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authorGravatar Tim Hosgood <timhosgood@gmail.com> 2021-04-20 00:34:24 +0100
committerGravatar Tim Hosgood <timhosgood@gmail.com> 2021-04-20 00:34:24 +0100
commit2b10710d5a3c5e8c8c4a3140e8e5b0070d409c23 (patch)
tree008db89104f9811546a09b054894ce8e0721acb8
parent5d163ddc1f57205761b0dd10d413209dc58e752c (diff)
parent9bae7b9043a74644e0942c6f7ec897f7a8aaf482 (diff)
downloadega-master.tar.gz
ega-master.zip
Merge branch 'master' of https://github.com/ryankeleti/egaHEADmaster
-rw-r--r--STYLE.md8
-rw-r--r--ega0/ega0-5.tex14
-rw-r--r--ega1/ega1-9.tex4
-rw-r--r--ega2/ega2-2.tex174
-rw-r--r--ega3/ega3-1.tex8
-rw-r--r--preamble-base.tex2
6 files changed, 175 insertions, 35 deletions
diff --git a/STYLE.md b/STYLE.md
index 654ff31..74cb3bf 100644
--- a/STYLE.md
+++ b/STYLE.md
@@ -221,7 +221,7 @@ if in EGA II, page 41 ends with `Hi! Schemes` and page 42 begins with `are cool.
### Miscellaneous
* `\vphi` --- phi `φ`
* `\emp` --- empty set `∅`
-* `\dual` --- for the dual sheaf, i.e., `\dual{\sh{F}}` for F<sup>v</sup>
+* `\dual` --- for the dual sheaf, i.e. `\dual{\sh{F}}` for F<sup>v</sup>
* `\rad` --- radical
* `\nilrad` --- nilradical
* `\setmin` --- set minus/difference
@@ -230,8 +230,10 @@ if in EGA II, page 41 ends with `Hi! Schemes` and page 42 begins with `are cool.
* `\RR` --- right derived R
* `\LL` --- left derived L
* `\kres` --- residue field k
-* `\op` --- opposite category, i.e., `\cat{C}\op` for C<sup>op</sup>
-* `\red` --- reduced, i.e., `X_\red` for X<sub>red</sub>
+* `\op` --- opposite category, i.e. `\cat{C}\op` for C<sup>op</sup>
+* `\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
## References
diff --git a/ega0/ega0-5.tex b/ega0/ega0-5.tex
index b56d7e6..eae4aa9 100644
--- a/ega0/ega0-5.tex
+++ b/ega0/ega0-5.tex
@@ -310,29 +310,29 @@ We conclude that if $\sh{L}$ is invertible, then $f^*(\sh{L}^{\otimes n})$ canon
\begin{env}[5.4.6]
\label{0.5.4.6}
-Let $\sh{L}$ be an invertible $\sh{O}_X$-module; we denote by $\Gamma_*(X,\sh{L})$ or simply $\Gamma_*(\sh{L})$ the abelian group direct sum $\bigoplus_{n\in\bb{Z}}\Gamma(X,\sh{L}^{\otimes n})$;
+Let $\sh{L}$ be an invertible $\sh{O}_X$-module; we denote by $\Gamma_\bullet(X,\sh{L})$ or simply $\Gamma_\bullet(\sh{L})$ the abelian group direct sum $\bigoplus_{n\in\bb{Z}}\Gamma(X,\sh{L}^{\otimes n})$;
we equip it with the structure of a \emph{graded ring}, by corresponding to a pair $(s_n,s_m)$, where $s_n\in\Gamma(X,\sh{L}^{\otimes n})$, $s_m\in\Gamma(X,\sh{L}^{\otimes m})$, the section of $\sh{L}^{\otimes(n+m)}$ over $X$ which corresponds canonically (5.4.4.1) to the section $s_n\otimes s_m$ of $\sh{L}^{\otimes n}\otimes_{\sh{O}_X}\sh{L}^{\otimes m}$;
the associativity of this multiplication is verified in an immediate way.
-It is clear that $\Gamma_*(X,\sh{L})$ is a covariant functor in $\sh{L}$, with values in the category of graded rings.
+It is clear that $\Gamma_\bullet(X,\sh{L})$ is a covariant functor in $\sh{L}$, with values in the category of graded rings.
If now $\sh{F}$ is any $\sh{O}_X$-module, then we set
\[
- \Gamma_*(\sh{L},\sh{F})=\bigoplus_{n\in\bb{Z}}\Gamma(X,\sh{F}\otimes_{\sh{O}_X}\sh{L}^{\otimes n}).
