Typos from already translated material

ega0/ega0-7.tex

@@ -1308,7 +1308,7 @@ \subsection{Completed tensor products}
 
 \begin{proposition}[7.7.8]
 \label{0.7.7.8}
-Let $A$ be a preadic ring, $\mathfrak{J}$ an ideal of defintion for $A$, $M$ an $A$-module
+Let $A$ be a preadic ring, $\mathfrak{J}$ an ideal of definition for $A$, $M$ an $A$-module
 \emph{of finite type}, equipped with the $\mathfrak{J}$-preadic topology. For every topological
 adic Noetherian $A$-algebra $B$, $B\otimes_A M$ identifies with the completed tensor product
 $(B\otimes_A M)^\wedge$.

ega2/ega2-2.tex

@@ -1029,7 +1029,7 @@ \subsection{The graded $S$-module associated to a sheaf on $\operatorname{Proj}(
 Recall \sref[0]{0.5.4.6} that $\Gamma_\bullet(\sh{O}_X)$ is endowed with the structure of a \emph{graded ring}, and $\Gamma_\bullet(\sh{F})$ with the structure of a \emph{graded $\Gamma_\bullet(\sh{O}_X)$-module}.
 
 Since $\sh{O}_X(n)$ is locally free, $\Gamma_\bullet(\sh{F})$ is a \emph{left exact} additive covariant functor in $\sh{F}$;
-in particular, if $\sh{J}$ is a sheaf of ideals of $\sh{O}_X$, then $\Gamma_\bullet(\sh{J})$ is canonically identified with a \emph{graded idea} of $\Gamma_\bullet(\sh{O}_X)$.
+in particular, if $\sh{J}$ is a sheaf of ideals of $\sh{O}_X$, then $\Gamma_\bullet(\sh{J})$ is canonically identified with a \emph{graded ideal} of $\Gamma_\bullet(\sh{O}_X)$.
 \end{env}
 
 \begin{env}[2.6.2]
@@ -1088,11 +1088,11 @@ \subsection{The graded $S$-module associated to a sheaf on $\operatorname{Proj}(
 
 \begin{env}[2.6.4]
 \label{II.2.6.4}
-Now let $\sh{F}$ be an $\sh{O}_X$-modules, and set $M=\Gamma_\bullet(\sh{F})$;
+Now let $\sh{F}$ be an $\sh{O}_X$-module, and set $M=\Gamma_\bullet(\sh{F})$;
 by the existence of the homomorphism of graded rings $\alpha: S\to\Gamma_\bullet(\sh{O}_X)$, we can consider $M$ as a graded $S$-module.
 For every $f\in S_d$ ($d>0$), it follows from \sref{II.2.6.3} that the restriction to $D_+(f)$ of the section $\alpha_d(f)$ of $\sh{O}_X(d)$ is invertible;
 thus so too is the restriction to $D_+(f)$ of the section $\alpha_d(f^n)$ of $\sh{O}_X(nd)$, for all $n>0$.
-So let $z\in M_{nd}=\Gamma(X,\sh{F}(nd)$ ($n>0$);
+So let $z\in M_{nd}=\Gamma(X,\sh{F}(nd))$ ($n>0$);
 if there exists an integer $k\geq0$ such that the restriction to $D_+(f)$ of $f^kz$, i.e. the
 \oldpage[II]{38}
 section $(z|D_+(f))(\alpha_d(f^k)|D_+(f))$ of $\sh{F}((n+k)d)$, is zero, then, by the above remark, we also have that $z|D_+(f)=0$.

ega2/ega2-3.tex

@@ -15,14 +15,15 @@ \subsection{Homogeneous spectrum of a quasi-coherent graded $\mathcal{O}_Y$-alge
 We denote by $\sh{M}(n)$ the graded $\sh{S}$-module such that $(\sh{M}(n))_k=\sh{M}_{n+k}$ for all $k\in\bb{Z}$;
 if $\sh{S}$ and $\sh{M}$ are quasi-coherent, then $\sh{M}(n)$ is a quasi-coherent graded $\sh{S}$-module \sref[I]{I.9.6.1}.
 
