Finished 0.11.4

CONTRIBUTORS

@@ -7,4 +7,4 @@ Tim Holzschuh
 Tim Hosgood
 Ryan Keleti
 Thiago Solovera \& Nery
-
+Wayne Yang

README.md

@@ -64,7 +64,7 @@ Here is the current status of the translation, along with who is currently worki
 + [x] 8. Representable functors _(@ryankeleti)_
 + [x] 9. Constructible sets _(@ryankeleti)_
 + [x] 10. Supplement on flat modules _(@thosgood)_
-+ [ ] 11. Supplement on homological algebra _(@ryankeleti)_
++ [ ] 11. Supplement on homological algebra _(@ryankeleti, @wayneyang108)_
 + [ ] 12. Supplement on sheaf cohomology (~25 pages)
 + [ ] 13. Projective limits in homological algebra (~10 pages)
 

ega0/ega0-3.tex

@@ -522,6 +522,7 @@ \subsection{Inverse images of presheaves}
 morphism $u_V:\sh{G}(V)\to\psi_*(\sh{F})(V)=\sh{F}(\psi^{-1}(V))$; it is
 immediate that these morphisms render commutative the diagrams
 \[
+\label{0.3.5.1.1}
   \xymatrix{
     \sh{G}(V)\ar[r]^{u_{U,V}}\ar[d] &
     \sh{F}(U)\ar[d]\\
@@ -603,6 +604,7 @@ \subsection{Inverse images of presheaves}
 each morphism $u:\sh{G}\to\psi_*(\sh{F})$ of presheaves factorizes in a unique
 way as
 \[
+\label{0.3.5.3.3}
   u:\sh{G}\xrightarrow{\rho_\sh{G}}\psi_*(\psi^*(\sh{G}))
   \xrightarrow{\psi_*(u^\sharp)}\psi_*(\sh{F}).
   \tag{3.5.3.3}
@@ -667,6 +669,7 @@ \subsection{Inverse images of presheaves}
 the morphism $(i_\sh{F})^\sharp$; the formula (3.5.4.3) gives in particular the
 factorization
 \[
+\label{0.3.5.4.4}
   u^\sharp:\psi^*(\sh{G})\xrightarrow{\psi^*(u)}\psi^*(\psi_*(\sh{F}))
   \xrightarrow{\sigma_\sh{F}}\sh{F}
   \tag{3.5.4.4}

ega0/ega0-4.tex

@@ -291,6 +291,7 @@ \subsection{Direct image of an $\mathcal{A}$-module}
 restriction operations, they give a canonical functorial homomorphism of
 $\sh{B}$-modules
 \[
+\label{0.4.2.2.1}
   \Psi_*(\sh{M})\otimes_\sh{B}\Psi_*(\sh{N})
   \to\Psi_*(\sh{M}\otimes_\sh{A}\sh{N})
   \tag{4.2.2.1}
@@ -441,6 +442,7 @@ \subsection{Inverse image of an $\mathcal{A}$-module}
 above homomorphism is in fact a \emph{isomorphism}. By tensoring with $\sh{A}$,
 we obtain a \emph{canonical functorial isomorphism}
 \[
+\label{0.4.3.3.1}
   \Psi^*(\sh{G}_1)\otimes_\sh{A}\Psi^*(\sh{G}_2)
   \isoto\Psi^*(\sh{G}_1\otimes_\sh{B}\sh{G}_2).
   \tag{4.3.3.1}
@@ -561,12 +563,14 @@ \subsection{Relation between direct and inverse images}
 If we take for $v$ the identity homomorphism of $\Psi^*(\sh{B})$, $v_\theta^\flat$ is a
 homomorphism
 \[
+\label{0.4.4.3.2}
   \rho_\sh{G}:\sh{G}\to\Psi_*(\Psi^*(\sh{G}));
   \tag{4.4.3.2}
 \]
 if we take for $u$ the identity homomorphism of $\Psi_*(\sh{F})$, $u_\theta^\sharp$ is a
 homomorphism
 \[
+\label{0.4.4.3.3}
   \sigma_\sh{F}:\Psi^*(\Psi_*(\sh{F}))\to\sh{F};
   \tag{4.4.3.3}
 \]

