{immersion}.
+It is equivalent \sref{II.4.2.3} to say that there exists a quasi-coherent $\sh{O}_Y$-module $\sh{E}$ and a \emph{surjective} homomorphism $\vphi:q^*(\sh{E})\to\sh{L}$ such that $r_{\sh{L},\vphi}:X\to\bb{P}(\sh{E})$ is an \emph{immersion}.
We note that the existence of a very ample (for $Y$) $\sh{O}_X$-module implies that $q$ is \emph{separated} (\sref{II.3.1.3} and \sref[I]{I.5.5.1}[(i) and (ii)]).
@@ -502,12 +502,12 @@ Then the following properties are equivalent:
Since $q$ is quasi-compact, we know that $q_*(\sh{L})$ is quasi-coherent if $q$ is separated \sref[I]{I.9.2.2}.
\oldpage[II]{80}
-We know \sref{II.3.4.7} that the existence of a surjective homomorphism $\varphi:q^*(\sh{E})\to\sh{L}$ (with $\sh{E}$ a quasi-coherent $\sh{O}_Y$-module) implies that $\sigma$ is surjective;
-furthermore, given the factorisation $q^*(\sh{E})\to q^*(q_*(\sh{L}))\xrightarrow{\sigma}\sh{L}$ of $\varphi$, there is a canonically corresponding factorisation
+We know \sref{II.3.4.7} that the existence of a surjective homomorphism $\vphi:q^*(\sh{E})\to\sh{L}$ (with $\sh{E}$ a quasi-coherent $\sh{O}_Y$-module) implies that $\sigma$ is surjective;
+furthermore, given the factorisation $q^*(\sh{E})\to q^*(q_*(\sh{L}))\xrightarrow{\sigma}\sh{L}$ of $\vphi$, there is a canonically corresponding factorisation
\[
q^*(\bb{S}(\sh{E})) \to q^*(\bb{S}(q_*(\sh{L}))) \to \bigoplus_{n\geq0}\sh{L}^{\otimes n}
\]
-and so \sref{II.3.8.3} the hypothesis that $r_{\sh{L},\varphi}$ is an immersion implies that so too is $j=r_{\sh{L},\sigma}$;
+and so \sref{II.3.8.3} the hypothesis that $r_{\sh{L},\vphi}$ is an immersion implies that so too is $j=r_{\sh{L},\sigma}$;
furthermore \sref{II.4.2.4}, $\sh{L}$ is isomorphic to $j^*(\sh{O}_{P'}(1))$, where $P'=\bb{P}(q_*(\sh{L}))$.
We thus see that (a) and (b) are equivalent.
\end{proof}
@@ -527,7 +527,7 @@ Indeed, condition (b) of \sref{II.4.4.4} is local on $Y$.
Let $Y$ be a quasi-compact scheme, or a prescheme whose underlying space is Noetherian, $q:X\to Y$ a morphism \emph{of finite type}, and $\sh{L}$ an invertible $\sh{O}_X$-module.
Then conditions (a) and (b) of \sref{II.4.4.4} are equivalent to the following:
\begin{enumerate}
- \item[\rm{(a')}] There exists a quasi-coherent $\sh{O}_Y$-module $\sh{E}$ \emph{of finite type} and a surjective homomorphism $\varphi:q^*(\sh{E})\to\sh{L}$ such that $r_{\sh{L},\varphi}$ is an immersion.
+ \item[\rm{(a')}] There exists a quasi-coherent $\sh{O}_Y$-module $\sh{E}$ \emph{of finite type} and a surjective homomorphism $\vphi:q^*(\sh{E})\to\sh{L}$ such that $r_{\sh{L},\vphi}$ is an immersion.
\item[\rm{(b')}] There exists a coherent sub-$\sh{O}_Y$-module $\sh{E}$ of $q_*(\sh{L})$ \emph{of finite type} that has the properties stated in condition~(a').
\end{enumerate}
\end{proposition}
@@ -546,7 +546,7 @@ If $\sh{L}$ is very ample for $q$, then there exists a graded quasi-coherent $\s
\begin{proof}
Indeed, condition~(b) of \sref{II.4.4.6} is satisfied;
the structure morphism $p:\bb{P}(\sh{E})=P'\to Y$ is then separated and of finite type \sref{II.3.1.3}, and so $P'$ is a quasi-compact scheme (resp. a Noetherian prescheme) if $Y$ is a quasi-compact scheme (resp. a Noetherian prescheme).
-Let $Z$ be the closure \sref[I]{I.9.5.11} of the subprescheme $X'$ of $P'$ associated to the immersion $j=r_{\sh{L},\varphi}$ from $X$ into $P'$;
+Let $Z$ be the closure \sref[I]{I.9.5.11} of the subprescheme $X'$ of $P'$ associated to the immersion $j=r_{\sh{L},\vphi}$ from $X$ into $P'$;
it is clear that $j$ factors as a dominant open immersion $i:X\to Z$ followed by the canonical injection $Z\to P'$.
But $Z$ can be identified with a prescheme $\Proj(\sh{S})$, where $\sh{S}$ is a graded $\sh{O}_Y$-algebra equal to the quotient of $\bb{S}(\sh{E})$ by a graded quasi-coherent sheaf of ideals \sref{II.3.6.2}, and it is clear that $\sh{S}_1$ is of finite type and generates $\sh{S}$;
furthermore, $\sh{O}_Z(1)$ is the inverse image of $\sh{O}_{P'}(1)$ by the canonical injection \sref{II.3.6.3}, and so $\sh{L}=i^*(\sh{O}_Z(1))$.
diff --git a/ega2/ega2-7.tex b/ega2/ega2-7.tex
index fbe7060..796442c 100644
--- a/ega2/ega2-7.tex
+++ b/ega2/ega2-7.tex
@@ -128,18 +128,18 @@ Then there exists a local scheme $Y'$, spectrum of a discrete valuation ring, a
\xymatrix{
& \kres(x) \ar[dl]_{\gamma}
\\ \kres(b)
- & \kres(y) \ar[u]_{\pi} \ar[l]^{\varphi}
+ & \kres(y) \ar[u]_{\pi} \ar[l]^{\vphi}
}
\tag{7.1.9.1}
\]
-(where $\pi$, $\varphi$, and $\gamma$ are the homomorphisms corresponding to $p$, $f$, and $g$, respectively) the morphism $\gamma$ is a bijection.
+(where $\pi$, $\vphi$, and $\gamma$ are the homomorphisms corresponding to $p$, $f$, and $g$, respectively) the morphism $\gamma$ is a bijection.
\end{proposition}
\begin{proof}
As in \sref{II.7.1.4}, we can restrict to the case where $Y$ is integral with generic point $y$ (taking \sref[I]{I.6.4.3}[iv] into account), and, since the question is local on $X$ and $Y$, we can assume that $p$ is of finite type;
we are then in the situation of \sref{II.7.1.4}, with the additional property that $\kres(x)$ is an extension \emph{of finite type} of $\kres(y)$ \sref[I]{I.6.4.11} and that $\sh{O}_{y'}$ is Noetherian;
this lets us apply \sref{II.7.1.7} and take $Y'=\Spec(A')$, where $A'$ is a discrete valuation ring that dominates $\sh{O}_{y'}$ and whose field of fractions is $\kres(x)$.
-We have thus defined a commutative diagram \sref{II.7.1.9.1} where $\gamma$ is a bijection, with $\pi$ and $\varphi$ corresponding to the morphisms $p$ and $f$.
+We have thus defined a commutative diagram \sref{II.7.1.9.1} where $\gamma$ is a bijection, with $\pi$ and $\vphi$ corresponding to the morphisms $p$ and $f$.
Furthermore, since $X$ and $Y$ are locally Noetherian \sref[I]{I.6.6.2} and since $Y'$ is integral, there exists exactly one rational $Y$-map $g$ from $Y'$ to $X$ to which corresponds the isomorphism $\gamma$ \sref[I]{I.7.1.15}, which finishes the proof.
\end{proof}
@@ -609,17 +609,17 @@ Since the Segre morphism \sref{II.4.3.3} gives an immersion of $P$ into $\bb{P}_
\begin{corollary}[7.4.11]
\label{II.7.4.11}
-Any normal algebraic curve $X$ is isomorphic to the scheme induced by some complete normal algebraic curve $\hat{X}$ on some everywhere dense open subset, and this $\hat{X}$ is unique up to unique isomorphism.