+ \Gamma_\bullet(\sh{L},\sh{F})=\bigoplus_{n\in\bb{Z}}\Gamma(X,\sh{F}\otimes_{\sh{O}_X}\sh{L}^{\otimes n}).
\]
-We equip this abelian group with the structure of a \emph{graded module} over the graded ring $\Gamma_*(\sh{L})$ in the following way:
+We equip this abelian group with the structure of a \emph{graded module} over the graded ring $\Gamma_\bullet(\sh{L})$ in the following way:
to a pair $(s_n,u_m)$, where $s_n\in\Gamma(X,\sh{L}^{\otimes n})$ and $u_m\in\Gamma(X,\sh{F}\otimes_{\sh{O}_X}\sh{L}^{\otimes m})$, we associate the section of $\sh{F}\otimes_{\sh{O}_X}\sh{L}^{\otimes(m+n)}$ which canonically corresponds (5.4.4.1) to $s_n\otimes u_m$;
the verification of the module axioms are immediate.
-For $X$ and $\sh{L}$ fixed, $\Gamma_*(\sh{L},\sh{F})$ is a covariant functor in $\sh{F}$ with values in the category of graded $\Gamma_*(\sh{L})$-modules;
+For $X$ and $\sh{L}$ fixed, $\Gamma_\bullet(\sh{L},\sh{F})$ is a covariant functor in $\sh{F}$ with values in the category of graded $\Gamma_\bullet(\sh{L})$-modules;
for $X$ and $\sh{F}$ fixed, it is a covariant functor in $\sh{L}$ with values in the category of abelian groups.
-If $f:Y\to X$ is a morphism of ringed spaces, the canonical homomorphism (4.4.3.2) $\rho:\sh{L}^{\otimes n}\to f_*(f^*(\sh{L}^{\otimes n}))$ defines a homomorphism of abelian groups $\Gamma(X,\sh{L}^{\otimes n})\to\Gamma(Y,f^*(\sh{L}^{\otimes n}))$, and as $f^*(\sh{L}^{\otimes n})=(f^*(\sh{L}))^{\otimes n})$, it follows from the definitions of the canonical homomorphisms (4.4.3.2) and (5.4.4.1) that the above homomorphisms define a \emph{functorial homomorphism of graded rings $\Gamma_*(\sh{L})\to\Gamma_*(f^*(\sh{L}))$}.
+If $f:Y\to X$ is a morphism of ringed spaces, the canonical homomorphism (4.4.3.2) $\rho:\sh{L}^{\otimes n}\to f_*(f^*(\sh{L}^{\otimes n}))$ defines a homomorphism of abelian groups $\Gamma(X,\sh{L}^{\otimes n})\to\Gamma(Y,f^*(\sh{L}^{\otimes n}))$, and as $f^*(\sh{L}^{\otimes n})=(f^*(\sh{L}))^{\otimes n})$, it follows from the definitions of the canonical homomorphisms (4.4.3.2) and (5.4.4.1) that the above homomorphisms define a \emph{functorial homomorphism of graded rings $\Gamma_\bullet(\sh{L})\to\Gamma_\bullet(f^*(\sh{L}))$}.
The same canonical homomorphism \sref{0.4.4.3} similarly defines a homomorphism of abelian groups $\Gamma(X,\sh{F}\otimes_{\sh{O}_X}\sh{L}^{\otimes n})\to\Gamma(Y,f^*(\sh{F}\otimes_{\sh{O}_X}\sh{L}^{\otimes n}))$, and as
\[
f^*(\sh{F}\otimes_{\sh{O}_X}\sh{L}^{\otimes n})=f^*(\sh{F})\otimes_{\sh{O}_Y}(f^*(\sh{L}))^{\otimes n}
\quad(4.3.3.1),
\]
\oldpage[0\textsubscript{I}]{51}
-these homomorphism (for $n$ variable) define a \emph{di-homomorphism of graded modules $\Gamma_*(\sh{L},\sh{F})\to\Gamma_*(f^*(\sh{L}),f^*(\sh{F}))$}.
+these homomorphism (for $n$ variable) define a \emph{di-homomorphism of graded modules $\Gamma_\bullet(\sh{L},\sh{F})\to\Gamma_\bullet(f^*(\sh{L}),f^*(\sh{F}))$}.