ega0/ega0-8.tex

@@ -194,7 +194,7 @@ \subsection{Representable functors}
 Let $\C$ be a category, $\{a\}$ a singleton set.
 Consider the contravariant functor $F:\C\to\Set$ which sends every object $X$ of $\C$ to the set $\{a\}$, and every morphism $X\to X'$ in $\C$ to the unique map $\{a\}\to\{a\}$.
 To say that this functor is \emph{representable} means that there exists an object $e\in\C$ such that for every $Y\in\C$, $\Hom(Y,e)=h_e(Y)$ is a \emph{singleton set}; we say that $e$ is an \emph{final object} of $\C$, and it is clear that two final objects of $\C$ are isomorphic (which allows us to define, in general with the axiom of choice, \emph{one} final object of $\C$ which we then denote $e_\C$).
-For example, in the category $\Set$, the final objects are the singleton sets; in the category of \emph{augmented algebras} over a field $K$ (where the morphisms are the algebra homomorphisms compatible with the augmentation), $K$ is a final object; in the category of \emph{$S$-preschemes} \sref[I]{1.2.5.1}, $S$ is a final object.
+For example, in the category $\Set$, the final objects are the singleton sets; in the category of \emph{augmented algebras} over a field $K$ (where the morphisms are the algebra homomorphisms compatible with the augmentation), $K$ is a final object; in the category of \emph{$S$-preschemes} \sref[I]{I.2.5.1}, $S$ is a final object.
 \end{env}
 
 \begin{env}[8.1.11]

ega0/ega0-9.tex

@@ -53,7 +53,7 @@ \subsection{Constructible sets}
 
 \begin{env}[9.1.6]
 \label{0.9.1.6}
-An important case is when every quasi-compact open subset of $X$ is retrocompact, in other words, when the intersection of two quasi-compact open subsets of $X$ is quasi-compact (cf.~\sref[I]{1.5.5.6}).
+An important case is when every quasi-compact open subset of $X$ is retrocompact, in other words, when the intersection of two quasi-compact open subsets of $X$ is quasi-compact (cf.~\sref[I]{I.5.5.6}).
 When $X$ is also quasi-compact, this implies that the retrocompact open subsets of $X$ are identical to the quasi-compact open subsets of $X$, and the constructible subsets of $X$ are finite unions of sets of the form $U\cap\complement{V}$, where $U$ and $V$ are quasi-compact open sets.
 \end{env}
 

ega1/ega1-1.tex

@@ -125,6 +125,7 @@ \subsection{The prime spectrum of a ring}
 \label{I.1.1.9}
 According to Proposition \sref{I.1.1.2}[iv], for two elements $f$, $g$ of $A$, we have
 \[
+\label{I.1.1.9.1}
   D(fg)=D(f)\cap D(g).
   \tag{1.1.9.1}
 \]
@@ -271,7 +272,7 @@ \subsection{Functorial properties of prime spectra of rings}
 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 $\mathfrak{a}'=\mathfrak{j}({}^a\vphi(Y))$, 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;
+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[I.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}
 
@@ -292,7 +293,7 @@ \subsection{Functorial properties of prime spectra of rings}
 
 \begin{proof}
 We show that for each subset $E\subset A$, there exists a subset $E'$ of $A'$ such that $V(E)=V(\vphi(E'))$;
-according to the ($T_0$) axiom \sref{I.1.1.8} and the formula \hyperref[1.1.2.2]{(1.2.2.1)}, this implies first of all that ${}^a\vphi$ is injective, and then, by \hyperref[1.1.2.2]{(1.2.2.1)}, that ${}^a\vphi$ is a homeomorphism.
+according to the ($T_0$) axiom \sref{I.1.1.8} and the formula \hyperref[I.1.2.2]{(1.2.2.1)}, this implies first of all that ${}^a\vphi$ is injective, and then, by \hyperref[I.1.2.2]{(1.2.2.1)}, that ${}^a\vphi$ is a homeomorphism.
 But it suffices, for each $f\in E$, to take $f'\in A'$ such that $h\vphi(f')=f$ with $h$ invertible in $A$;
 the set $E'$ of these elements $f'$ is exactly what we are searching for.
 \end{proof}
@@ -319,7 +320,7 @@ \subsection{Functorial properties of prime spectra of rings}
 \end{corollary}
 