+Any normal algebraic curve $X$ is isomorphic to the scheme induced by some complete normal algebraic curve $\widehat{X}$ on some everywhere dense open subset, and this $\widehat{X}$ is unique up to unique isomorphism.
\end{corollary}
\begin{proof}
If $X_1$ and $X_2$ are complete normal curves, then it follows from \sref{II.7.4.9} that every isomorphism from any dense open $U_1$ in $X_1$ to any dense open $U_2$ in $X_2$ can be uniquely extended to an isomorphism from $X_1$ to $X_2$;
whence the uniqueness claim.
-To prove the existence of $\hat{X}$, it suffices to note that we can consider $X$ as a subscheme of a projective bundle $\bb{P}_k^n$ \sref{II.7.4.10}.
+To prove the existence of $\widehat{X}$, it suffices to note that we can consider $X$ as a subscheme of a projective bundle $\bb{P}_k^n$ \sref{II.7.4.10}.
Let $\overline{X}$ be the closure of $X$ in $\bb{P}_k^n$ \sref[I]{I.9.5.11};
since $X$ is induced by $\overline{X}$ on a dense open subset of $\overline{X}$ \sref[I]{I.9.5.10}, the generic points $x_i$ of the irreducible components of $X$ are also the generic points of the irreducible components of $\overline{X}$, and the $\kres(x_i)$ are the same for both of these schemes, and so \sref{II.7.4.1} $\overline{X}$ is an algebraic curve over $k$ that is reduced \sref[I]{I.9.5.9} and projective over $k$ \sref{II.5.5.1}, whence complete \sref{II.5.5.3}.
-So we take for $\hat{X}$ the \emph{normalisation} of $\overline{X}$, which is again complete \sref{II.7.4.8};
-furthermore, if $h:\hat{X}\to\overline{X}$ is the canonical morphism, then the restriction of $h$ to $h^{-1}(X)$ is an isomorphism to $X$, since $X$ is normal \sref{II.6.3.4}, and since $h^{-1}(X)$ contains the generic points of the irreducible components of $\hat{X}$ \sref{II.6.3.8}, it is dense in $\hat{X}$, which finishes the proof.
+So we take for $\widehat{X}$ the \emph{normalisation} of $\overline{X}$, which is again complete \sref{II.7.4.8};
+furthermore, if $h:\widehat{X}\to\overline{X}$ is the canonical morphism, then the restriction of $h$ to $h^{-1}(X)$ is an isomorphism to $X$, since $X$ is normal \sref{II.6.3.4}, and since $h^{-1}(X)$ contains the generic points of the irreducible components of $\widehat{X}$ \sref{II.6.3.8}, it is dense in $\widehat{X}$, which finishes the proof.
\end{proof}
\oldpage[II]{151}
diff --git a/ega2/ega2-8.tex b/ega2/ega2-8.tex
index 7b8bcb8..de80491 100644
--- a/ega2/ega2-8.tex
+++ b/ega2/ega2-8.tex
@@ -837,7 +837,7 @@ The gluing of $C$ and $\widehat{C}_f$ along $C_f$ is thus determined by the inje
\begin{proposition}[8.3.5]
\label{II.8.3.5}
-With the notation of \sref{II.8.3.1} and \sref{II.8.3.4}, the morphism associated \sref{II.3.5.1} to the canonical injection $\varphi:\sh{S} \to \widehat{\sh{S}} = \sh{S}[\bb{z}]$ is a surjective affine morphism (called the canonical retraction)
+With the notation of \sref{II.8.3.1} and \sref{II.8.3.4}, the morphism associated \sref{II.3.5.1} to the canonical injection $\vphi:\sh{S} \to \widehat{\sh{S}} = \sh{S}[\bb{z}]$ is a surjective affine morphism (called the canonical retraction)
\[
\label{II.8.3.5.1}
p:\widehat{E} \to X
@@ -853,7 +853,7 @@ such that
\begin{proof}
To prove the proposition, we can restrict to the case where $Y$ is affine.
-Taking into account the expression in \sref{II.8.3.4.4} for $\widehat{E}$, the fact that the domain of definition $G(\varphi)$ of $p$ is equal to $\widehat{E}$ will follow from the first of the following claims:
+Taking into account the expression in \sref{II.8.3.4.4} for $\widehat{E}$, the fact that the domain of definition $G(\vphi)$ of $p$ is equal to $\widehat{E}$ will follow from the first of the following claims:
\begin{env}[8.3.5.3]
\label{II.8.3.5.3}
If $Y=\Spec(A)$ is affine, and $\sh{S}=\widetilde{S}$, then, for all homogeneous $f\in S_+$,
@@ -866,7 +866,7 @@ and the restriction of $p$ to $\widehat{C}_f=\Spec(S_f^\leq)$, thought of as a m
If, further, $f\in S_1$, then $\widehat{C}_f$ is isomorphic to $X_f\otimes_{\bb{Z}}\bb{Z}[T]$ (where $T$ is an indeterminate).
\end{env}
-Indeed, Equation~\sref{II.8.3.5.4} is exactly a particular case of \sref{II.2.8.1.1}, and the second claim is exactly the definition of $\Proj(\varphi)$ whenever $Y$ is affine \sref{II.2.8.1}.
+Indeed, Equation~\sref{II.8.3.5.4} is exactly a particular case of \sref{II.2.8.1.1}, and the second claim is exactly the definition of $\Proj(\vphi)$ whenever $Y$ is affine \sref{II.2.8.1}.
Then Equation~\sref{II.8.3.5.2} and the fact that $p$ is surjective show that the composition $\sh{S}\to\widehat{\sh{S}}\to\sh{S}$ of the canonical homomorphisms is the identity on $\sh{S}$.
Finally, the last claim of \sref{II.8.3.5.3} follows from the fact that $S_f^\leq$ is isomorphic to $S_{(f)}[T]$ whenever $f\in S_1$ \sref{II.2.2.1}.
\end{proof}
@@ -1094,13 +1094,13 @@ Let $Y$ and $Y'$ be prescheme, $q:Y'\to Y$ a morphism, and $\sh{S}$ (resp. $\sh{
Consider a $q$-morphism of graded algebras
\[
\label{eq:2.8.5.1.1}
- \varphi: \sh{S} \to \sh{S}'.
+ \vphi: \sh{S} \to \sh{S}'.
\tag{8.5.1.1}
\]
We know \sref{II.1.5.6} that this corresponds, canonically, to a morphism
\[
- \Phi = \Spec(\varphi): \Spec(\sh{S}') \to \Spec(\sh{S})
+ \Phi = \Spec(\vphi): \Spec(\sh{S}') \to \Spec(\sh{S})
\]
such that the diagram
\[
@@ -1140,7 +1140,7 @@ which corresponds to the diagram
\[
\xymatrix{
\sh{S}
- \ar[r]^{\varphi}
+ \ar[r]^{\vphi}
\ar[d]
& \sh{S}'
\ar[d]
@@ -1149,39 +1149,39 @@ which corresponds to the diagram
& \sh{O}_{Y'}
}
\]
-where the vertical arrows are the augmentation homomorphisms, and so the commutativity follows from the hypothesis that $\varphi$ is assumed to be a homomorphism of \emph{graded} algebras.
+where the vertical arrows are the augmentation homomorphisms, and so the commutativity follows from the hypothesis that $\vphi$ is assumed to be a homomorphism of \emph{graded} algebras.
\end{env}
\begin{proposition}[8.5.2]
\label{II.8.5.2}
If $E$ (resp. $E'$) is the pointed affine cone defined by $\sh{S}$ (resp. $\sh{S}'$), then $\Phi^{-1}(E)\subset E'$;
-if, further, $\Proj(\varphi):G(\varphi)\to\Proj(\sh{S})$ is everywhere defined (or, equivalently, if $G(\varphi)=\Proj(\sh{S}')$), then $\Phi^{-1}(E)=E'$, and conversely.