\end{env}
\begin{env}[5.4.7]
diff --git a/ega1/ega1-9.tex b/ega1/ega1-9.tex
index 528d458..1be4fac 100644
--- a/ega1/ega1-9.tex
+++ b/ega1/ega1-9.tex
@@ -367,8 +367,8 @@ The following corollaries give an interpretation of Theorem~\sref{I.9.3.1} in a
algebraic language:
\begin{corollary}[9.3.2]
\label{I.9.3.2}
-With the hypotheses of \sref{I.9.3.1}, consider the graded ring $A_*=\Gamma_*(\sh{L})$
-and the graded $A_*$-module $M_*=\Gamma_*(\sh{L},\sh{F})$ \sref[0]{0.5.4.6}. If $f\in A_n$,
+With the hypotheses of \sref{I.9.3.1}, consider the graded ring $A_\bullet=\Gamma_\bullet(\sh{L})$
+and the graded $A_\bullet$-module $M_*=\Gamma_\bullet(\sh{L},\sh{F})$ \sref[0]{0.5.4.6}. If $f\in A_n$,
where $n\in\bb{Z}$, then there is a canonical isomorphism
$\Gamma(X_f,\sh{F})\isoto((M_*)_f)_0$ (\emph{the subgroup of the module of
fractions $(M_*)_f$ consisting of elements of degree $0$}).
diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex
index cacea42..cc96971 100644
--- a/ega2/ega2-2.tex
+++ b/ega2/ega2-2.tex
@@ -675,7 +675,7 @@ In fact, these two schemes are both canonically isomorphic to $\Proj(S')$, using
\begin{env}[2.5.1]
\label{II.2.5.1}
Let $M$ be a \emph{graded} $S$-module.
-The, for every homogeneous $f$ in $S_+$, $M_{(f)}$ is an $S_{(f)}$-module, and thus has a corresponding quasi-coherent associated sheaf $(M_{(f)})^\supertilde$ on the affine scheme $\Spec(S_{(f)})$, identified with $D_+(f)$ \sref[I]{I.1.3.4}.
+The, for every homogeneous $f$ in $S_+$, $M_{(f)}$ is an $S_{(f)}$-module, and thus has a corresponding quasi-coherent associated sheaf $(M_{(f)})\supertilde$ on the affine scheme $\Spec(S_{(f)})$, identified with $D_+(f)$ \sref[I]{I.1.3.4}.
\end{env}
\oldpage[II]{31}
@@ -686,13 +686,13 @@ There exists on $X=\Proj(S)$ exactly one quasi-coherent $\sh{O}_X$-module $\wide
\begin{proof}
Suppose that $f\in S_d$ and $g\in S_e$.
-Since $D_+(fg)$ can be identified with the prime spectrum of $(S_{(f)})_{g^d/f^e}$ by \sref{II.2.2.2}, the restriction to $D_+(fg)$ of the sheaf $(M_{(f)})^\supertilde$ on $D_+(f)$ is canonically identified with the sheaf associated to the module $(M_{(f)})_{g^d/f^e}$ \sref[I]{I.1.3.6}, and thus also with $(M_{(fg)})^\supertilde$ \sref{II.2.2.2};
+Since $D_+(fg)$ can be identified with the prime spectrum of $(S_{(f)})_{g^d/f^e}$ by \sref{II.2.2.2}, the restriction to $D_+(fg)$ of the sheaf $(M_{(f)})\supertilde$ on $D_+(f)$ is canonically identified with the sheaf associated to the module $(M_{(f)})_{g^d/f^e}$ \sref[I]{I.1.3.6}, and thus also with $(M_{(fg)})\supertilde$ \sref{II.2.2.2};
we thus conclude that there exists a canonical isomorphism
\[
- \theta_{g,f}: (M_{(f)})^\supertilde|D_+(fg) \xrightarrow{\sim} (M_{(g)})^\supertilde|D_+(fg)
+ \theta_{g,f}: (M_{(f)})\supertilde|D_+(fg) \xrightarrow{\sim} (M_{(g)})\supertilde|D_+(fg)
\]
such that, if $h$ is a third homogeneous element of $S_+$, then $\theta_{f,h}=\theta_{f,g}\circ\theta_{g,h}$ in $D_+(fgh)$.
-Consequently \sref[0]{0.3.3.1} there exists a quasi-coherent $\sh{O}_X$-module $\sh{F}$ on $X$, and, for every homogeneous $f$ in $S_+$, an isomorphism $\eta_f$ from $\sh{F}|D_+(f)$ to $(M_{f})^\supertilde$ such that $\theta_{g,f}=\eta_g\circ\eta_f^{-1}$.
+Consequently \sref[0]{0.3.3.1} there exists a quasi-coherent $\sh{O}_X$-module $\sh{F}$ on $X$, and, for every homogeneous $f$ in $S_+$, an isomorphism $\eta_f$ from $\sh{F}|D_+(f)$ to $(M_{f})\supertilde$ such that $\theta_{g,f}=\eta_g\circ\eta_f^{-1}$.