 \begin{proof}
-Applying Equation~\hyperref[1.1.2.2]{(1.2.2.3)} to the ideal $\mathfrak{a}=(0)$, we have $\widetilde{{}^a\vphi(X)}=V(\Ker\vphi)$, and for \erratum[II]{$V(\Ker\vphi)=X'$} to hold, it is necessary and sufficient for $\Ker\vphi$ to be contained in all the prime ideals of \erratum[II]{$A'$, or, equivalently, in the nilradical $\rad'$ of $A'$.}
+Applying Equation~\hyperref[I.1.2.2]{(1.2.2.3)} to the ideal $\mathfrak{a}=(0)$, we have $\widetilde{{}^a\vphi(X)}=V(\Ker\vphi)$, and for \erratum[II]{$V(\Ker\vphi)=X'$} to hold, it is necessary and sufficient for $\Ker\vphi$ to be contained in all the prime ideals of \erratum[II]{$A'$, or, equivalently, in the nilradical $\rad'$ of $A'$.}
 \end{proof}
 
 \subsection{Sheaf associated to a module}
@@ -1155,7 +1156,7 @@ \subsection{Morphisms from locally ringed spaces to affine schemes}
 
 \begin{lemma}[1.8.5]
 \label{I.1.8.5}
-With the notation of \sref{I.1.8.4}, for each $A$-module $M$, the canonical functorial $\sh{O}_S$-homomorphism \sref[0]{4.4.3.3}
+With the notation of \sref{I.1.8.4}, for each $A$-module $M$, the canonical functorial $\sh{O}_S$-homomorphism \sref[0]{0.4.4.3.3}
 \[
 \label{I.1.8.5.1}
   i^*(i_*(\widetilde{M}))\to\widetilde{M}

ega1/ega1-10.tex

@@ -673,7 +673,7 @@ \subsection{Formal completion of a prescheme along a closed subset}
 It is immediate that, for every open $U\subset X$, we have $(\sh{F}|U)_{/(U\cap X')}=(\sh{F}_{/X'})|(U\cap X')$.
 
 By passing to the projective limit, it is clear that $(\sh{O}_X)_{/X'}$ is a sheaf of rings, and that $\sh{F}_{/X'}$ can be considered as an $(\sh{O}_X)_{/X'}$-module.
-In addition, since there exists a basis for the topology of $X'$ consisting of quasi-compact open sets, we can consider $(\sh{O}_X)_{/X'}$ (resp. $\sh{F}_{/X'}$) as a \emph{sheaf of topological rings} (resp. of \emph{topological groups}), the projective limit of the \emph{pseudo-discrete} sheaves of rings (resp. groups) $\sh{O}_X/\sh{F}$ (resp. $\sh{F}\otimes_{\sh{O}_X}(\sh{O}_X/\sh{F})=\sh{F}/\sh{J}\sh{F}$), and, by passing to the projective limit, $\sh{F}_{/X'}$ then becomes a \emph{topological $(\sh{O}_X)_{/X'}$-module} (\sref[0]{0.3.8.1} and \sref[0]{3.8.2}); recall that, for every \emph{quasi-compact} open $U\subset X$, $\Gamma(U\cap X',(\sh{O}_X)_{/X'})$ (resp. $\Gamma(U\cap X',\sh{F}_{/X'})$) is then the projective limit of the discrete rings (resp. groups) $\Gamma(U,\sh{O}_X/\sh{J})$ (resp. $\Gamma(U,\sh{F}/\sh{J}\sh{F})$).
+In addition, since there exists a basis for the topology of $X'$ consisting of quasi-compact open sets, we can consider $(\sh{O}_X)_{/X'}$ (resp. $\sh{F}_{/X'}$) as a \emph{sheaf of topological rings} (resp. of \emph{topological groups}), the projective limit of the \emph{pseudo-discrete} sheaves of rings (resp. groups) $\sh{O}_X/\sh{F}$ (resp. $\sh{F}\otimes_{\sh{O}_X}(\sh{O}_X/\sh{F})=\sh{F}/\sh{J}\sh{F}$), and, by passing to the projective limit, $\sh{F}_{/X'}$ then becomes a \emph{topological $(\sh{O}_X)_{/X'}$-module} (\sref[0]{0.3.8.1} and \sref[0]{0.3.8.2}); recall that, for every \emph{quasi-compact} open $U\subset X$, $\Gamma(U\cap X',(\sh{O}_X)_{/X'})$ (resp. $\Gamma(U\cap X',\sh{F}_{/X'})$) is then the projective limit of the discrete rings (resp. groups) $\Gamma(U,\sh{O}_X/\sh{J})$ (resp. $\Gamma(U,\sh{F}/\sh{J}\sh{F})$).
 