+if, further, $\Proj(\vphi):G(\vphi)\to\Proj(\sh{S})$ is everywhere defined (or, equivalently, if $G(\vphi)=\Proj(\sh{S}')$), then $\Phi^{-1}(E)=E'$, and conversely.
\end{proposition}
\begin{proof}
The first claim follows from the commutativity of \eref{eq:2.8.5.1.3}.
To prove the second, we can restrict to the case where $Y=\Spec(A)$ and $Y'=\Spec(A')$ are affine, and $\sh{S}=\widetilde{S}$ and $\sh{S}'=\widetilde{S'}$.
-For every homogeneous $f$ in $S_+$, writing $f'=\varphi(f)$, we have that $\Phi^{-1}(C_f)=C'_{f'}$ \sref[I]{I.2.2.4.1};
-saying that $G(\varphi)=\Proj(S')$ implies that the radical (in $S'_+$) of the ideal generated by the $f'=\varphi(f)$ is $S'_+$ itself (\sref{II.2.8.1} and \sref{II.2.3.14}), and this is equivalent to saying that the $C'_{f'}$ cover $E'$ \eref{II.8.3.4.4}.
+For every homogeneous $f$ in $S_+$, writing $f'=\vphi(f)$, we have that $\Phi^{-1}(C_f)=C'_{f'}$ \sref[I]{I.2.2.4.1};
+saying that $G(\vphi)=\Proj(S')$ implies that the radical (in $S'_+$) of the ideal generated by the $f'=\vphi(f)$ is $S'_+$ itself (\sref{II.2.8.1} and \sref{II.2.3.14}), and this is equivalent to saying that the $C'_{f'}$ cover $E'$ \eref{II.8.3.4.4}.
\end{proof}
\begin{env}[8.5.3]
\label{II.8.5.3}
-The $q$-morphism $\varphi$ canonically extends to a $q$-morphism of graded algebras
+The $q$-morphism $\vphi$ canonically extends to a $q$-morphism of graded algebras
\[
\label{II.8.5.3.1}
- \widehat{\varphi}: \widehat{\sh{S}} \to \widehat{\sh{S}'}
+ \widehat{\vphi}: \widehat{\sh{S}} \to \widehat{\sh{S}'}
\tag{8.5.3.1}
\]
-by letting $\widehat{\varphi}(\bb{z})=\bb{z}$.
+by letting $\widehat{\vphi}(\bb{z})=\bb{z}$.
This induces a morphism
\[
- \widehat{\Phi} = \Proj(\widehat{\varphi}) : G(\widehat{\varphi}) \to \widehat{C} = \Proj(\widehat{\sh{S}})
+ \widehat{\Phi} = \Proj(\widehat{\vphi}) : G(\widehat{\vphi}) \to \widehat{C} = \Proj(\widehat{\sh{S}})
\]
such that the diagram
\[
\xymatrix{
- G(\widehat{\varphi})
+ G(\widehat{\vphi})
\ar[r]^{\widehat{\Phi}}
\ar[d]
& \widehat{C}
@@ -1192,7 +1192,7 @@ such that the diagram
}
\]
commutes \sref{II.3.5.6}.
-It follows immediately from the definitions that, if we write $i:C\to\widehat{C}$ and $i':C'\to\widehat{C'}$ to mean the canonical open immersions \sref{II.8.3.2}, then $i'(C')\subset G(\widehat{\varphi})$, and the diagram
+It follows immediately from the definitions that, if we write $i:C\to\widehat{C}$ and $i':C'\to\widehat{C'}$ to mean the canonical open immersions \sref{II.8.3.2}, then $i'(C')\subset G(\widehat{\vphi})$, and the diagram
\[
\label{eq:2.8.5.3.2}
\xymatrix{
@@ -1201,24 +1201,24 @@ It follows immediately from the definitions that, if we write $i:C\to\widehat{C}
\ar[d]_{i}
& C
\ar[d]^{i'}
- \\G(\widehat{\varphi})
+ \\G(\widehat{\vphi})
\ar[r]_{\widehat{\Phi}}
& \widehat{C}
}
\tag{8.5.3.2}
\]
commutes.
-Finally, if we let $X=\Proj(\sh{S})$ and $X'=\Proj(\sh{S}')$, and if $j:X\to\widehat{C}$ and $j':X'\to\widehat{C'}$ are the canonical closed immersions \sref{II.8.3.2}, then it follows from the definition of these immersions that $j'(G(\varphi))\subset G(\widehat{\varphi})$, and that the diagram
+Finally, if we let $X=\Proj(\sh{S})$ and $X'=\Proj(\sh{S}')$, and if $j:X\to\widehat{C}$ and $j':X'\to\widehat{C'}$ are the canonical closed immersions \sref{II.8.3.2}, then it follows from the definition of these immersions that $j'(G(\vphi))\subset G(\widehat{\vphi})$, and that the diagram
\oldpage[II]{171}
\[
\label{II.8.5.3.3}
\xymatrix{
- G(\varphi)
- \ar[r]^{\Proj(\varphi)}
+ G(\vphi)
+ \ar[r]^{\Proj(\vphi)}
\ar[d]_{j'}
& X
\ar[d]^{j}
- \\G(\widehat{\varphi})
+ \\G(\widehat{\vphi})
\ar[r]_{\widehat{\Phi}}
& \widehat{C}
}
@@ -1230,7 +1230,7 @@ commutes.
\begin{proposition}[8.5.4]
\label{II.8.5.4}
If $\widehat{E}$ (resp. $\widehat{E'}$) is the pointed projective cone defined by $\sh{S}$ (resp. by $\sh{S}'$), then $\widehat{\Phi}^{-1}(\widehat{E}) \subset \widehat{E'}$;
-furthermore, if $p:\widehat{E}\to X$ and $p':\widehat{E'}\to X'$ are the canonical retractions, then $p'(\widehat{\Phi}^{-1}(\widehat{E})) \subset G(\widehat{\varphi})$, and the diagram
+furthermore, if $p:\widehat{E}\to X$ and $p':\widehat{E'}\to X'$ are the canonical retractions, then $p'(\widehat{\Phi}^{-1}(\widehat{E})) \subset G(\widehat{\vphi})$, and the diagram
\[
\label{II.8.5.4.1}
\xymatrix{
@@ -1239,27 +1239,27 @@ furthermore, if $p:\widehat{E}\to X$ and $p':\widehat{E'}\to X'$ are the canonic
\ar[d]_{p'}
& \widehat{E}
\ar[d]^{p}
- \\G(\varphi)
- \ar[r]_{\Proj(\varphi)}
+ \\G(\vphi)
+ \ar[r]_{\Proj(\vphi)}
& X
}
\tag{8.5.4.1}
\]
commutes.
-If $\Proj(\varphi)$ is everywhere defined, then so too is $\widehat{\Phi}$, and we have that $\widehat{\Phi}^{-1}(\widehat{E}) = \widehat{E'}$
+If $\Proj(\vphi)$ is everywhere defined, then so too is $\widehat{\Phi}$, and we have that $\widehat{\Phi}^{-1}(\widehat{E}) = \widehat{E'}$
\end{proposition}
\begin{proof}
-The first claim follows from the commutativity of Diagrams \sref{II.8.5.1.3} and \sref{II.8.5.3.2}, and the two following claims from the definition of the canonical retractions \sref{II.8.3.5} and the definition of $\widehat{\varphi}$.
-To see that $\widehat{\Phi}$ is everywhere defined whenever $\Proj(\varphi)$ is, we can restrict to the case where $Y=\Spec(A)$ and $Y'=\Spec(A')$ are affine, and where $\sh{S}=\widetilde{S}$ and $\sh{S}'=\widetilde{S'}$;
-the hypothesis is that, when $f$ runs over the set of homogeneous elements of $S_+$, the radical in $S'_+$ of the ideal generated in $S'_+$ by the $\varphi(f)$ is $S'_+$ itself;
-we thus immediately conclude that the radical in $(S'[\bb{z}])_+$ of the ideal generated by $\bb{z}$ and the $\varphi(f)$ is $(S'[\bb{z}])_+$ itself, whence our claim;
-this also shows that $\widehat{E'}$ is the union of the $\widehat{C'}_{\varphi(f)}$, and hence equal to $\widehat{\Phi}^{-1}(\widehat{E})$.