-We say that $\sh{M}$ is a graded $\sh{S}$-module \emph{of finite type} (resp. admitting a \emph{finite presentation})  if, for all $y\in Y$, there exists an open neighbourhood $U$ of $y$, along with integers $n_i$ (resp. integers $m_i$ and $n_j$) such that there is a surjective degree~$0$ homomorphism $\bigoplus_{i=1}^r(\sh{S}(n_i)|U)\to\sh{M}|U$ (resp. such that $\sh{M}|U$ is isomorphic to the cokernel of a degree~$0$ homomorphism $\bigoplus_{i=1}^r(\sh{S}(m_i)|U)\to\bigoplus_{j 1}^s(\sh{S}(n_J)|U)$).
+We say that $\sh{M}$ is a graded $\sh{S}$-module \emph{of finite type} (resp. admitting a \emph{finite presentation})  if, for all $y\in Y$, there exists an open neighbourhood $U$ of $y$, along with integers $n_i$ (resp. integers $m_i$ and $n_j$) such that there is a surjective degree~$0$ homomorphism $\bigoplus_{i=1}^r(\sh{S}(n_i)|U)\to\sh{M}|U$ (resp. such that $\sh{M}|U$ is isomorphic to the cokernel of a degree~$0$ homomorphism $\bigoplus_{i=1}^r(\sh{S}(m_i)|U)\to\bigoplus_{j=1}^s(\sh{S}(n_j)|U)$).
 
 Let $U$ be an affine open of $Y$, with ring $A=\Gamma(U,\sh{O}_Y)$;
 by hypothesis, the graded $(\sh{O}_Y|U)$-algebra $\sh{S}|U$ is isomorphic to $\widetilde{S}$, where $S=\Gamma(U,\sh{S})$ is a graded $A$-algebra \sref[I]{I.1.4.3};
 \oldpage[II]{50}
 set $X_U=\Proj(\Gamma(U,\sh{S}))$.
 Let $U'\subset U$ be another affine open of $Y$, with ring $A'$, and let $j:U'\to U$ be the canonical injection, which corresponds to the restriction homomorphism $A\to A'$;
-we have that $\sh{S}|U'=j^*(\sh{S}|U)$, and so $S'=\Gamma(U',\sh{S})$ is canonically identified with $X_U\times_U U'$, and thus also with $f_U^{-1}(U')$, where we denote by $f_U$ the structure morphism $X_U\to U$ \sref[I]{I.4.4.1}.
+we have that $\sh{S}|U'=j^*(\sh{S}|U)$, and so $S'=\Gamma(U',\sh{S})$ is canonically identified with $S\otimes_AA'$ \sref[I]{I.1.6.5}.
+We conclude \sref{II.2.8.10} that $X_{U'}$ is canonically identified with $X_U\times_U U'$, and thus also with $f_U^{-1}(U')$, where we denote by $f_U$ the structure morphism $X_U\to U$ \sref[I]{I.4.4.1}.
 We denote by $\sigma_{U',U}$ the canonical isomorphism $f_U^{-1}(U')\simto X_{U'}$ thus defined, and by $\rho_{U',U}$ the open immersion $X_{U'}\to X_U$ obtained by composing $\sigma_{U',U}^{-1}$ with the canonical injection $f_U^{-1}(U')\to X_U$.
 It is immediate that, if $U''\subset U'$ is another affine open of $Y$, then $\rho_{U'',U}=\rho_{U'',U'}\circ\rho_{U',U}$.
 \end{env}
@@ -131,7 +132,7 @@ \subsection{Homogeneous spectrum of a quasi-coherent graded $\mathcal{O}_Y$-alge
 \end{corollary}
 
 \begin{proof}
-By applying \sref{II.3.1.6} to the unique section $f\in\Gamma(Y,\sh{S})$ that is equal to $T$ at each point of $Y$< we see that $X_f=X$.
+By applying \sref{II.3.1.6} to the unique section $f\in\Gamma(Y,\sh{S})$ that is equal to $T$ at each point of $Y$ we see that $X_f=X$.
 Further, here we have $d=1$, and $\sh{S}^{(1)}/(f-1)\sh{S}^{(1)}=\sh{S}/(f-1)\sh{S}$ is canonically isomorphic to $\sh{A}$, whence the corollary \sref{II.1.2.2}.
 \end{proof}
 

ega2/ega2-5.tex

@@ -671,7 +671,7 @@ \subsection{Chow's lemma}
 \end{enumerate}
 Under these hypotheses,
 \begin{enumerate}
-  \item[{\rm(i)}] there exists a \emph{quasi-projective} $S$-scheme $X'$, and an $S$-morphism $f:X'\to X$ that is both\emph{projective} and \emph{surjective};
+  \item[{\rm(i)}] there exists a \emph{quasi-projective} $S$-scheme $X'$, and an $S$-morphism $f:X'\to X$ that is both \emph{projective} and \emph{surjective};
   \item[{\rm(ii)}] we can take $X'$ and $f$ to be such that there exists an open subset $U\subset X$ for which $U'=f^{-1}(U)$ is dense in $X'$, and for which the restriction of $f$ to $U'$ is an isomorphism $U'\isoto U$; and
   \item[{\rm(iii)}] if $X$ is reduced (resp. irreducible, integral), then we can assume that $X'$ is reduced (resp. irreducible, integral).
 \end{enumerate}