 Now, if $u:\sh{F}\to\sh{G}$ is a homomorphism of $\sh{O}_X$-modules, then there are canonically induced homomorphisms $u_\sh{J}:\sh{F}\otimes_{\sh{O}_X}(\sh{O}_X/\sh{J})\to\sh{G}\otimes_{\sh{O}_X}(\sh{O}_X/\sh{J})$ for all $\sh{J}\in\Phi$, and these homomorphisms form a projective system.
 By passing to the projective limit and restricting to $X'$, these give a continuous $(\sh{O}_X)_{/X'}$-homomorphism $\sh{F}_{/X'}\to\sh{G}_{/X'}$, denoted $u_{/X'}$ or $\widehat{u}$, and called the \emph{completion} of the homomorphism $u$ along $X'$.
@@ -743,7 +743,7 @@ \subsection{Formal completion of a prescheme along a closed subset}
 \[
   \Gamma(U\cap X',\sh{F}\otimes_{\sh{O}_X}\sh{O}_X/\sh{J}^n)=M\otimes_A(A/\mathfrak{J}^n)
 \]
-\sref{I.3.12}; so, by definition of the projective limit, we have
+\sref{I.1.3.12}; so, by definition of the projective limit, we have
 \[
   \Gamma(U\cap X',\sh{F}_{/X'})=\varprojlim_n(M\otimes_A(A/\mathfrak{J}^n))=\widehat{M},
 \]
@@ -864,7 +864,7 @@ \subsection{Extension of morphisms to completions}
 It can be checked immediately that this morphism does not depend on the choice of sheaves of ideals $\sh{J}$ and $\sh{K}$ satisfying the above conditions.
 It suffices to consider the case where $X$ and $Y$ are Noetherian affine schemes with rings $A$ and $B$ (respectively); then $\sh{J}=\widetilde{\mathfrak{J}}$ and $\sh{K}=\widetilde{\mathfrak{K}}$, where $\mathfrak{J}$ (resp. $\mathfrak{K}$) is an ideal of $A$ (resp. $B$), $f$ corresponds to a ring homomorphism $\vphi:B\to A$ such that $\vphi(\mathfrak{K})\subset\mathfrak{J}$ (\sref{I.4.4.6} and \sref{I.1.7.4}); \emph{$\widehat{f}$ is then the morphism corresponding \sref{I.10.2.2} to the continuous homomorphism $\widehat{\vphi}:\widehat{B}\to\widehat{A}$}, where $\widehat{A}$ (resp. $\widehat{B}$) is the separated completion of $A$ (resp. $B$) with respect to the $\mathfrak{J}$-preadic (resp. $\mathfrak{K}$-preadic) topology \sref{I.10.6.8}; we know that, if we replace $\sh{J}$ by another
 \oldpage[I]{199}
-sheaf of ideals $\sh{J}'=\widetilde{\mathfrak{J}'}$ such that the support of $\sh{O}_X/\sh{J}'$ is $X'$, then the $\mathfrak{J}$-preadic and $\mathfrak{J}'$-preadic topologies on $A$ are the same \sref{I.10.82}.
+sheaf of ideals $\sh{J}'=\widetilde{\mathfrak{J}'}$ such that the support of $\sh{O}_X/\sh{J}'$ is $X'$, then the $\mathfrak{J}$-preadic and $\mathfrak{J}'$-preadic topologies on $A$ are the same \sref{I.10.8.2}.
 
 We note that, by definition, the continuous map $X'\to Y'$ of the underlying spaces of $X_{/X'}$ and $Y_{/Y'}$ corresponding to $\widehat{f}$ is exactly the restriction to $X'$ of $f$.
 \end{env}
@@ -1238,12 +1238,14 @@ \subsection{Adic morphisms of formal preschemes}
 The (locally Noetherian) adic $\mathfrak{S}$-preschemes clearly form a \emph{category}.
 We say that an inductive system $(X_n)$ of locally Noetherian (usual) $S_n$-preschemes is an \emph{adic inductive $(S_n)$-system} if the structure morphisms $f_n:X_n\to S_n$ are such that, for $m\leq n$, the diagrams
 \[
+\label{I.10.12.2.1}
   \xymatrix{
     X_n \ar[d]_{f_n}
     & X_m \ar[l] \ar[d]^{f_m}\\
     S_n
     & S_m \ar[l]
   }
+  \tag{10.12.2.1}
 \]
 commute and \emph{identify $X_m$ with the product $X_n\times_{S_n}S_m=(X_n)_{(S_m)}$}.
 The adic inductive systems form a \emph{category}:

ega1/ega1-5.tex

@@ -220,9 +220,20 @@ \subsection{Existence of a subprescheme with a given underlying space}
 
 \begin{proposition}[5.2.2]
 \label{I.5.2.2}
-Let $X$ be a reduced subprescheme of a prescheme $Y$; if $Z$ is the closed reduced subprescheme of $Y$ that has $\overline{X}$ as its underlying space, then $X$ is a subprescheme induced on an open subset of $Z$.
+Let $X$ be reduced, $f:X \to Y$ a morphism, and $Z$ a sub-prescheme closed over $Y$ such that $f(X) \subset Z$.
+Then, $f$ factors as $X \xrightarrow{g} Z \xrightarrow{j} Y$, where $j$ is an injective morphism.
 \end{proposition}
 
+\begin{proof}
+It follows from the hypotheses that the closed subprescheme $f^{-1}(Z)$ of $X$ has all of $X$ as its underlying space \sref{I.4.4.1};
+since $X$ is reduced, this closed subprescheme agrees with $X$ \sref{I.5.1.2}, and the proposition then follows from \sref{I.4.4.1}.
+\end{proof}
+
+\begin{corollary}[5.2.3]
+\label{I.5.2.3}
+Let $X$ be a reduced subprescheme of a prescheme $Y$; if $Z$ is the closed reduced subprescheme of $Y$ that has $\overline{X}$ as its underlying space, then $X$ is a subprescheme induced on an open subset of $Z$.
+\end{corollary}
+
 \begin{proof}
 \oldpage[I]{132}
 There is indeed an open subset $U$ of $Y$ such that $X=U\cap\overline{X}$;
@@ -296,6 +307,7 @@ \subsection{Diagonal; graph of a morphism}
 Let $f:X\to S$ and $g:Y\to S$ be the structure morphisms, $p$ and $q$ the projections of $X\times_S Y$, and $\pi=f\circ p=g\circ q$ the structure morphism $X\times_S Y\to S$.
 Then the diagram
 \[
+\label{I.5.3.5.1}
   \xymatrix{
     X\times_S Y\ar[r]^{(p,q)_T}\ar[d]_\pi &
     X\times_T Y\ar[d]^{f\times_T g}\\
@@ -347,7 +359,7 @@ \subsection{Diagonal; graph of a morphism}
 
 \begin{proof}
 Indeed, to say that $f$ is a monomorphism implies that, for every $Y$-prescheme $Z$, the corresponding map $f':X(Z)_Y\to Y(Z)_Y$ is an injection, and, since $Y(Z)_Y$ consists of a single element, this implies that $X(Z)_Y$ consists of a single element as well.
-But this can also be expressed by saying that $X(Z)_Y\times X(Z)_Y$ is canonically isomorphic to $X(Z)_Y$; the former is exactly the set $(X\times_Y X)(Z)_Y$ \hyperref[1.3.4.3]{(3.4.3.1)}, which implies that $\Delta_{X|Y}$ is an isomorphism.
+But this can also be expressed by saying that $X(Z)_Y\times X(Z)_Y$ is canonically isomorphic to $X(Z)_Y$; the former is exactly the set $(X\times_Y X)(Z)_Y$ \hyperref[I.3.4.3]{(3.4.3.1)}, which implies that $\Delta_{X|Y}$ is an isomorphism.
 \end{proof}
 
 \begin{proposition}[5.3.9]
@@ -606,7 +618,7 @@ \subsection{Separation criteria}
 \oldpage[I]{137}
 proves~(ii).
 Given~(i) and~(ii), (iii) and~(iv) are equivalent \sref{I.3.5.1}, so it suffices to prove (iv).
-But $X_{(S')}\times_{Y_{(S')}}X_{(S')}$ is canonically identified with $(X\times_Y X)\times_Y Y_{(S')}$ by \sref{I.3.3.11} and \hyperref[1.3.3.9]{(3.3.9.1)}, and we immediately see that the diagonal morphism $\Delta_{X_{(S')}}$ can then be identified with $\Delta_X\times_Y 1_{Y_{(S')}}$;
+But $X_{(S')}\times_{Y_{(S')}}X_{(S')}$ is canonically identified with $(X\times_Y X)\times_Y Y_{(S')}$ by \sref{I.3.3.11} and \hyperref[I.3.3.9]{(3.3.9.1)}, and we immediately see that the diagonal morphism $\Delta_{X_{(S')}}$ can then be identified with $\Delta_X\times_Y 1_{Y_{(S')}}$;
 the proposition then follows from \sref{I.4.3.1}.
 