+The first claim follows from the commutativity of Diagrams \sref{II.8.5.1.3} and \sref{II.8.5.3.2}, and the two following claims from the definition of the canonical retractions \sref{II.8.3.5} and the definition of $\widehat{\vphi}$.
+To see that $\widehat{\Phi}$ is everywhere defined whenever $\Proj(\vphi)$ is, we can restrict to the case where $Y=\Spec(A)$ and $Y'=\Spec(A')$ are affine, and where $\sh{S}=\widetilde{S}$ and $\sh{S}'=\widetilde{S'}$;
+the hypothesis is that, when $f$ runs over the set of homogeneous elements of $S_+$, the radical in $S'_+$ of the ideal generated in $S'_+$ by the $\vphi(f)$ is $S'_+$ itself;
+we thus immediately conclude that the radical in $(S'[\bb{z}])_+$ of the ideal generated by $\bb{z}$ and the $\vphi(f)$ is $(S'[\bb{z}])_+$ itself, whence our claim;
+this also shows that $\widehat{E'}$ is the union of the $\widehat{C'}_{\vphi(f)}$, and hence equal to $\widehat{\Phi}^{-1}(\widehat{E})$.
\end{proof}
\begin{corollary}[8.5.5]
\label{II.8.5.5}
-Whenever $\Proj(\varphi)$ is everywhere defined, the inverse image under $\widehat{\Phi}$ of the underlying space of the part at infinity (resp. of the vertex prescheme) of $\widehat{C'}$ is the underlying space of the part at infinity (resp. of the vertex prescheme) of $\widehat{C}$.
+Whenever $\Proj(\vphi)$ is everywhere defined, the inverse image under $\widehat{\Phi}$ of the underlying space of the part at infinity (resp. of the vertex prescheme) of $\widehat{C'}$ is the underlying space of the part at infinity (resp. of the vertex prescheme) of $\widehat{C}$.
\end{corollary}
\begin{proof}
@@ -1548,11 +1548,11 @@ We then \sref{II.1.5.2} have that $C_{U'} = C_U\times_X X' = C_U\times_U U'$, an
\tag{8.7.4.1}
\]
- Now let $\varphi:\sh{S}''\to\sh{S}$ be a homomorphism of graded $\sh{O}_Y$-algebras such that, if we write $X''=\Proj(\sh{S}'')$, then $u=\Proj(\varphi):X\to X''$ is everywhere defined;
+ Now let $\vphi:\sh{S}''\to\sh{S}$ be a homomorphism of graded $\sh{O}_Y$-algebras such that, if we write $X''=\Proj(\sh{S}'')$, then $u=\Proj(\vphi):X\to X''$ is everywhere defined;
we also have
\oldpage[II]{176}
- a $Y$-morphism $v:C\to C''$ (with $C''=\Spec(\sh{S}'')$) such that $\sh{A}(v)=\varphi$, and, since $\varphi$ is a homomorphism of graded algebras, $\varphi$ induces a $v$-morphism of graded algebras $\psi:\sh{S}^{\prime\prime\natural}\to\sh{S}^\natural$ \sref{II.1.4.1}.
- Furthermore, it follows from \sref{II.8.2.10}[iv] and from the hypothesis on $\varphi$ that $\Proj(\psi)$ is everywhere defined.
+ a $Y$-morphism $v:C\to C''$ (with $C''=\Spec(\sh{S}'')$) such that $\sh{A}(v)=\vphi$, and, since $\vphi$ is a homomorphism of graded algebras, $\vphi$ induces a $v$-morphism of graded algebras $\psi:\sh{S}^{\prime\prime\natural}\to\sh{S}^\natural$ \sref{II.1.4.1}.
+ Furthermore, it follows from \sref{II.8.2.10}[iv] and from the hypothesis on $\vphi$ that $\Proj(\psi)$ is everywhere defined.
Finally, taking \sref{II.3.5.6.1} into account, there is a canonical $u$-morphism $\sh{S}_{X''}\to\sh{S}_X$, whence \sref{II.1.5.6} a morphism $w:C_{X''}\to C_X$.
With this in mind, the diagram
\[
@@ -1840,17 +1840,17 @@ It remains only to prove that $g$ is \emph{projective}.
Since $f$ is of finite type, by hypothesis, we can apply \sref{II.3.8.5} to the homomorphism
\oldpage[II]{180}
$\tau$ from \sref{II.8.8.1.2}:
-there is an integer $d>0$ and a quasi-coherent $\sh{O}_Y$-submodule $\sh{E}$ of finite type of $\sh{S}_d$ such that, if $\sh{S}'$ is the $\sh{O}_Y$-subalgebra of $\sh{S}$ generated by $\sh{E}$, and $\tau'=\tau\circ q^*(\varphi)$ (where $\varphi$ is the canonical injection $\sh{S}'\to\sh{S}$), then $r'=r_{\sh{L},\tau'}$ is an immersion
+there is an integer $d>0$ and a quasi-coherent $\sh{O}_Y$-submodule $\sh{E}$ of finite type of $\sh{S}_d$ such that, if $\sh{S}'$ is the $\sh{O}_Y$-subalgebra of $\sh{S}$ generated by $\sh{E}$, and $\tau'=\tau\circ q^*(\vphi)$ (where $\vphi$ is the canonical injection $\sh{S}'\to\sh{S}$), then $r'=r_{\sh{L},\tau'}$ is an immersion
\[
X\to P'=\Proj(\sh{S}').
\]
-Furthermore, since $\varphi$ is injective, $r'$ is also a \emph{dominant immersion} \sref{II.3.7.6};
+Furthermore, since $\vphi$ is injective, $r'$ is also a \emph{dominant immersion} \sref{II.3.7.6};
the same argument as for $r$ then shows that $r'$ is a \emph{surjective closed immersion};
-since $r'$ factors as $X\xrightarrow{r}\Proj(\sh{S})\xrightarrow{\Phi}\Proj(\sh{S}')$, where $\Phi=\Proj(\varphi)$, we thus conclude that $\Phi$ is also a \emph{surjective closed immersion}.
+since $r'$ factors as $X\xrightarrow{r}\Proj(\sh{S})\xrightarrow{\Phi}\Proj(\sh{S}')$, where $\Phi=\Proj(\vphi)$, we thus conclude that $\Phi$ is also a \emph{surjective closed immersion}.
But this implies that $\Phi$ is an \emph{isomorphism};
we can restrict to the case where $Y=\Spec(A)$ is affine, and $\sh{S}=\widetilde{S}$ and $\sh{S}'=\widetilde{S'}$, with $S$ a graded $A$-algebra and $S'$ a graded subalgebra of $S$.
For every homogeneous element $t\in S'$, we have that $S'_{(t)}$ is a subring of $S_{(t)}$;
-if we return to the definition of $\Proj(\varphi)$ \sref{II.2.8.1}, we see that it suffices to prove that, if $B'$ is a subring of a ring $B$, and if the morphism $\Spec(B)\to\Spec(B')$ corresponding to the canonical injection $B'\to B$ is a closed immersion, then this morphism is necessarily an \emph{isomorphism};
+if we return to the definition of $\Proj(\vphi)$ \sref{II.2.8.1}, we see that it suffices to prove that, if $B'$ is a subring of a ring $B$, and if the morphism $\Spec(B)\to\Spec(B')$ corresponding to the canonical injection $B'\to B$ is a closed immersion, then this morphism is necessarily an \emph{isomorphism};
but this follows from \sref[I]{I.4.2.3}.
Furthermore, $\Phi^*(\sh{O}_{P'}(n))=\sh{O}_P(n)$ (\sref{II.3.5.2}[ii] and \sref{II.3.5.4}), and so $r^{'*}(\sh{O}_{P'}(n))$ is isomorphic to $\sh{L}^{\otimes n}$ \sref{II.4.6.3}.
Let $\sh{S}''=\sh{S}^{'(d)}$, so that \sref{II.3.1.8}[i] $X$ is canonically identified with $P''=\Proj(\sh{S}'')$, and $\sh{L}''=\sh{L}^{\otimes d}$ with $\sh{O}_{P''}(1)$ \sref{II.3.2.9}[ii].
