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authorGravatar Tim Hosgood <timhosgood@googlemail.com> 2020-03-05 17:50:17 +0100
committerGravatar Tim Hosgood <timhosgood@googlemail.com> 2020-03-06 01:37:18 +0100
commit1cd366ab158dba94504b05135910ab794dd4faf8 (patch)
tree40c4ffa3e8068bb55bcb45d4876ebed4aaa41223
parent129d053d367f5102b7f61237d3f8abc5651c060b (diff)
downloadega-1cd366ab158dba94504b05135910ab794dd4faf8.tar.gz
ega-1cd366ab158dba94504b05135910ab794dd4faf8.zip
finished 2.8.13
-rw-r--r--ega2/ega2-8.tex108
1 files changed, 92 insertions, 16 deletions
diff --git a/ega2/ega2-8.tex b/ega2/ega2-8.tex
index 7df9832..79d752e 100644
--- a/ega2/ega2-8.tex
+++ b/ega2/ega2-8.tex
@@ -2130,7 +2130,7 @@ Then the following all hold true.
\end{lemma}
\begin{proof}
-\label{proof-2.8.10.1}
+\label{proof-II.8.10.1}
Note first of all that the support of the $\sh{O}_V$-module $\sh{O}_V/\sh{J}^{n+1}$ is $j(X)$, and the support of $\sh{J}/\sh{J}^{n+1}$ is contained in $j(X)$.
In the case of (ii), $j(X)$ is a closed point $j(\xi)$ of $V$,
\oldpage[II]{185}
@@ -2193,7 +2193,7 @@ Finally, $\sh{J}'$ is an $\sh{O}_{V'}$-module that is canonically isomorphic to
\end{lemma}
\begin{proof}
-\label{proof-2.8.10.2}
+\label{proof-II.8.10.2}
The fact that $j'$ is a closed immersion follows from \sref[I]{I.4.3.1}, and it is an $X'$-section of $V'$ by functoriality of extension of the base prescheme.
Furthermore, if $Z$ (resp. $Z'$) is the closed subprescheme of $V$ (resp. $V'$) associated to $j$ (resp. $j'$), then $Z'=g^{'-1}(Z)$ \sref[I]{I.4.3.1}, and the second claim then follows from \sref[I]{I.4.4.5}.
To prove the other claims, we see, as in \sref{II.8.10.1}, that we can restrict to the case where $X$, $V$, and $X'$ (and thus also $V'$) are affine;
@@ -2211,7 +2211,7 @@ furthermore, there exists exactly one irreducible component of $\pi^{-1}(x)$ tha
\end{corollary}
\begin{proof}
-\label{proof-2.8.10.3}
+\label{proof-II.8.10.3}
Since $\pi^{-1}(x)=V\times_X\Spec(\kres(x))$, we are led, by \sref{II.8.10.2}, to the case where $X$ is the spectrum of a field $K$.
Since $\pi$ is of finite type \sref[I]{I.6.4.3}[iv], $\sh{O}_{j(x)}$ is a Noetherian local ring, and thus separated for the $\fk{m}_{j(x)}$-preadic topology \sref[0]{0.7.3.5};
it follows from \sref{II.8.10.1}[ii and iii] that the completion of this ring is isomorphic to $K[[T]]$, and so $\sh{O}_{j(x)}$ is regular and of dimension 1 (\cite[p.~17-01, th.~1]{I-1});
@@ -2227,7 +2227,7 @@ Then $\sh{K}_V$ is the largest quasi-coherent ideal of $\sh{O}_V$ whose restrict
\end{corollary}
\begin{proof}
-\label{proof-2.8.10.4}
+\label{proof-II.8.10.4}
Indeed, we see as in \sref{II.8.10.1} that the question is local on $X$ and $V$;
we can thus reuse the notation from the proof of \sref{II.8.10.1}, with $\fk{J}=Bt$, where $t$ is not a zero divisor in $B$.