If we then consider the sheaf $\sh{G}$ associated to the presheaf (on the base of the topology of $X$ given by the $D_+(f)$) defined by $D_+(f)\mapsto M_{(f)}$, with the canonical homomorphisms $M_{(f)}\to M_{(fg)}$ as restriction homomorphisms, then the above proves that $\sh{F}$ and $\sh{G}$ are isomorphic (taking \sref[I]{I.1.3.7} into account);
the sheaf $\sh{G}$ is denoted by $\widetilde{M}$, and indeed satisfies the conditions of the statement.
We have, in particular, $\widetilde{S}=\sh{O}_X$.
@@ -712,7 +712,7 @@ The functor $M\mapsto\widetilde{M}$ is an exact additive covariant functor from
\end{proposition}
\begin{proof}
-Indeed, since these properties are local, it suffices to show that they are satisfied for the sheaves of the form $\widetilde{M}|D_+(f)=(M_{(f)})^\supertilde$;
+Indeed, since these properties are local, it suffices to show that they are satisfied for the sheaves of the form $\widetilde{M}|D_+(f)=(M_{(f)})\supertilde$;
but the functors $M\mapsto M_f$, $N\mapsto N_0$ (to the category of graded $S_f$-modules), and $P\mapsto\widetilde{P}$ (to the category of $S_{(f)}$-modules) all have the three properties of exactness and of commutativity with inductive limits and direct sums (\sref[I]{I.1.3.5} and \sref[I]{I.1.3.9});
whence the proposition.
\end{proof}
@@ -751,7 +751,7 @@ the condition is \emph{a fortiori} satisfied for all homogeneous $f$ in $S_+$ if
\begin{proposition}[2.5.7]
\label{II.2.5.7}
Let $d>0$ be an integer, and let $f\in S_d$.
-Then, for all $n\in\bb{Z}$, the $(\sh{O}_X|D_+(f))$-module $(S(nd))^\supertilde|D_+(f)$ is canonically isomorphic to $\sh{O}_X|D_+(f)$.
+Then, for all $n\in\bb{Z}$, the $(\sh{O}_X|D_+(f))$-module $(S(nd))\supertilde|D_+(f)$ is canonically isomorphic to $\sh{O}_X|D_+(f)$.
\end{proposition}
\begin{proof}
@@ -761,12 +761,12 @@ in other words, the $S_{(f)}$-modules $S_{(f)}$ and $S(nd)_{(f)}$ are canonicall
\begin{corollary}[2.5.8]
\label{II.2.5.8}
-On the open subset $U=\bigcup_{f\in S_d}D_+(f)$, the restriction of the $\sh{O}_X$-module $(S(nd))^\supertilde$ is an invertible $(\sh{O}_X|U)$-module \sref[0]{0.5.4.1}.
+On the open subset $U=\bigcup_{f\in S_d}D_+(f)$, the restriction of the $\sh{O}_X$-module $(S(nd))\supertilde$ is an invertible $(\sh{O}_X|U)$-module \sref[0]{0.5.4.1}.
\end{corollary}
\begin{corollary}[2.5.9]
\label{II.2.5.9}
-If the ideal $S_+$ of $S$ is generated by the set $S_1$ of homogeneous elements of degree~$1$, then the $\sh{O}_X$-module $(S(n))^\supertilde$ is invertible for all $n\in\bb{Z}$.
+If the ideal $S_+$ of $S$ is generated by the set $S_1$ of homogeneous elements of degree~$1$, then the $\sh{O}_X$-module $(S(n))\supertilde$ is invertible for all $n\in\bb{Z}$.
\end{corollary}
\begin{proof}
@@ -778,7 +778,7 @@ It suffices to remark that $X=\bigcup_{f\in S_1}D_+(f)$, by the hypothesis \sref
We set, for the rest of this section,
\[
\label{II.2.5.10.1}
- \sh{O}_X(n) = (S(n))^\supertilde
+ \sh{O}_X(n) = (S(n))\supertilde
\tag{2.5.10.1}
\]
for all $n\in\bb{Z}$, and, for every open subset $U$ of $X$, and every $(\sh{O}_X|U)$-module $\sh{F}$,
@@ -820,7 +820,7 @@ It immediately follows from this definition that, if $g\in S_e$ ($e>0$), then th
Thus $\lambda$ induces a canonical functorial homomorphism of $\sh{O}_X$-modules
\[
\label{II.2.5.11.2}
- \lambda: \widetilde{M}\otimes_{\sh{O}_X}\widetilde{N} \to (M\otimes_S N)^\supertilde.
+ \lambda: \widetilde{M}\otimes_{\sh{O}_X}\widetilde{N} \to (M\otimes_S N)\supertilde.