 To prove (v), consider, as in \sref{I.5.3.13}, the factorisation $X\xrightarrow{\Gamma_f}X\times_Z Y\xrightarrow{p_2}Y$ of $f$, noting that $p_2=(g\circ f)\times_Z 1_Y$;
@@ -653,7 +665,7 @@ \subsection{Separation criteria}
 
 \begin{proof}
 The necessity follows from \sref{I.5.5.1}[i, ii, and v].
-Conversely, if the condition of the statement is satisfied, then each of the restrictions $X_k\to Y$ of $f$ is separated \hyperref[1.5.5.1]{(5.5.1, (i) and (ii))};
+Conversely, if the condition of the statement is satisfied, then each of the restrictions $X_k\to Y$ of $f$ is separated \hyperref[I.5.5.1]{(5.5.1, (i) and (ii))};
 if $p_1$ and $p_2$ are the projections of $X\times_Y X$, then the subspace $\Delta_{X_k}(X_k)$ can be identified with the subspace $\Delta_X(X)\cap p_1^{-1}(X_k)$ of the underlying space of $X\times_Y X$ \sref{I.5.3.16};
 these subspaces are closed in $X\times_Y X$, and thus so too is their union $\Delta_X(X)$.
 \end{proof}

ega1/ega1-6.tex

@@ -518,7 +518,7 @@ \subsection{Algebraic preschemes}
   n(X\times_K Y)=n(X)n(Y)
   \tag{6.4.8.2}
 \]
-because of the interpretation of $n(X)$ as the number of elements of $X(\Omega)_K$ and Equation~\hyperref[1.3.4.3]{(3.4.3.1)}.
+because of the interpretation of $n(X)$ as the number of elements of $X(\Omega)_K$ and Equation~\hyperref[I.3.4.3]{(3.4.3.1)}.
 \end{env}
 
 \begin{proposition}[6.4.9]

ega1/ega1-7.tex

@@ -87,7 +87,7 @@ \subsection{Rational maps and rational functions}
 \end{lemma}
 
 \begin{proof}
-To show \hyperref[1.7.1.9.1]{(7.1.9.1)}, we again denote by $X_i$ ($1\leq i\leq n$) the irreducible components of $X$; if $D(f)$ is dense in $X$ then $D(f)\cap X_i\neq\emp$ for $1\leq i\leq n$, and vice-versa; but this means that $f\not\in\mathfrak{p}_i$ for $1\leq i\leq n$, where we set $\mathfrak{p}_i=\mathfrak{j}(X_i)$, and since the $\mathfrak{p}_i$ are the minimal prime ideals of $A$ \hyperref[1.1.1.14]{(1.1.14)}, the conditions $f\not\in\mathfrak{p}_i$ ($1\leq i\leq n$) are equivalent to $f\in Q$, whence the first claim of the lemma.
+To show \hyperref[I.7.1.9]{(7.1.9.1)}, we again denote by $X_i$ ($1\leq i\leq n$) the irreducible components of $X$; if $D(f)$ is dense in $X$ then $D(f)\cap X_i\neq\emp$ for $1\leq i\leq n$, and vice-versa; but this means that $f\not\in\mathfrak{p}_i$ for $1\leq i\leq n$, where we set $\mathfrak{p}_i=\mathfrak{j}(X_i)$, and since the $\mathfrak{p}_i$ are the minimal prime ideals of $A$ \hyperref[I.1.1.14]{(1.1.14)}, the conditions $f\not\in\mathfrak{p}_i$ ($1\leq i\leq n$) are equivalent to $f\in Q$, whence the first claim of the lemma.
 For the other claim, if $U$ is a dense open subset of $X$, the complement of $U$ is a set of the form $V(\mathfrak{a})$, where $\mathfrak{a}$ is an ideal which is not contained in any of the $\mathfrak{p}_i$; it is thus not contained in their union (\cite[p.~13]{I-10}), and there thus exists some $f\in\mathfrak{a}$ belonging to $Q$; whence $D(f)\subset U$, which finishes the proof.
 \end{proof}