Furthermore, we have $W=\Spec(B_t)$ and $\sh{K}=\widetilde{\fk{K}}$, where $\fk{K}$ is an ideal of $A$;
@@ -2256,7 +2256,7 @@ let $W=V\setmin j(X)$, $x$ be a point of $X$, $\sh{K}$ a quasi-coherent ideal of
\end{corollary}
\begin{proof}
-\label{proof-2.8.10.5}
+\label{proof-II.8.10.5}
\begin{enumerate}
\item[\rm{(i)}] Since the ideal of $\sh{O}_W$ generated by $g|W$ is contained in $\sh{K}_W$ by hypothesis, the ideal of $\sh{O}_V$ generated by $g$ is contained in $\sh{K}_V$ by \sref{II.8.10.4}, or, in other words, $g$ is a section of $\sh{K}_V$.
To prove the second claim of (i), we can again assume that $X$ and $V$ are affine, and reuse the notation from \sref{II.8.10.1};
@@ -2335,7 +2335,7 @@ It is immediate that the ideal $\sh{J}$ of $\sh{O}_V$ that defines $X$ is again
\subsection{Uniqueness of contractions}
-\label{subsection:2.8.11}
+\label{subsection:II.8.11}
\begin{lemma}[8.11.1]
\label{II.8.11.1}
@@ -2355,7 +2355,7 @@ that, to each morphism $v=(v_0,\nu)$ from $V$ to $W$, associates the morphism $u
\end{lemma}
\begin{proof}
-\label{proof-2.8.11.1}
+\label{proof-II.8.11.1}
It is clear that, if $u=v\circ h$, so that $u_0=v_0\circ h_0$, then $u_0$ is constant on every set $h_0^{-1}(x)$.
Conversely, if $u$ has this property, we will show that there exists exactly one $v\in\Hom(V,W)$ such that $u=v\circ h$.
The existence and uniqueness of the continuous map $v_0:V\to W$ such that $u_0=v_0\circ h_0$ follows from the hypotheses, since $h_0$ can be identified with the canonical map from $U$ to $U/R$.
@@ -2380,7 +2380,7 @@ then every morphism $h:U\to V$ that is universally closed, birational, and radic
\end{corollary}
\begin{proof}
-\label{proof-2.8.11.2}
+\label{proof-II.8.11.2}
If $h=(h_0,\lambda)$, then it follows from the hypotheses that $h_0$ is injective and closed, and that $h_0(U)$
\oldpage[II]{190}
is dense in $V$, and so $h_0$ is a \emph{homeomorphism} from $U$ to $V$.
@@ -2412,7 +2412,7 @@ Then there exists exactly one $Y$-morphism $u:C'\to C$ such that $q=u\circ q'$.
\end{lemma}
\begin{proof}
-\label{proof-2.8.11.5}
+\label{proof-II.8.11.5}
The hypothesis on $\theta$ implies, in particular, that $p$ is surjective;
since, by \sref{II.8.8.4}, the restriction of $q'$ to $\bb{V}(\sh{L})\setmin j(X)$ is an \emph{isomorphism} to $C'\setmin\varepsilon'(Y)$ (where $\varepsilon$ is the vertex section of $C'$), it follows from \sref{II.8.8.4} that $q'$ is \emph{proper} and \emph{surjective};
furthermore, by \sref{II.8.8.6}, if we let $q'=(q'_0,\tau)$, then $\tau:\sh{O}_{C'}\to q'_*(\sh{O}_V)$ is an isomorphism.
@@ -2429,7 +2429,7 @@ Then the morphism $u$ is universally closed, surjective, and radicial, and its r
\end{corollary}
\begin{proof}
-\label{proof-2.8.11.6}
+\label{proof-II.8.11.6}
Since $q'$ is an isomorphism from $\bb{V}(\sh{L})\setmin j(X)$ to $C'\setmin\varepsilon'(Y)$ \sref{II.8.8.4}, the last claim follows immediately from the fact that $q=u\circ q'$.
Furthermore, the commutativity of the diagrams
\oldpage[II]{191}
@@ -2449,7 +2449,7 @@ Then the unique $Y$-morphism $u:C'\to C$ such that $q=u\circ q'$ is an \emph{iso
\end{proposition}
\begin{proof}
-\label{proof-2.8.11.7}
+\label{proof-II.8.11.7}
It follows from \sref{II.8.8.6} that $C'$ is integral;
since $u$ is a homeomorphism of the underlying subspaces $C'\to C$ ($u$ being bijective and closed, by \sref{II.8.11.6}), $C$ is irreducible, thus integral, and, since the restriction of $u$ to a non-empty open subset of $C'$ is an isomorphism to an open subset of $C$, $u$ is birational.