\tag{2.5.11.2}
\]
@@ -831,7 +831,7 @@ the diagram
\label{II.2.5.11.3}
\xymatrix{
\widetilde{\mathfrak{J}}\otimes_{\sh{O}_X}\widetilde{\mathfrak{K}} \ar[rr]^\lambda \ar[dr]
- && (\widetilde{\mathfrak{J}}\otimes_S\widetilde{\mathfrak{K}})^\supertilde \ar[dl]
+ && (\widetilde{\mathfrak{J}}\otimes_S\widetilde{\mathfrak{K}})\supertilde \ar[dl]
\\&\sh{O}_X&
}
\tag{2.5.11.3}
@@ -844,9 +844,9 @@ Finally, note that, if $M$, $N$, and $P$ are graded $S$-modules, then the diagra
\label{II.2.5.11.4}
\xymatrix{
\widetilde{M}\otimes_{\sh{O}_X}\widetilde{N}\otimes_{\sh{O}_X}\widetilde{P} \ar[r]^{\lambda\otimes1} \ar[d]_{1\otimes\lambda}
- & (M\otimes_S N)^\supertilde\otimes_{\sh{O}_X}\widetilde{P} \ar[d]^\lambda
- \\\widetilde{M}\otimes_{\sh{O}_X}(N\otimes_S P)^\supertilde \ar[r]_\lambda
- & (M\otimes_S N\otimes_S P)^\supertilde
+ & (M\otimes_S N)\supertilde\otimes_{\sh{O}_X}\widetilde{P} \ar[d]^\lambda
+ \\\widetilde{M}\otimes_{\sh{O}_X}(N\otimes_S P)\supertilde \ar[r]_\lambda
+ & (M\otimes_S N\otimes_S P)\supertilde
}
\tag{2.5.11.4}
\]
@@ -877,7 +877,7 @@ For $g\in S_e$ ($e>0$), we again have a commutative diagram:
We thus again conclude (taking \sref[I]{I.1.3.8} into account) that the $\mu_f$ define a functorial canonical homomorphism of $\sh{O}_X$-modules
\[
\label{II.2.5.12.2}
- \mu: (\Hom_S(M,N))^\supertilde \to \shHom_{\sh{O}_X}(\widetilde{M},\widetilde{N})
+ \mu: (\Hom_S(M,N))\supertilde \to \shHom_{\sh{O}_X}(\widetilde{M},\widetilde{N})
\tag{2.5.12.2}
\]
\end{env}
@@ -924,7 +924,7 @@ is the isomorphism $z/f^h\mapsto z/f^{h-n}$ from $(N(-n))_{(f)}$ to $N_{(f)}$, a
If the ideal $S_+$ is generated by $S_1$, then we deduce from \sref{II.2.5.13} that, for every graded ideal $\mathfrak{J}$ of $S$, and for every graded $S$-module $M$, we have
\[
\label{II.2.5.13.1}
- \widetilde{\mathfrak{J}}\cdot\widetilde{M} = (\mathfrak{J}\cdot M)^\supertilde
+ \widetilde{\mathfrak{J}}\cdot\widetilde{M} = (\mathfrak{J}\cdot M)\supertilde
\tag{2.5.13.1}
\]
up to canonical isomorphism;
@@ -932,7 +932,7 @@ this follows from the commutativity of the diagram
\[
\xymatrix{
\widetilde{\mathfrak{J}}\otimes_{\sh{O}_X}\widetilde{M} \ar[rr]^\lambda \ar[dr]
- && (\mathfrak{J}\otimes_S M)^\supertilde
+ && (\mathfrak{J}\otimes_S M)\supertilde
\\&\widetilde{M}
}
\]
@@ -967,7 +967,7 @@ Suppose that $S$ is generated by $S_1$.
Then, for every graded $S$-module $M$, and for every $n\in\bb{Z}$, we have
\[
\label{II.2.5.15.1}
- (M(n))^\supertilde = \widetilde{M}(n)
+ (M(n))\supertilde = \widetilde{M}(n)
\tag{2.5.15.1}
\]
up to canonical isomorphism.
@@ -1010,4 +1010,142 @@ By \sref{II.2.5.16}, we can restrict to the case where $d=1$, and the conclusion
\begin{env}[2.6.1]
\label{II.2.6.1}
+The $\sh{O}_X$-module $\sh{O}_X(1)$ is \emph{invertible} \sref{II.2.5.9};
+we thus define, for every $\sh{O}_X$-module $\sh{F}$ \sref[0]{0.5.4.6},
+\[
+\label{II.2.6.1.1}
+ \Gamma_\bullet(\sh{F})
+ = \Gamma_\bullet(\sh{O}_X(1),\sh{F})
+ = \bigoplus_{n\in\bb{Z}} \Gamma(X,\sh{F}(n))
+\tag{2.6.1.1}
+\]
+taking \sref{II.2.5.14.2} into account.