@@ -2079,10 +2079,10 @@ the canonical injection, then let $\sh{L} = j^*(\sh{J}) = \sh{J}\otimes_{\sh{O}_
Assume that the structure morphism $p:X\to Y$ is separated and quasi-compact, and that the following conditions are satisfied:
\begin{enumerate}
\item[\rm{(i)}] there exists a $Y$-morphism $\pi:V\to X$ of finite type such that $\pi\circ j=1_X$, and so $\pi_*(\sh{J}/\sh{J}^2)=\sh{L}$;
- \item[\rm{(ii)}] there exists a homomorphism of $\sh{O}_X$-modules $\varphi:\sh{L}\to\varprojlim\pi_*(\sh{J}/\sh{J}^{n+1})$ such that the composition
+ \item[\rm{(ii)}] there exists a homomorphism of $\sh{O}_X$-modules $\vphi:\sh{L}\to\varprojlim\pi_*(\sh{J}/\sh{J}^{n+1})$ such that the composition
\[
\sh{L}
- \xrightarrow{\varphi}
+ \xrightarrow{\vphi}
\varprojlim \pi_*(\sh{J}/\sh{J}^{n+1})
\xrightarrow{\alpha}
\pi_*(\sh{J}/\sh{J}^2) = \sh{L}
@@ -2124,8 +2124,8 @@ Then the following all hold true.
\[
\sh{L} = j^*(\sh{J}) = \pi_*(\sh{J}/\sh{J}^2)
\]
- is an invertible $\sh{O}_X$-module), and that there exists a homomorphism $\varphi:\sh{L}\to\varprojlim\pi_*(\sh{J}/\sh{J}^{n+1})$ such that the composition $\sh{L} \xrightarrow{\varphi} \varprojlim\pi_*(\sh{J}/\sh{J}^{n+1}) \xrightarrow{\alpha} \pi_*(\sh{J}/\sh{J}^2)$ (where $\alpha$ is the canonical homomorphism) is the identity.
- If we write $\sh{S}=\bigoplus_{n\geq0}\sh{L}^{\otimes n}$, then $\varphi$ canonically induces an isomorphism of $\sh{O}_X$-algebras from the completion $\widehat{\sh{S}}$ of $\sh{S}$ relative to its canonical filtration (the completion being isomorphic to the product $\prod_{n\geq0}\sh{L}^{\otimes n}$) to $\varprojlim\pi_*(\sh{O}_V/\sh{J}^{n+1})$.
+ is an invertible $\sh{O}_X$-module), and that there exists a homomorphism $\vphi:\sh{L}\to\varprojlim\pi_*(\sh{J}/\sh{J}^{n+1})$ such that the composition $\sh{L} \xrightarrow{\vphi} \varprojlim\pi_*(\sh{J}/\sh{J}^{n+1}) \xrightarrow{\alpha} \pi_*(\sh{J}/\sh{J}^2)$ (where $\alpha$ is the canonical homomorphism) is the identity.
+ If we write $\sh{S}=\bigoplus_{n\geq0}\sh{L}^{\otimes n}$, then $\vphi$ canonically induces an isomorphism of $\sh{O}_X$-algebras from the completion $\widehat{\sh{S}}$ of $\sh{S}$ relative to its canonical filtration (the completion being isomorphic to the product $\prod_{n\geq0}\sh{L}^{\otimes n}$) to $\varprojlim\pi_*(\sh{O}_V/\sh{J}^{n+1})$.
\end{enumerate}
\end{lemma}
@@ -2148,7 +2148,7 @@ In other words, if $\pi'$ is the restriction of $\pi$ to $U'$, then the $(\sh{O}
Since $U'$ and $U_0$ are affine, and since the $U_0$ cover $X$, we thus conclude \sref[I]{I.1.6.3} that $\pi_*(\sh{O}_V/\sh{J}^{n+1})$ is quasi-coherent, and the proof is identical for $\pi_*(\sh{J}/\sh{J}^{n+1})$.
Finally, to prove (iii), note that $\sh{S}$ is exactly $\bb{S}_{\sh{O}_X}(\sh{L})$;
-so $\varphi$ canonically induces a homomorphism of $\sh{O}_X$-algebras $\psi:\sh{S}\to\varprojlim\pi_*(\sh{O}_V/\sh{J}^{n+1})$ \sref{II.1.7.4};
+so $\vphi$ canonically induces a homomorphism of $\sh{O}_X$-algebras $\psi:\sh{S}\to\varprojlim\pi_*(\sh{O}_V/\sh{J}^{n+1})$ \sref{II.1.7.4};
furthermore, this homomorphism sends $\sh{L}^{\otimes n}$ to $\varprojlim_m\pi_*(\sh{J}^n/\sh{J}^{n+1})$, and is thus continuous for the topologies considered, and indeed then extends to a homomorphism $\widehat{\psi}:\widehat{\sh{S}}\to\varprojlim\pi_*(\sh{O}_V/\sh{J}^{n+1})$.
To see that this is indeed an isomorphism, we can, as in the proof of (i), restrict to the case where $X=\Spec(A)$ and $V=\Spec(B)$ are affine, with $\sh{J}=\widetilde{\mathfrak{J}}$, where $\mathfrak{J}$ is an ideal of $B$;
there is an injection $A\to B$ corresponding to $\pi$ that identifies $A$ with a subring of $B$ that is \emph{complementary} to $B$, and $\sh{L}$ (resp. $\pi_*(\sh{O}_V/\sh{J}^{n+1})$) is the quasi-coherent $\sh{O}_X$-module associated to the $A$-module $L=\mathfrak{J}/\mathfrak{J}^2$ (resp. $B/\mathfrak{J}^{n+1}$).
@@ -2158,7 +2158,7 @@ From the fact that $B=A\oplus Bt$, we deduce that, for all $n>0$,
B = A \oplus At \oplus At^2 \oplus \ldots \oplus At^n \oplus Bt^{n+1}
\]
and so there exists a canonical $A$-isomorphism from the ring of formal series $A[[T]]$ to $C=\varprojlim B/\mathfrak{J}^{n+1}$ that sends $T$ to $t$.
-We also have that $L=A\bar{t}$, where $\bar{t}$ is the class of $t$ modulo $Bt^2$, and the homomorphism $\varphi$ sends, by hypothesis, $\bar{t}$ to an element $t'\in C$ that is congruent to $t$ modulo $Ct^2$.
+We also have that $L=A\bar{t}$, where $\bar{t}$ is the class of $t$ modulo $Bt^2$, and the homomorphism $\vphi$ sends, by hypothesis, $\bar{t}$ to an element $t'\in C$ that is congruent to $t$ modulo $Ct^2$.
We thus deduce, by induction on $n$, that
\[
A \oplus At' \oplus \ldots \oplus At^{'n} \oplus Ct^{n+1}
@@ -2380,8 +2380,8 @@ is dense in $V$, and so $h_0$ is a \emph{homeomorphism} from $U$ to $V$.
To prove the corollary, it will suffice to show that $\lambda:\sh{O}_V\to(h_0)_*(\sh{O}_U)$ is an isomorphism: we can then apply \sref{II.8.11.1}, which proves that the map \sref{II.8.11.1.1} is bijective (the fibres $h_0^{-1}(x)$ each consisting of a single point);
thus $h$ will be an isomorphism.
The question clearly being local on $V$, we can suppose that $V=\Spec(A)$ is affine, of an integral and integrally closed ring \sref{II.8.8.6.1};
-$h$ then corresponds \sref[I]{I.2.2.4} to a homomorphism $\varphi:A\to\Gamma(U,\sh{O}_U)$, and everything reduces to showing that $\varphi$ is an isomorphism.
-But, if $K$ is the field of fractions of $A$, then $\Gamma(U,\sh{O}_U)$ has, by hypothesis, $K$ as its field of fractions, and $A$ is a subring of $\Gamma(U,\sh{O}_U)$, with $\varphi$ being the canonical injection \sref[I]{I.8.2.7}.
+$h$ then corresponds \sref[I]{I.2.2.4} to a homomorphism $\vphi:A\to\Gamma(U,\sh{O}_U)$, and everything reduces to showing that $\vphi$ is an isomorphism.
