From cb27fcefd6dd2139ec7633e0e14952a6249abf3f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Tue, 21 Oct 2025 16:01:40 +0200 Subject: [PATCH 1/9] Update ega2-3.tex --- ega2/ega2-3.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ega2/ega2-3.tex b/ega2/ega2-3.tex index d07bc3d5..29753ebe 100644 --- a/ega2/ega2-3.tex +++ b/ega2/ega2-3.tex @@ -131,7 +131,7 @@ \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} From 3655a2d4fccf7f62456322ddbb5744fe852300d9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Tue, 21 Oct 2025 16:32:35 +0200 Subject: [PATCH 2/9] Update ega2-3.tex --- ega2/ega2-3.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ega2/ega2-3.tex b/ega2/ega2-3.tex index 29753ebe..9575fc80 100644 --- a/ega2/ega2-3.tex +++ b/ega2/ega2-3.tex @@ -15,7 +15,7 @@ \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}; From 25be7e057cd23f007966ecef32c7df22997d86ee Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Thu, 23 Oct 2025 16:30:09 +0200 Subject: [PATCH 3/9] Transcription typo --- ega2/ega2-3.tex | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/ega2/ega2-3.tex b/ega2/ega2-3.tex index 9575fc80..31254c85 100644 --- a/ega2/ega2-3.tex +++ b/ega2/ega2-3.tex @@ -22,7 +22,8 @@ \subsection{Homogeneous spectrum of a quasi-coherent graded $\mathcal{O}_Y$-alge \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.1.6.4}. +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} From 318341809cef5b3e8bdd27937020102cbc7d253f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Thu, 23 Oct 2025 16:40:13 +0200 Subject: [PATCH 4/9] Update ega2-3.tex Grothendieck cites I.1.6.4, but upon inspection it is clear he really meant I.1.6.5. I have checked his errata lists in later EGA volumes and this typo is not mentioned. --- ega2/ega2-3.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ega2/ega2-3.tex b/ega2/ega2-3.tex index 31254c85..4a52f2f3 100644 --- a/ega2/ega2-3.tex +++ b/ega2/ega2-3.tex @@ -22,7 +22,7 @@ \subsection{Homogeneous spectrum of a quasi-coherent graded $\mathcal{O}_Y$-alge \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 $S\otimes_AA'$ \sref[I]{I.1.6.4}. +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 $\rho_{U'',U}=\rho_{U'',U'}\circ\rho_{U',U}$. From 18cfa6d15272ddfb1f6f799f6a7bff8204571a28 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 24 Oct 2025 14:45:23 +0200 Subject: [PATCH 5/9] Update ega2-2.tex --- ega2/ega2-2.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex index d6d9003a..8ebb74dd 100644 --- a/ega2/ega2-2.tex +++ b/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] From 010cc6f0cc067398b1ef1c785439edaeeaf24df1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 24 Oct 2025 15:15:37 +0200 Subject: [PATCH 6/9] Update ega2-2.tex --- ega2/ega2-2.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex index 8ebb74dd..65df7e92 100644 --- a/ega2/ega2-2.tex +++ b/ega2/ega2-2.tex @@ -1092,7 +1092,7 @@ \subsection{The graded $S$-module associated to a sheaf on $\operatorname{Proj}( 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 $z|D_+(f)=0$. From 9bc4af1424940ec42d5d9c251f276a44183197da Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 24 Oct 2025 17:24:21 +0200 Subject: [PATCH 7/9] Update ega2-2.tex --- ega2/ega2-2.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex index 65df7e92..e8db489f 100644 --- a/ega2/ega2-2.tex +++ b/ega2/ega2-2.tex @@ -1088,7 +1088,7 @@ \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 $n>0$. From e10c66cb364d651eda5db1fc8a85223f5b7121a7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Fri, 7 Nov 2025 11:01:42 +0100 Subject: [PATCH 8/9] Update ega2-5.tex --- ega2/ega2-5.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ega2/ega2-5.tex b/ega2/ega2-5.tex index 3d9fbfc5..ac44a2dd 100644 --- a/ega2/ega2-5.tex +++ b/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} From 884c1cc40f758379e62e69d71443d9eada50ee05 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?El=C3=ADas=20Guisado=20Villalgordo?= Date: Tue, 2 Dec 2025 16:35:49 +0100 Subject: [PATCH 9/9] Update ega0-7.tex --- ega0/ega0-7.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ega0/ega0-7.tex b/ega0/ega0-7.tex index ec78f54e..1aab3d2b 100644 --- a/ega0/ega0-7.tex +++ b/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 $(B\otimes_A M)^\wedge$.