+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 $\Gamma_\bullet(\sh{O}_X)$.
+\end{env}
+
+\begin{env}[2.6.2]
+\label{II.2.6.2}
+Let $M$ be a graded $S$-module;
+for every $f\in S_d$ ($d>0$), $x\mapsto x/1$ is a homomorphism of abelian groups $M_0\to M_{(f)}$, and, since $M_{(f)}$ is canonically identified
+\oldpage[II]{37}
+with $\Gamma(D_+(f),\widetilde{M})$, we thus obtain a homomorphism of abelian groups $\alpha_0^f\colon M_0\to\Gamma(D_+(f),\widetilde{M})$.
+It is clear that, for every $g\in S_e$ ($e>0$), the diagram
+\[
+ \xymatrix{
+ & \Gamma(D_+(f),\widetilde{M}) \ar[dd]
+ \\M_0 \ar[ur]^{\alpha_0^f} \ar[dr]_{\alpha_0^{fg}}
+ \\& \Gamma(D_+(fg),\widetilde{M})
+ }
+\]
+commutes;
+this implies that, for all $x\in M_0$, the sections $\alpha_0^f(x)$ and $\alpha_0^g(x)$ of $M$ agree on $D_+(f)\cap D_+(g)$, and thus there exists a unique section $\alpha_0(x)\in\Gamma(X,\widetilde{M})$ whose restriction to each $D_+(f)$ is $\alpha_0^f(x)$.
+We have thus defined (without using the hypothesis that $S$ be generated by $S_1$) a homomorphism of abelian groups
+\[
+\label{II.2.6.2.1}
+ \alpha_0\colon M_0\to\Gamma(X,\widetilde{M}).
+\tag{2.6.2.1}
+\]
+
+Applying this result to the graded $S$-module $M(n)$ (for each $n\in\bb{Z}$), we obtain, for each $n\in\bb{Z}$, a homomorphism of abelian groups
+\[
+\label{II.2.6.2.2}
+ \alpha_n\colon M_n=(M(n))_0\to\Gamma(X,\widetilde{M}(n))
+\tag{2.6.2.2}
+\]
+(taking \sref{II.2.5.15});
+whence we obtain a functorial homomorphism (of degree~$0$) of graded abelian groups
+\[
+\label{II.2.6.2.3}
+ \alpha\colon M\to\Gamma_\bullet(\widetilde{M})
+\tag{2.6.2.3}
+\]
+(also denoted by $\alpha_M$) which, on each $M_n$, agrees with $\alpha_n$.
+
+If we take, in particular, $M=S$, then we immediately see (taking into account the definition \sref[0]{0.5.4.6} of multiplication in $\Gamma_\bullet(\sh{O}_X)$) that $\alpha\colon S\to\Gamma_\bullet(\sh{O}_X)$ is a homomorphism of graded rings, and that, for every graded $S$-module $M$, \sref{II.2.6.2.3} is a di-homomorphism of graded modules.
+\end{env}
+
+\begin{proposition}[2.6.3]
+\label{II.2.6.3}
+For every $f\in S_d$ ($d>0$), $D_+(f)$ is identical to the set of $\mathfrak{p}\in X$ on which the section $\alpha_d(f)$ of $\sh{O}_X(d)$ does not vanish \sref[0]{0.5.5.2}.
+\end{proposition}
+
+\begin{proof}
+Since $X=\bigcup_{g\in S_1}D_+(g)$ by hypothesis, it suffices to show that, for all $g\in S_1$, the set of $\mathfrak{p}\in D_+(g)$ on which $\alpha_d(f)$ does not vanish is identical to $D_+(fg)$.
+But the restriction of $\alpha_d(f)$ to $D_+(g)$ is, by definition, the section corresponding to the element $f/1$ of $(S(d))_{(g)}$;
+under the canonical isomorphism $(S(d))_{(g)}\xrightarrow{\sim}S_{(g)}$ \sref{II.2.5.7}, this section of $\sh{O}_X(d)$ over $D_+(g)$ is identified with the section of $\sh{O}_X$ over $D_+(g)$ that corresponds to the element $f/g^d$ of $S_{(g)}$;
+to say that this section vanishes at $\mathfrak{p}\in D_+(g)$ implies that $f/g^d\in\mathfrak{q}$, where $\mathfrak{q}$ is the prime ideal of $S_{(g)}$ corresponding to $\mathfrak{p}$ \sref{II.2.3.6};
+by definition, this implies that $f\in\mathfrak{p}$, whence the proposition.