+But, if $K$ is the field of fractions of $A$, then $\Gamma(U,\sh{O}_U)$ has, by hypothesis, $K$ as its field of fractions, and $A$ is a subring of $\Gamma(U,\sh{O}_U)$, with $\vphi$ being the canonical injection \sref[I]{I.8.2.7}.
Since the morphism $h$ satisfies the hypotheses of \sref{II.7.3.11}, $\Gamma(U,\sh{O}_U)$ is a subring of the integral closure of $A$ in $K$, and is thus identical to $A$ by hypothesis.
\end{proof}
@@ -2562,16 +2562,16 @@ It follows from \sref{II.1.5.6} that, for every quasi-coherent graded $\sh{S}$-m
\tag{8.12.4.1}
\]
of $\sh{O}_{C'}$-modules;
-on the other hand, \sref{II.3.5.6} implies the existence of a canonical $\Proj(\varphi)$-morphism
+on the other hand, \sref{II.3.5.6} implies the existence of a canonical $\Proj(\vphi)$-morphism
\[
\label{II.8.12.4.2}
- \shProj_0\sh{M} \to (\shProj_0(q^*(\sh{M}))\otimes_{q^*(\sh{S})}\sh{S}')|G(\varphi)
+ \shProj_0\sh{M} \to (\shProj_0(q^*(\sh{M}))\otimes_{q^*(\sh{S})}\sh{S}')|G(\vphi)
\tag{8.12.4.2}
\]
and also of a canonical $\widehat{\Phi}$-morphism
\[
\label{II.8.12.4.3}
- \shProj_0\widehat{\sh{M}} \to (\shProj_0(q^*(\widehat{\sh{M}}))\otimes_{q^*(\widehat{\sh{S}})}\widehat{\sh{S}}')|G(\widehat{\varphi}).
+ \shProj_0\widehat{\sh{M}} \to (\shProj_0(q^*(\widehat{\sh{M}}))\otimes_{q^*(\widehat{\sh{S}})}\widehat{\sh{S}}')|G(\widehat{\vphi}).
\tag{8.12.4.3}
\]
\end{env}
@@ -2579,7 +2579,7 @@ and also of a canonical $\widehat{\Phi}$-morphism
\begin{env}[8.12.5]
\label{II.8.12.5}
Consider now the setting of \sref{II.8.6.1}, with the same notation;
-we thus take $Y'=X$, the morphism $q:X\to Y$ being the structure morphism, and $\varphi$ the canonical $q$-morphism \sref{II.8.6.1.2}.
+we thus take $Y'=X$, the morphism $q:X\to Y$ being the structure morphism, and $\vphi$ the canonical $q$-morphism \sref{II.8.6.1.2}.
We then have a canonical isomorphism
\[
\label{II.8.12.5.1}
diff --git a/ega4/ega4-16.tex b/ega4/ega4-16.tex
index e3391cf..9a313b9 100644
--- a/ega4/ega4-16.tex
+++ b/ega4/ega4-16.tex
@@ -66,8 +66,8 @@ which coincide in degrees $0$ and $1$ with the identities.
\begin{env}[16.1.4]
\label{IV.16.1.4}
The example \sref{IV.16.1.3}[(ii)] shows that in general the $\sh{O}_{Y^{(n)}}$ are \emph{not canonically endowed with a structure of an $\sh{O}_Y$-module}, or \emph{a fortiori} with a structure of an $\sh{O}_Y$-algebra.
-The data of such structure is equivalent to the data of a homomorphism of sheaves of rings $\lambda_n:\sh{O}_Y \to \sh{O}_{Y^{(n)}}$, right inverse to the augmentation morphism $\varphi_{0n}$;
-it is also equivalent to the data of a morphism of ringed spaces $(1_Y, \lambda_n): Y^{(n)} \to Y$ left inverse to the canonical morphism $(1_Y, \varphi_{0n}): Y \to Y^{(n)}$.
+The data of such structure is equivalent to the data of a homomorphism of sheaves of rings $\lambda_n:\sh{O}_Y \to \sh{O}_{Y^{(n)}}$, right inverse to the augmentation morphism $\vphi_{0n}$;
+it is also equivalent to the data of a morphism of ringed spaces $(1_Y, \lambda_n): Y^{(n)} \to Y$ left inverse to the canonical morphism $(1_Y, \vphi_{0n}): Y \to Y^{(n)}$.
\end{env}
\begin{proposition}[16.1.5]
@@ -78,7 +78,7 @@ Then:
\item[{\rm(i)}] $\shGr_\bullet(f)$ is a quasi-coherent graded $\sh{O}_Y$-algebra.
\oldpage[IV-4]{7}
\item[{\rm(ii)}] The $Y^{(n)}$ are preschemes, canonically isomorphic to subpreschemes of $X$.
- \item[{\rm(iii)}] Every homomorphism of sheaves of rings $\lambda_n: \sh{O}_Y \to \sh{O}_{Y^{(n)}}$, right inverse to the augmentation homomorphism $\varphi_{0n}$, makes the $\sh{O}_{Y^{(n)}}$ and $\sh{O}_{Y^{(k)}}$ for $k\leq n$ quasi-coherent $\sh{O}_Y$-algebras;
+ \item[{\rm(iii)}] Every homomorphism of sheaves of rings $\lambda_n: \sh{O}_Y \to \sh{O}_{Y^{(n)}}$, right inverse to the augmentation homomorphism $\vphi_{0n}$, makes the $\sh{O}_{Y^{(n)}}$ and $\sh{O}_{Y^{(k)}}$ for $k\leq n$ quasi-coherent $\sh{O}_Y$-algebras;
the $\sh{O}_Y$-module structures induced from the above structures on the $\shGr_k(f)$ for $k \leq n$ coincide with the ones defined in \sref{IV.16.1.2}.
\end{enumerate}
\end{proposition}
@@ -86,7 +86,7 @@ Then:
\begin{proof}
(i) Since the question is local on $X$ and $Y$, we can reduce to the case where $Y$ is a closed subpreschemes of $X$ defined by an quasi-coherent ideal $\sh{I}$ of $\sh{O}_X$;
since $\sh{O}_Y$ is the restriction to $Y$ of $\sh{O}_X/\sh{I}$ the assertion (i) is evident, and $Y^{(n)}$ is the closed subprescheme of $X$ defined by the quasi-coherent ideal $\sh{I}^{n+1}$ of $\sh{O}_X$.
-Finally, to prove (iii) we notice that the data of $\lambda_n$ makes the ideal $\sh{I}/\sh{I}^n$ of the augmentation $\varphi_{0n}$ and their quotients $\sh{I}/\sh{I}^{k+1} (1\leq k \leq n)$ $\sh{O}_Y$-modules, and it suffices to prove by induction on $k$ that the $\sh{I}/\sh{I}^{k+1}$ are quasi-coherent $\sh{O}_Y$-modules and the structure of quotient $\sh{O}_Y$-module induced on $\sh{I}^k/\sh{I}^{k+1}$ is the same as defined on \sref{IV.16.1.2}.
+Finally, to prove (iii) we notice that the data of $\lambda_n$ makes the ideal $\sh{I}/\sh{I}^n$ of the augmentation $\vphi_{0n}$ and their quotients $\sh{I}/\sh{I}^{k+1} (1\leq k \leq n)$ $\sh{O}_Y$-modules, and it suffices to prove by induction on $k$ that the $\sh{I}/\sh{I}^{k+1}$ are quasi-coherent $\sh{O}_Y$-modules and the structure of quotient $\sh{O}_Y$-module induced on $\sh{I}^k/\sh{I}^{k+1}$ is the same as defined on \sref{IV.16.1.2}.
The second assertion is immediate, $\sh{I}^k/\sh{I}^{k+1}$ being killed by $\sh{I}/\sh{I}^{n+1}$;
the first result, by induction on $k$, is trivial for $k=1$ and for $\sh{I}/\sh{I}^{k+1}$ being an extension of $\sh{I}/\sh{I}^{k}$ by $\sh{I}^k/\sh{I}^{k+1}$ \sref{III.1.4.17}.
