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author | Tim Hosgood <thosgood@users.noreply.github.com> | 2020-03-02 18:58:05 +0100 |
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committer | GitHub <noreply@github.com> | 2020-03-02 18:58:05 +0100 |

commit | 625c6e2a3b0cfdaa4dc14f7b2f05ac0c342be9b0 (patch) | |

tree | 2a8372c47728e66a96e29eb8e6b263ee3197ef76 | |

parent | 1baedc662d1e8ca3b75d5e7f609a85d99a13ebcc (diff) | |

parent | 3c3cb2aee8a6bd74aeced15300d84b4314991a2a (diff) | |

download | ega-625c6e2a3b0cfdaa4dc14f7b2f05ac0c342be9b0.tar.gz ega-625c6e2a3b0cfdaa4dc14f7b2f05ac0c342be9b0.zip |

Merge pull request #134 from ryankeleti/ega2-8

almost finished II.8

-rw-r--r-- | README.md | 4 | ||||

-rw-r--r-- | ega2.tex | 4 | ||||

-rw-r--r-- | ega2/ega2-8.tex | 571 |

3 files changed, 561 insertions, 18 deletions

@@ -1,6 +1,6 @@ # EGA -![EGA0status](https://img.shields.io/badge/EGA%200-46%25-yellow) ![EGA1status](https://img.shields.io/badge/EGA%20I-100%25-brightgreen) ![EGA2status](https://img.shields.io/badge/EGA%20II-30%25-red) ![EGA3status](https://img.shields.io/badge/EGA%20III-3%25-red) ![EGA4status](https://img.shields.io/badge/EGA%20IV-1%25-red) +![EGA0status](https://img.shields.io/badge/EGA%200-46%25-yellow) ![EGA1status](https://img.shields.io/badge/EGA%20I-100%25-brightgreen) ![EGA2status](https://img.shields.io/badge/EGA%20II-35%25-orange) ![EGA3status](https://img.shields.io/badge/EGA%20III-3%25-red) ![EGA4status](https://img.shields.io/badge/EGA%20IV-1%25-red) Community translation (French to English) of A. Grothendieck's EGA. S’il-vous plaît pardonnez-nous, Grothendieck. @@ -28,7 +28,7 @@ All the PDFs are auto-compliled every hour if any changes have been made since t | - | - | - | | ![EGA0](https://img.shields.io/badge/EGA-0-lightgrey) | ![EGA0fd](https://img.shields.io/badge/-46%25-yellow) | ![EGA0p](https://img.shields.io/badge/-7%25-red)| | ![EGA1](https://img.shields.io/badge/EGA-1-lightgrey) | ![EGA1fd](https://img.shields.io/badge/-100%25-green) | ![EGA1p](https://img.shields.io/badge/-100%25-green)| -| ![EGA2](https://img.shields.io/badge/EGA-2-lightgrey) | ![EGA2fd](https://img.shields.io/badge/-30%25-orange) | ![EGA2p](https://img.shields.io/badge/-0%25-red)| +| ![EGA2](https://img.shields.io/badge/EGA-2-lightgrey) | ![EGA2fd](https://img.shields.io/badge/-35%25-orange) | ![EGA2p](https://img.shields.io/badge/-0%25-red)| | ![EGA3](https://img.shields.io/badge/EGA-3-lightgrey) | ![EGA3fd](https://img.shields.io/badge/-3%25-red) | ![EGA3p](https://img.shields.io/badge/-0%25-red)| | ![EGA4](https://img.shields.io/badge/EGA-4-lightgrey) | ![EGA4fd](https://img.shields.io/badge/-1%25-red) | ![EGA4p](https://img.shields.io/badge/-0%25-red)| @@ -24,8 +24,8 @@ \bigskip \oldpage[II]{5} -The various classes of morphisms studied in this chapter are used extensively in cohomological methods; further study, using these methods, will be done in Chapter~III, where we use especially \textsection\textsection2, 4, and 5 of Chapter~II. -Section \textsection8 can be omitted on a first reading: it gives some supplements to the formalism developed in \textsection\textsection1 and 3, reducing to easy applications of this formalism, and we will use it less consistently than the other results of this chapter. +The various classes of morphisms studied in this chapter are used extensively in cohomological methods; further study using these methods will be done in Chapter~III, where we make particular use of \textsection\textsection2, 4, and 5 of