+\end{proof}
+
+\begin{env}[2.4.6]
+\label{II.2.4.6}
+Now let $\sh{F}$ be an $\sh{O}_X$-modules, and set $M=\Gamma_\bullet(\sh{F})$;
+by the existence of the homomorphism of graded rings $\alpha\colon S\to\Gamma_\bullet(\sh{O}_X)$, 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 $z\in M_{nd}=\Gamma(X,\sh{F}(nd)$ ($n>0$);
+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 $z|D_+(f)=0$.
+This shows that we have defined an $S_{(f)}$-homomorphism $\beta_f\colon M_{(f)}\to\Gamma(D_+(f),\sh{F})$ by sending the element $z/f^n$ to the section $(z|D_+(f))(\alpha_d(f^n)|D_+(f))^{-1}$ of $\sh{F}$ over $D_+(f)$.
+We can further immediately show that the diagram
+\[
+\label{II.2.6.4.1}
+ \xymatrix{
+ M_{(f)} \ar[r]^{\beta_f} \ar[d]
+ & \Gamma(D_+(f),\sh{F}) \ar[d]
+ \\M_{(fg)} \ar[r]_{\beta_{fg}}
+ & \Gamma(D_+(fg),\sh{F})
+ }
+\tag{2.6.4.1}
+\]
+commutes for $g\in S_e$ ($e>0$).
+If we recall that $M_{(f)}$ is canonically identified with $\Gamma(D_+(f),\widetilde{M})$, and that the $D_+(f)$ form a base for the topology of $X$ \sref{II.2.3.4}, then we see that the $\beta_f$ come from a unique canonical homomorphism of $\sh{O}_X$-modules
+\[
+\label{II.2.6.4.2}
+ \beta\colon(\Gamma_\bullet(\sh{F}))\supertilde\to\sh{F}
+\tag{2.6.4.2}
+\]
+(also denoted by $\beta_{\sh{F}}$) which is evidently functorial.
\end{env}
+
+\begin{proposition}[2.6.5]
+\label{II.2.6.5}
+Let $M$ be a graded $S$-module, and $\sh{F}$ an $\sh{O}_X$-module;
+then the composite homomorphisms
+\[
+\label{II.2.6.5.1}
+ \widetilde{M}
+ \xrightarrow{\widetilde{\alpha}} (\Gamma_\bullet(\widetilde{M}))\supertilde
+ \xrightarrow{\beta} \widetilde{M}
+\tag{2.6.5.1}
+\]
+\[
+\label{II.2.6.5.2}
+ \Gamma_\bullet(\sh{F})
+ \xrightarrow{\alpha} \Gamma_\bullet((\Gamma_\bullet(\sh{F}))\supertilde)
+ \xrightarrow{\Gamma_\bullet(\beta)} \Gamma_\bullet(\sh{F})
+\tag{2.6.5.2}
+\]
+are the identity isomorphisms.
+\end{proposition}
+
+\begin{proof}
+The proof for \sref{II.2.6.5.1} is local:
+in an open subset $D_+(f)$, it follows immediately from the definitions, along with the fact that $\beta$, applied to quasi-coherent sheaves, is determined by its action on the sections over $D_+(f)$ \sref[I]{I.1.3.8}.
+The proof for \sref{II.2.6.5.2} is done for each degree separately:
+if we set $M=\Gamma_\bullet(\sh{F})$, then $M_n=\Gamma(X,\sh{F}(n))$, and $(\Gamma_\bullet(\widetilde{M}))_n=\Gamma(X,\widetilde{M}(n))=\Gamma(X,(M(n))\supertilde)$.
+But if $f\in S_1$ and $z\in M_n$, then $\alpha_n^f(z)$ is the element $z/1$ of $(M(n))_{(f)}$, equal to $(f/1)^n(z/f^n)$;
+it corresponds, via $\beta_f$, to the section
+\[
+ \Big(\big(\alpha_1(f)\big)^n|D_+(f)\Big) \Big(\big(z|D_+(f)\big)\big((\alpha_1(f))^n|D_+(f)\big)^{-1}\Big)
+\]
+over $D_+(f)$, i.e. the restriction of $z$ to $D_+(f)$, which finishes the proof for \sref{II.2.6.5.2}.