\end{proof}
@@ -103,8 +103,8 @@ Indeed, with the notation from the proof of \sref{IV.16.1.5}, $\sh{I}$ is an ide
\begin{corollary}[16.1.7]
\label{IV.16.1.7}
Under the general hypotheses of \sref{IV.16.1.5}, let $g:X \to Y$ be a morphism of preschemes, left inverse to $f$.
-Therefore, for every $n$, the composite morphism $(1, \lambda_n): Y^{(n)}\xrightarrow{h_n} X \xrightarrow{g} Y$ defines a homomorphism of sheaves of rings $\lambda_n: \sh{O}_Y \to \sh{O}_{Y^{(n)}}$ right inverse to the augmentation $\varphi_{0n}$, making $\sh{O}_{Y^{(n)}}$ a quasi-coherent $\sh{O}_Y$-algebra;
-via these homomorphisms, the transition homomorphism $\varphi_{nm}:\sh{O}_{Y^{(m)}} \to \sh{O}_{Y^{(n)}}$ ($n\leq m$) are homomorphisms of $\sh{O}_Y$-algebras.
+Therefore, for every $n$, the composite morphism $(1, \lambda_n): Y^{(n)}\xrightarrow{h_n} X \xrightarrow{g} Y$ defines a homomorphism of sheaves of rings $\lambda_n: \sh{O}_Y \to \sh{O}_{Y^{(n)}}$ right inverse to the augmentation $\vphi_{0n}$, making $\sh{O}_{Y^{(n)}}$ a quasi-coherent $\sh{O}_Y$-algebra;
+via these homomorphisms, the transition homomorphism $\vphi_{nm}:\sh{O}_{Y^{(m)}} \to \sh{O}_{Y^{(n)}}$ ($n\leq m$) are homomorphisms of $\sh{O}_Y$-algebras.
Also, if $g$ is locally of finite type, then the $\sh{O}_{Y^{(n)}}$ are quasi-coherent $\sh{O}_Y$-modules of finite type.
\end{corollary}
@@ -132,12 +132,12 @@ Let $X$ be a prescheme, $j: Y \to X$ an immersion locally of finite presentation
\item[(a)] There exists an open neighborhood $U$ of y in $Y$ such that $j|U$ is a homeomorphism of $U$ onto an open set of $X$.
\item[(b)] There is an integer $n>0$ such that the canonical homomorphism
\[
- (\varphi_{n-1,n})_y: \sh{O}_{Y^{(n)},y} \to \sh{O}_{Y^{(n-1)},y}
+ (\vphi_{n-1,n})_y: \sh{O}_{Y^{(n)},y} \to \sh{O}_{Y^{(n-1)},y}
\]
is bijective.
\item[(c)] There is an integer $n>0$ such that $(\shGr_n(j))_y = 0$.
- In addition, if the integer $n$ satisfies \emph{(b)} or \emph{(c)}, then there is a neighborhood $V$ of $y$ in $Y$ such that $\shGr_m(j)|V = 0$ for $m \geq n$ and that $\varphi_{nm}|V: \sh{O}_{Y^{(m)}}|V \to \sh{O}_{Y^{(n)}}|V$ is bijective for $m \geq n$.
+ In addition, if the integer $n$ satisfies \emph{(b)} or \emph{(c)}, then there is a neighborhood $V$ of $y$ in $Y$ such that $\shGr_m(j)|V = 0$ for $m \geq n$ and that $\vphi_{nm}|V: \sh{O}_{Y^{(m)}}|V \to \sh{O}_{Y^{(n)}}|V$ is bijective for $m \geq n$.
\end{enumerate}
\end{proposition}
@@ -166,7 +166,7 @@ The condition is clearly necessary, and the previous reasoning applied to $n=1$
\label{IV.16.1.11}
\medskip\noindent
\begin{enumerate}
- \item[(i)] Under the conditions of the definition \sref{IV.16.1.1}, the projective limit of the projective system $(\sh{O}_{Y^{(n)}}, \varphi_{nm})$ of sheaves of rings over $Y$ is called the \emph{normal invariant of infinite order} of $f$, and sometimes denoted by $\sh{O}_{Y^{(\infty)}}$.
+ \item[(i)] Under the conditions of the definition \sref{IV.16.1.1}, the projective limit of the projective system $(\sh{O}_{Y^{(n)}}, \vphi_{nm})$ of sheaves of rings over $Y$ is called the \emph{normal invariant of infinite order} of $f$, and sometimes denoted by $\sh{O}_{Y^{(\infty)}}$.
When $X$ is a locally noetherian prescheme, $j:Y \to X$ a closed immersion, $Y$ then is a closed subprescheme of $X$ defined by a coherent ideal $\sh{I}$ and $\sh{O}_{Y^{(\infty)}}$ is exactly the \emph{formal completion} of $\sh{O}_X$ along $Y$ \sref[I]{I.10.8.4}, and $Y^{(\infty)} = (Y, \sh{O}_{Y^{(\infty)}})$ is the formal prescheme that is the \emph{completion} of $X$ along $Y$ \sref[I]{I.10.8.5}.
In all cases, we could say that $Y^{(\infty)}$ is the \emph{formal neighborhood} of $Y$ in $X$ (via the morphism $f$).
In the particular case we have just considered, it is the formal prescheme that is the inductive limit of the infinitesimal neighborhoods of order $n$.
@@ -1832,7 +1832,7 @@ This is an immediate consequence of \sref{IV.16.7.5.3} and of the particular cas
\label{IV.16.7.7}
The canonical homomorphisms of sheaves of rings
\[
- \varphi_{nm}:\sh{P}_{X/S}^m \to \sh{P}_{X/S}^n
+ \vphi_{nm}:\sh{P}_{X/S}^m \to \sh{P}_{X/S}^n
\]
for $n \leq m$ \sref{IV.16.1.2} define, because of \sref{IV.16.7.2.1}, canonical homomorphisms
\[
@@ -2159,7 +2159,7 @@ It remains to prove the lemma \sref{IV.16.8.9.3}.
Considering \sref{IV.16.7.6}, which proves the uniqueness of $\delta$, we are brought back to the case where $S = \Spec(A)$ and $X = \Spec(B)$ are affines;
letting $\mathfrak{I} = \mathfrak{I}_{B/A}$, it suffices to define a canonical homomorphism of $B$-modules
\[
- \varphi: (B \otimes_A B)/\mathfrak{I}^{n+n'+1} \to ((B \otimes_A B)/\mathfrak{I}^{n' + 1}) \otimes_B ((B \otimes_A B)/\mathfrak{I}^{n + 1})
+ \vphi: (B \otimes_A B)/\mathfrak{I}^{n+n'+1} \to ((B \otimes_A B)/\mathfrak{I}^{n' + 1}) \otimes_B ((B \otimes_A B)/\mathfrak{I}^{n + 1})
\]
the $B$-module structure of the two members coming from the first $B$ factor;
recall that on tensor product of the second member, $(B \otimes_A B)/\mathfrak{I}^{n' + 1}$ must be considered
@@ -2167,18 +2167,18 @@ recall that on tensor product of the second member, $(B \otimes_A B)/\mathfrak{I
as a right $B$-module by its second $B$ factor , and $(B \otimes_A B)/\mathfrak{I}^{n + 1}$ as a left $B$-module by its first $B$ factor \sref{IV.16.7.2}.
It is the same to define a homomorphism of $B$-modules
\[
- \varphi_0: B \otimes_A B \to ((B \otimes_A B)/\mathfrak{I}^{n' + 1}) \otimes_B ((B \otimes_A B)/\mathfrak{I}^{n + 1})
+ \vphi_0: B \otimes_A B \to ((B \otimes_A B)/\mathfrak{I}^{n' + 1}) \otimes_B ((B \otimes_A B)/\mathfrak{I}^{n + 1})
\]
and prove it is zero on $\mathfrak{I}^{n+n'+1}$.
Now, we immediately define a homomorphism by the condition that
\[
- \varphi_0(b \otimes b') = \pi_{n'}(b \otimes 1) \otimes \pi_n(1 \otimes b') \quad \text{for $b$, $b'$ in $B$}
+ \vphi_0(b \otimes b') = \pi_{n'}(b \otimes 1) \otimes \pi_n(1 \otimes b') \quad \text{for $b$, $b'$ in $B$}
\]
under the notations of \sref{IV.16.3.7}.