Since $C$ is assumed to be normal, it suffices to apply \sref{II.8.11.2} to obtain the conclusion.
@@ -2475,7 +2475,7 @@ Since $C$ is assumed to be normal, it suffices to apply \sref{II.8.11.2} to obta
\subsection{Quasi-coherent sheaves on based cones}
-\label{subsection:2.8.12}
+\label{subsection:II.8.12}
\begin{env}[8.12.1]
\label{II.8.12.1}
@@ -2645,7 +2645,7 @@ The restrictions to $\widehat{E}_X$ and $E_X$ of the canonical $r$-morphism \sre
\end{proposition}
\begin{proof}
-\label{proof-2.8.12.6}
+\label{proof-II.8.12.6}
We restrict to the case where $Y$ is affine, as in the proof of \sref{II.8.6.2} (whose notation we adopt);
by reducing to definitions \sref{II.2.8.8}, we have to show that the canonical homomorphism
\[
@@ -2662,7 +2662,7 @@ With the identifications of \sref{II.8.6.3}, the restriction of $(\sh{M}_X^\geq)
\end{corollary}
\begin{proof}
-\label{proof-2.8.12.7}
+\label{proof-II.8.12.7}
We can restrict to the affine case, and this follows from the identification of $(M_f^\geq)_{f/1}^\leq$ with $M_f^\leq$, and of $(M_f^\geq)_{f/1}$ with $M_f$ \sref{II.8.2.9.2}.
\end{proof}
@@ -2672,7 +2672,7 @@ Under the hypotheses of \sref{II.8.6.4}, the canonical homomorphism \sref{II.8.1
\end{proposition}
\begin{proof}
-\label{proof-2.8.12.8}
+\label{proof-II.8.12.8}
Taking into account the fact that $\Proj(\sh{S}_X^\geq)\to X$ is an isomorphism \sref{II.8.6.2}, and the
\oldpage[II]{195}
isomorphisms \sref{II.8.12.5.4} and \sref{II.8.12.6.1}, we are led to proving the corresponding proposition for the canonical homomorphism $p_X^*(\shProj_0(\sh{M}_X^\geq))\to(\sh{M}_X^\geq)^\square|E_X$, or, in other words, we can restrict to the case where $\sh{S}_1$ is an invertible $\sh{O}_Y$-module, and where $\sh{S}$ is generated by $\sh{S}_1$.
@@ -2707,7 +2707,83 @@ There exists a canonical $h$-isomorphism
\end{proposition}
\begin{proof}
-\label{proof-2.8.12.10}
+\label{proof-II.8.12.10}
We argue as in \sref{II.8.7.3}, this time using the existence of the di-isomorphism \sref{II.8.2.9.3} instead of \sref{II.8.2.7.3}.
We leave the details to the reader.
\end{proof}
+
+
+\subsection{Projective closures of subsheaves and closed subschemes}
+\label{subsection:II.8.13}
+
+\begin{env}[8.13.1]
+\label{II.8.13.1}
+With hypotheses and notation as in \sref{II.8.12.1}, consider a \emph{not-necessarily graded} quasi-coherent sub-$\sh{S}$-module $\sh{N}$ of $\sh{M}$.
+We can then consider the quasi-coherent $\sh{O}_C$-module $\widetilde{\sh{N}}$ associated to $\sh{N}$, which is a sub-$\sh{O}_C$-module of $\widetilde{\sh{M}}$.
+We have seen elsewhere \sref{II.8.12.2.1} that $\widetilde{\sh{M}}$ can be identified with the restriction of $\sh{M}^\square$ to $C$.
+Since the canonical injection $i:C\to\widehat{C}$ is an affine morphism \sref{II.8.3.2}, and \emph{a fortiori} quasi-compact, the \emph{canonical extension} $(\widetilde{\sh{N}})^-$, the largest sub-$\sh{O}_{\widehat{C}}$-module contained in $\sh{M}^\square$ and inducing $\widetilde{\sh{N}}$ on $C$, is a \emph{quasi-coherent} $\sh{O}_{\widehat{C}}$-module \sref[I]{I.9.4.2}.