+\end{proof}
+
+
+\subsection{Finiteness conditions}
+\label{subsection:II.2.7}
diff --git a/ega3/ega3-1.tex b/ega3/ega3-1.tex
index 6f521e1..f7fec43 100644
--- a/ega3/ega3-1.tex
+++ b/ega3/ega3-1.tex
@@ -444,24 +444,24 @@ corresponding to an exact sequence $0\to\sh{F}'\to\sh{F}\to\sh{F}''\to 0$ of qua
\begin{proposition}[1.4.5]
\label{III.1.4.5}
\oldpage[III]{90}
-Let $X$ be a quasi-compact scheme, $\sh{L}$ an invertible $\sh{O}_X$-module, and consider the graded ring $A_*=\Gamma_*(\sh{L})$ \sref[0]{0.5.4.6}; then $\HH^\bullet(\sh{F},\sh{L})=\bigoplus_{n\in\bb{Z}}\HH^\bullet(X,\sh{F}\otimes\sh{L}^{\otimes n})$ is a graded $A_*$-module, and for all $f\in A_n$, we have a canonical isomorphism
+Let $X$ be a quasi-compact scheme, $\sh{L}$ an invertible $\sh{O}_X$-module, and consider the graded ring $A_\bullet=\Gamma_\bullet(\sh{L})$ \sref[0]{0.5.4.6}; then $\HH^\bullet(\sh{F},\sh{L})=\bigoplus_{n\in\bb{Z}}\HH^\bullet(X,\sh{F}\otimes\sh{L}^{\otimes n})$ is a graded $A_\bullet$-module, and for all $f\in A_n$, we have a canonical isomorphism
\[
\HH^\bullet(X_f,\sh{F})\isoto(\HH^\bullet(\sh{F},\sh{L}))_{(f)}
\tag{1.4.5.1}
\]
-of $(A_*)_{(f)}$-modules.
+of $(A_\bullet)_{(f)}$-modules.
\end{proposition}
\begin{proof}
As $X$ is a quasi-compact scheme, we can calculate the cohomology of all the $\sh{O}_X$-modules $\sh{F}\otimes\sh{L}^{\otimes n}$ using the same finite cover $\mathfrak{U}=(U_i)$ consisting of the affine open sets such that the restriction $\sh{L}|U_i$ is isomorphic to $\sh{O}_X|U_i$ for each $i$ \sref{III.1.4.1}.
It is then immediate that the $U_i\cap X_f$ are affine open sets \sref[I]{3.1.3.6}, and we can thus calculate the cohomology $\HH^\bullet(X_f,\sh{F}\otimes\sh{L}^{\otimes n})$ using the cover $\mathfrak{U}|X_f=(U_i\cap X_f)$ \sref{III.1.4.1}.
It is immediate that for all $f\in A_n$, multiplication by $f$ defines a homomorphism $C^\bullet(\mathfrak{U},\sh{F}\otimes\sh{L}^m)\to C^\bullet(\mathfrak{U},\sh{F}\otimes\sh{L}^{\otimes(m+n)})$, hence a homomorphism $\HH^\bullet(\mathfrak{U},\sh{F}\otimes\sh{L}^{\otimes m})\to\HH^\bullet(\mathfrak{U},\sh{F}\otimes\sh{L}^{\otimes(m+n)})$, which establishes the first assertion.
-On the other hand, for a given $f\in A_n$, it follows from \sref[I]{I.9.3.2} that we have an isomorphism of complexes of $(A_*)_{(f)}$-modules
+On the other hand, for a given $f\in A_n$, it follows from \sref[I]{I.9.3.2} that we have an isomorphism of complexes of $(A_\bullet)_{(f)}$-modules
\[
C^\bullet(\mathfrak{U}|X_f,\sh{F})\isoto\bigg(\!\!C^\bullet\bigg(\mathfrak{U},\bigoplus_{n\in\bb{Z}}\sh{F}\otimes\sh{L}^{\otimes n}\bigg)\!\!\bigg)_{(f)},
\]
taking into account \sref[I]{I.1.3.9}[ii].
-Passing to the cohomology of these complexes, we induce the isomorphism (1.4.5.1), recalling that the functor $M\mapsto M_{(f)}$ is exact on the category of graded $A_*$-modules.
+Passing to the cohomology of these complexes, we induce the isomorphism (1.4.5.1), recalling that the functor $M\mapsto M_{(f)}$ is exact on the category of graded $A_\bullet$-modules.
\end{proof}
\begin{corollary}[1.4.6]
diff --git a/preamble-base.tex b/preamble-base.tex
index d6e2c47..ce008e1 100644
--- a/preamble-base.tex
+++ b/preamble-base.tex
@@ -18,7 +18,7 @@
\def\op{^\cat{op}} % opposite category
\def\Set{\cat{Set}} % category of sets
\def\CHom{\cat{Hom}} % functor category
-\def\supertilde{{\,\widetilde{\,}\,}} % use ^\supertilde instead of ^\sim
+\def\supertilde{{\,\widetilde{\,}\,}} % use \supertilde instead of ^\sim
\def\GL{\bb{GL}}
\def\red{\mathrm{red}}
\def\rg{\operatorname{rg}}