Also, it is immediate that $\phi_0$ is a homomorphism of \emph{rings}.
Now, we can write
\[
- \varphi_0(b \otimes 1 - 1 \otimes b) = \pi_{n'}(b \otimes 1 - 1 \otimes b) \otimes \pi_n(1 \otimes 1) + \pi_{n'}(1 \otimes b) \otimes \pi_n(1 \otimes 1) - \pi_{n'}(1 \otimes 1) \otimes \pi_n(1 \otimes b)
+ \vphi_0(b \otimes 1 - 1 \otimes b) = \pi_{n'}(b \otimes 1 - 1 \otimes b) \otimes \pi_n(1 \otimes 1) + \pi_{n'}(1 \otimes b) \otimes \pi_n(1 \otimes 1) - \pi_{n'}(1 \otimes 1) \otimes \pi_n(1 \otimes b)
\]
and we have
\[
@@ -2187,7 +2187,7 @@ and we have
from which, finally
\[
\label{IV.16.8.9.4}
- \varphi_0(b \otimes 1 - 1 \otimes b) = \pi_{n'}(b \otimes 1 - 1 \otimes b) \otimes \pi_n(1 \otimes 1) + \pi_{n'}(1 \otimes 1) \otimes \pi_n(b \otimes 1 - 1 \otimes b).
+ \vphi_0(b \otimes 1 - 1 \otimes b) = \pi_{n'}(b \otimes 1 - 1 \otimes b) \otimes \pi_n(1 \otimes 1) + \pi_{n'}(1 \otimes 1) \otimes \pi_n(b \otimes 1 - 1 \otimes b).
\tag{16.8.9.4}
\]
A product of $n + n' + 1$ of terms of the form \sref{IV.16.8.9.4} is therefore necessarily zero, because the same is true for the product of $n+1$ terms of the form $\pi_n(b \otimes 1 - 1 \otimes b)$ and of $n' + 1$ terms of the form $\pi_{n'}(b \otimes 1 - 1 \otimes b)$.
@@ -2574,8 +2574,8 @@ since by hypothesis $B = (B \otimes_A B)/\mathfrak{I}$ is a formally smooth $A$-
\begin{proposition}[16.10.3]
\label{IV.16.10.3}
-For a morphism $f:X \to S$ to be differentially smooth, it is necessary and sufficient that for every $x \in X$, there is an open affine neighborhood of $x$, of ring $A$, such that $\Gamma(U, \sh{P}_{X/S}^\infty)$ is an augmented topological $A$-algebra isomorphic to the completion $\hat{B}$, where $B = \bb{S}_A(V)$, $V$ being a projective $A$-module and $B$ being endowed with the $B^+$-preadic topology (where $B^+$ is the augmentation ideal).
-If $\Omega_{X/S}^1$ is locally free of finite rank, we can replace $\hat{B}$ with the ring of formal series $A[[T_1, \dots, T_n]]$.
+For a morphism $f:X \to S$ to be differentially smooth, it is necessary and sufficient that for every $x \in X$, there is an open affine neighborhood of $x$, of ring $A$, such that $\Gamma(U, \sh{P}_{X/S}^\infty)$ is an augmented topological $A$-algebra isomorphic to the completion $\widehat{B}$, where $B = \bb{S}_A(V)$, $V$ being a projective $A$-module and $B$ being endowed with the $B^+$-preadic topology (where $B^+$ is the augmentation ideal).
+If $\Omega_{X/S}^1$ is locally free of finite rank, we can replace $\widehat{B}$ with the ring of formal series $A[[T_1, \dots, T_n]]$.
\end{proposition}
\begin{proof}
diff --git a/ega4/ega4-17.tex b/ega4/ega4-17.tex
index 56a862a..ced33a7 100644
--- a/ega4/ega4-17.tex
+++ b/ega4/ega4-17.tex
@@ -30,8 +30,8 @@ It is clear that for $f$ to be formally \'etale, it is necessary and sufficient
\label{IV.17.1.2}
\medskip\noindent
\begin{enumerate}
- \item[(i)] Suppose that $Y=\Spec(A)$ and $X=\Spec(B)$ are affine, so that $f$ comes from a homomorphism of rings $\varphi:A\to B$.
- According to \sref[0]{0.19.3.1} and \sref[0]{0.19.10.1}, saying that $f$ is formally smooth (resp. formally unramified, resp. formally \'etale) means that, via $\varphi$, $B$ is a \emph{formally smooth} (resp. \emph{formally unramified}, resp. \emph{formally \'etale}) $A$-algebra, for the \emph{discrete} topologies on $A$ and $B$.
+ \item[(i)] Suppose that $Y=\Spec(A)$ and $X=\Spec(B)$ are affine, so that $f$ comes from a homomorphism of rings $\vphi:A\to B$.
+ According to \sref[0]{0.19.3.1} and \sref[0]{0.19.10.1}, saying that $f$ is formally smooth (resp. formally unramified, resp. formally \'etale) means that, via $\vphi$, $B$ is a \emph{formally smooth} (resp. \emph{formally unramified}, resp. \emph{formally \'etale}) $A$-algebra, for the \emph{discrete} topologies on $A$ and $B$.
\item[(ii)] To verify that $f$ is formally smooth (resp. formally unramified, resp. formally \'etale), we can, in Definition~\sref{IV.17.1.1}, restrict to the case where $\sh{J}^2=0$.
To see this, if $f$ satisfies the corresponding condition of Definition~\sref{IV.17.1.1} in the particular case $\sh{J}^2=0$, and if we have $\sh{J}^n=0$, then we consider the closed subscheme $Y_j'$ of $Y'$ defined by the sheaf of ideals $\sh{J}^{j+1}$ for $0\leq j\leq n-1$, so that $Y_j'$ is a closed subscheme of $Y_{j+1}'$ defined by a square-zero sheaf of ideals;
the hypotheses imply that each of the maps
diff --git a/intro.tex b/intro.tex
index 64c9893..169b7dc 100644
--- a/intro.tex
+++ b/intro.tex
@@ -76,7 +76,7 @@ It is suitable, however, to say some words here about the works which have most
We absolutely must mention the fundamental work (FAC) of J.-P.~Serre first, which has served as an introduction to algebraic geometry for more that one young student (the author of this treatise being one), deterred by the dryness of the classic \emph{Foundations} of A.~Weil~\cite{I-18}.
It is there that it is shown, for the first time, that the ``Zariski topology'' of an ``abstract'' algebraic variety is perfectly suited to applying certain techniques from algebraic topology, and notably to be able to define a cohomology theory.
Further, the definition of an algebraic variety given therein is that which translates most naturally to the idea that we develop here\footnote{Just as J.-P.~Serre informed us, it is right to note that the idea of defining the structure of a manifold by the data of a sheaf of rings is due to H.~Cartan, who took this idea as the starting point of his theory of analytic spaces.
-Of course, just as in algebraic geometry, it would be important in ``analytic geometry'' to give the allow the use of nilpotent elements in local rings of analytic spaces.
+Of course, just as in algebraic geometry, it would be important in ``analytic geometry'' to allow the use of nilpotent elements in local rings of analytic spaces.
This extension of the definition of H.~Cartan and J.-P.~Serre has recently been broached by H.~Grauert~\cite{I-5}, and there is room to hope that a systematic report of analytic geometry in this setting will soon see the light of day.
It is also evident that the ideas and techniques developed in this treatise retain a sense of analytic geometry, even though one must expect more considerable technical difficulties in this latter theory.
We can foresee that algebraic geometry, by the simplicity of its methods, will be able to serve as a sort of formal model for future developments in the theory of analytic spaces.}.
diff --git a/preamble-base.tex b/preamble-base.tex
index ce008e1..166c97b 100644
--- a/preamble-base.tex
+++ b/preamble-base.tex
@@ -47,3 +47,6 @@
\let\mapstoo\mapsto
\renewcommand{\mapsto}{\mathchoice{\longmapsto}{\mapstoo}{\mapstoo}{\mapstoo}}
\def\isoto{\simeq} % isomorphism
+
+\renewcommand{\hat}[1]{\widehat{#1}}
+\renewcommand{\tilde}[1]{\widetilde{#1}}