+We will give a more explicit description by using a graded $\widehat{\sh{S}}$-module.
+\end{env}
+
+\begin{env}[8.13.2]
+\label{II.8.13.2}
+For this, consider, for every integer $n\geq0$, the homomorphism $\bigoplus_{i\leq n}\sh{M}_i\to\sh{M}$ which, for every open $U$ of $Y$, sends the family
+\[
+ (s_i) \in \bigoplus_{i\leq n}\Gamma(U,\sh{M}_i)
+\]
+to the section $\sum_i s_i\in\Gamma(U,\sh{M})$.
+Denote by $\sh{N}'_n$ the inverse image of $\sh{N}$ by this homomorphism, which is a quasi-coherent sub-$\sh{S}$-module of $\bigoplus_{i\leq n}\sh{M}_i$.
+Now consider the homomorphism $\bigoplus_{i\leq n}\sh{M}_i\to\widehat{\sh{M}}=\sh{M}[\bb{z}]$ which sends $(s_i)$ to the section $\sum_{i\leq n}s_i\bb{z}^{n-i}\in\Gamma(U,\widehat{\sh{M}}_n)$, and let $\sh{N}_n$ be the image of $\sh{N}'_n$ under this homomorphism;
+we immediately have that $\overline{\sh{N}}=\bigoplus_{n\geq0}\sh{N}_n$ is a (quasi-coherent) sub-$\widehat{\sh{S}}$-module of $\widehat{\sh{M}}$;
+we say that $\overline{\sh{N}}$ is induced from $\sh{N}$ by \emph{homogenisation}, via the ``homogenising variable'' $\bb{z}$.
+We note
+\oldpage[II]{196}
+that, if $\sh{N}$ is already a \emph{graded} sub-$\sh{S}$-module of $\sh{M}$, then $\overline{\sh{N}}$ can be identified with the direct sum of the components $\widehat{\sh{N}}_n$ of degree $n\geq0$ in $\widehat{\sh{N}}=\sh{N}[\bb{z}]$.
+\end{env}
+
+\begin{proposition}[8.13.3]
+\label{II.8.13.3}
+The $\sh{O}_{\widehat{C}}$-module $\shProj_0(\overline{\sh{N}})$ is the canonical extension $(\widetilde{\sh{N}})^-$ of $\widetilde{\sh{N}}$ to $\widehat{C}$.
+\end{proposition}
+
+\begin{proof}
+\label{proof-II.8.13.3}
+The question is local on $Y$ and $\widehat{C}$ by the definition of the canonical extension \sref[I]{I.9.4.1}.
+We can thus already suppose that $Y=\Spec(A)$ is affine, with $\sh{S}=\widetilde{S}$, $\sh{M}=\widetilde{M}$, and $\sh{N}=\widetilde{N}$, where $N$ is a non-necessarily-graded sub-$S$-module of $M$.
+Furthermore \sref{II.8.3.2.6}, $\widehat{C}$ is a union of affine opens $\widehat{C}_z=C$ and $\widehat{C}_f=\Spec(S_f^\leq)$ (with $f$ homogeneous in $S_+$).
+It thus suffices to show that: (1) the restriction of $\shProj_0(\overline{\sh{N}})$ to $C$ is $\widetilde{\sh{N}}$; (2) the restriction of $\shProj_0(\overline{\sh{N}})$ to each $\widehat{C}_f$ is the canonical extension of the restriction of $\sh{N}$ to $C\cap\widehat{C}_f=\Spec(S_f)$ \sref{II.8.3.2.6}.
+For the first point, note that $\shProj_0(\overline{\sh{N}})|C$ can be identified with $(\overline{N}_{(\bb{z})})^\sim$ \sref{II.8.3.2.4};
+but $\overline{N}_{(\bb{z})}$ is canonically identified \sref{II.2.2.5} with the image of $\overline{N}$ in $\widehat{M}/(\bb{z}-1)\widehat{M}$, and by the canonical isomorphism of the latter with $M$ \sref{II.8.2.5}, this image can be identified with $N$, by the definition of $\overline{N}$ given in \sref{II.8.13.2}.
+
+To prove the second point, note that the injection $i:C\cap\widehat{C}_f\to\widehat{C}$ corresponds to the canonical injection $S_f^\leq\to S_f$ \sref{II.8.3.2.6};
+we also have that $\Gamma(\widehat{C}_f,\sh{M}^\square)=M_f^\leq$, that $\Gamma(\widehat{C}_f,i_*(\widetilde{\sh{N}}))=N$, and, by \sref{II.8.12.2.1}, that $\Gamma(\widehat{C}_f,i_*(i^*(\sh{M}^\square)))=M_f$.
+Taking \sref[I]{I.9.4.2} into account, we are thus led to showing that $\overline{N}_{(f)}\subset\widehat{M}_{(f)}=M_f^\leq$ is canonically identified with the inverse image of $N_f$ under the canonical injection $M_f^\leq\to M_f$.
+Indeed, let $d=\deg(f)>0$, and suppose that an element $(\sum_{k\leq md}x_k)/f^m$ of $M_f$ (with $x_k\in M_k$) is of the form $y/f^m$ with $y\in N$.
+By multiplying $y$ and the $x_k$ by one single suitable $f^h$, we can already assume that $\sum_{k\leq md}x_k=y$.
+But in the identification of \sref{II.8.2.5.2}, $(\sum_{k\leq md}x_k)/f^m$ corresponds to $\sum_{k\leq md}x_k\bb{z}^{md-k}/f^m$, and this is indeed an element of $\overline{N}_{(f)}$, since $\sum_{k\leq md}x_k\in N$;
+the converse is evident.
+\end{proof}
+
+\begin{remark}[8.13.4]
+\label{II.8.13.4}
+\begin{enumerate}
+ \item[\rm{(i)}] The most important case of application of \sref{II.8.13.3} is that where $\sh{M}=\sh{S}$, with $\widetilde{\sh{N}}$ then being an \emph{arbitrary} quasi-coherent sheaf of ideals $\sh{J}$ of $\sh{O}_C$ \sref{II.1.4.3}, corresponding bijectively to a \emph{closed subprescheme} $Z$ of $C$.
+ Then the canonical extension $\overline{\sh{J}}$ of $\sh{J}$ is the quasi-coherent sheaf of ideals of $\sh{O}_{\widehat{C}}$ that defines the \emph{closure} $\overline{Z}$ of $Z$ in $\widehat{C}$ \sref[I]{I.9.5.10};
+ Proposition~\sref{II.8.13.3} gives a canonical way of defining $\overline{Z}$ by using a graded ideal in $\widehat{\sh{S}}=\sh{S}[\bb{z}]$.
+ \item[\rm{(ii)}] Suppose, to simplify things, that $Y$ is affine, and adopt the notation from the proof of \sref{II.8.13.3}.
+ For every non-zero $x\in N$, let $d(x)$ be the largest degree of the homogeneous components $x_i$ of $x$ in $M$;
+ by definition, $\overline{N}$ is the submodule of $\widehat{M}$ consisting of $0$ and elements of the form $h(x,k)=\bb{z}^k\sum_{i\leq d(x)}x_i\bb{z}^{d(x)-i}$ (for integral $k\geq0$);
+ it is thus generated, as a module over $\widehat{S}=S[\bb{z}]$, by the elements of the form
+ \[
+ h(x,0) = \sum_{i\leq d(x)}x_i\bb{z}^{d(x)-i}.
+ \]
+ \oldpage[II]{197}
+ We say that $h(x,0)$ is induced from $x$ by \emph{homogenisation} via the ``homogenising variable'' $\bb{z}$.
+ But since $h(x,0)$ does not depend additively on $x$ (nor \emph{a fortiori} $S$-linearly), \emph{we will refrain from believing} (even when $M=S$) that the $h(x,0)$ form a \emph{system of generators} of the graded $\widehat{S}$-module $\overline{N}$ when we let $x$ run over a \emph{system of generators} of the $S$-module $N$.
+ This is, however, the case (considered only in elementary algebraic geometry) when $N$ is a \emph{free cyclic} $S$-module, since, if $t$ is a basis of $N$, then $h(t,0)$ generates the $\widehat{S}$-module $\overline{N}$.
+\end{enumerate}
+\end{remark}
+
+
+\subsection{Supplement on sheaves associated to graded $\sh{S}$-modules}
+\label{subsection:II.8.14}