Chapter~II. +On a first reading, \textsection8 can be omitted: it supplements the formalism developed in \textsection\textsection1 and 3, reducing to easy applications of this formalism, and we will use it less consistently than the other results of this chapter. \bigskip \input{ega2/ega2-1} diff --git a/ega2/ega2-8.tex b/ega2/ega2-8.tex index 994a690..7df9832 100644 --- a/ega2/ega2-8.tex +++ b/ega2/ega2-8.tex @@ -40,7 +40,7 @@ If we then replace the $\sh{I}_n$ by the $\sh{I}_n\sh{J}^n$, and, in doing so, r \begin{env}[8.1.2] \label{II.8.1.2} -Suppose that $Y$ is \emph{locally integral}, so that the sheaf $\sh{R}(Y)$ of rational functions is a quasi-coherent $\sh{O}_Y$-algebra \sref[1]{1.7.3.7}. +Suppose that $Y$ is \emph{locally integral}, so that the sheaf $\sh{R}(Y)$ of rational functions is a quasi-coherent $\sh{O}_Y$-algebra \sref[1]{I.7.3.7}. We say that a $\sh{O}_Y$-submodule $\sh{I}$ of $\sh{R}(Y)$ is a \emph{fractional ideal} of $\sh{R}(Y)$ if it is of \emph{finite type} \sref[0]{0.5.2.1}. Suppose we have, for all $n\geq0$, a quasi-coherent fractional ideal $\sh{I}_n$ of $\sh{R}(Y)$, such that $\sh{I}_0 = \sh{O}_Y$, and such that condition \sref{II.8.1.1.2} (but not necessarily the second condition \sref{II.8.1.1.1}) is satisfied; we can then again define a quasi-coherent graded $\sh{O}_Y$-algebra by Equation~\sref{II.8.1.1.4}, and the corresponding $Y$-scheme $X = \Proj(\sh{S})$; @@ -90,7 +90,7 @@ Let $Y$ be an integral prescheme. \end{enumerate} \end{proof} -We show a \emph{converse} of \sref{II.8.1.4} in \sref[III]{3.2.3.8}. +We show a \emph{converse} of \sref{II.8.1.4} in \sref[III]{III.2.3.8}. \begin{env}[8.1.5] \label{II.8.1.5} @@ -1814,13 +1814,13 @@ The claims of \sref{II.8.8.2} immediately follow from these facts, by taking $g$ \label{II.8.8.3} Assume further that $Y$ is a \emph{Noetherian} prescheme, and that $f$ is a \emph{proper} morphism. Since $r$ is then \emph{proper}\sref{II.5.4.4}, and thus closed, and since it is also a dominant open immersion, $r$ is necessarily an \emph{isomorphism} $X\xrightarrow{\sim}P$. -Furthermore, we will see, in Chapter~III \sref[III]{3.2.3.5.1}, that $\sh{S}$ is then necessarily an $\sh{O}_Y$-algebra \emph{of finite type}. +Furthermore, we will see, in Chapter~III \sref[III]{III.2.3.5.1}, that $\sh{S}$ is then necessarily an $\sh{O}_Y$-algebra \emph{of finite type}. It then follows that $\sh{S}^\natural$ is an $\sh{S}_0^\natural$-algebra \emph{of finite type} (\sref{II.8.2.10}[i] and \sref{II.8.7.2.7}); since $C_P$ is $C$-isomorphic to $\Proj(\sh{S}^\natural)$ \sref{II.8.7.3}, we see that the morphism $h:C_P\to C$ is \emph{projective}; since the morphism $r$ is an isomorphism, so too is $q:\bb{V}(\sh{L})\to C_P$, and we thus conclude that the morphism $g:\bb{V}(\sh{L})\to C$ is \emph{projective}. Furthermore, since the restriction of $h$ to $E_P$ is an isomorphism to $E$, and since $q$ is an isomorphism, the restriction \sref{II.8.8.2.3} of $g$ is an isomorphism $\bb{V}(\sh{L})\setmin j(X)\xrightarrow{\sim}E$. -If we further assume that $L$ is \emph{very ample} for $f$, then, as we will also see in Chapter~III \sref[III]{3.2.3.5.1}, there exists some integer $n_0>0$ such that $\sh{S}_n=\sh{S}_1^n$ for $n\geq n_0$. +If we further assume that $L$ is \emph{very ample} for $f$, then, as we will also see in Chapter~III \sref[III]{III.2.3.5.1}, there exists some integer $n_0>0$ such that $\sh{S}_n=\sh{S}_1^n$ for $n\geq n_0$. We then conclude, by \sref{II.8.7.7}, that $\bb{V}(\sh{L})$ can be identified with the prescheme $Z$ given by \emph{blowing up the vertex prescheme \emph{(identified with $Y$)} in the affine cone $C$}, and that the \emph{null section} of $\bb{V}(\sh{L})$ (identified with $Y$) is the \emph{inverse image} of the vertex subprescheme $Y$ of $C$. Some of the above results can in fact be proven even without the Noetherian hypothesis: @@ -2115,33 +2115,385 @@ Then $\sh{L}$ is ample relative to $p$. \begin{lemma}[8.10.1] \label{II.8.10.1} +Let $\pi:V\to X$ be a morphism, $j:X\to V$ an $X$-section of $V$ that is also a closed immersion, and $\sh{J}$ a quasi-coherent ideal of $\sh{O}_V$ that defines the closed subprescheme of $V$ associated to $j$. +Then the following all hold true. +\begin{enumerate} + \item[\rm{(i)}] For all $n\geq0$, $\pi_*(\sh{O}_V/\sh{J}^{n+1})$ and $\pi_*(\sh{J}/\sh{J}^{n+1})$ are quasi-coherent $\sh{O}_X$-modules, and $\pi_*(\sh{O}_V/\sh{J})=\sh{O}_X$ and $\pi_*(\sh{J}/\sh{J}^2)=j^*(\sh{J})$. + \item[\rm{(ii)}] If $X=\{\xi\}=\Spec(k)$, where $k$ is a field, then $\varprojlim\pi_*(\sh{O}_V/\sh{J}^{n+1})$ is isomorphic to the separated completion of the local ring $\sh{O}_{j(\xi)}$ for the $\fk{m}_{j(\xi)}$-preadic topology. + \item[\rm{(iii)}] Assume that $\sh{J}$ is an invertible $\sh{O}_V$-module (which implies that + \[ + \sh{L} = j^*(\sh{J}) = \pi_*(\sh{J}/\sh{J}^2) + \] + is an invertible $\sh{O}_X$-module), and that there exists a homomorphism $\varphi:\sh{L}\to\varprojlim\pi_*(\sh{J}/\sh{J}^{n+1})$ such that the composition $\sh{L} \xrightarrow{\varphi} \varprojlim\pi_*(\sh{J}/\sh{J}^{n+1}) \xrightarrow{\alpha} \pi_*(\sh{J}/\sh{J}^2)$ (where $\alpha$ is the canonical homomorphism) is the identity. + If we write $\sh{S}=\bigoplus_{n\geq0}\sh{L}^{\otimes n}$, then $\varphi$ canonically induces an isomorphism of $\sh{O}_X$-algebras from the completion $\widehat{\sh{S}}$ of $\sh{S}$ relative to its canonical filtration (the completion being isomorphic to the product $\prod_{n\geq0}\sh{L}^{\otimes n}$) to $\varprojlim\pi_*(\sh{O}_V/\sh{J}^{n+1})$. +\end{enumerate} +\end{lemma} + +\begin{proof} +\label{proof-2.8.10.1} +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} +and, by definition, $\pi_*(\sh{O}_V/\sh{J}^{n+1})$ is the fibre of $\sh{O}_V/\sh{J}^{n+1}$ at the point $j(\xi)$, or, equivalently, setting $C=\sh{O}_{j(\xi)}$, and denoting by $\fk{m}$ the maximal ideal of $C$, the $C$-module $C/\fk{m}^{n+1}$; +claim (ii) is then evident. + +To prove (i), note that the question is local on $X$; +we can thus restrict to the case where $X$ is affine. +Let $U$ be an affine open subset of $V$; +then $j(X)\cap U$ is an affine open subset of $j(X)$, so $U_0=\pi(j(X)\cap U)$, which is isomorphic to it, is an affine open subset of $X$; +for every affine open subset $W_0\subset U_0$ in $X$, $W=\pi^{-1}(W_0)\cap U$ is an affine open subset of $V$, since $X$ is a scheme \sref[I]{I.5.5.10}; +in particular, $U'=U\cap\pi^{-1}(U_0)$ is an affine open subset of $V$, and clearly $\pi(U')=U_0$ and $j(U_0)=j(X)\cap U$. +Then, by definition, $\Gamma(W_0,\pi_*(\sh{O}_V/\sh{J}^{n+1})) = \Gamma(\pi^{-1}(W_0),\sh{O}_V/\sh{J}^{n+1})$; +but since every point of $\pi^{-1}(W_0)$ not belonging to $j(W_0)$ has an open neighbourhood in $\pi^{-1}(W_0)$ not intersecting $j(X)$, and in which $\sh{O}_V/\sh{J}^{n+1}$ is thus zero, it is clear that the sections of $\sh{O}_V/\sh{J}^{n+1}$ over $\pi^{-1}(W_0)$ and over $W$ are in bijective correspondence. +In other words, if $\pi'$ is the restriction of $\pi$ to $U'$, then the $(\sh{O}_X|U_0)$-modules $\pi_*(\sh{O}_V/\sh{J}^{n+1})|U_0$ and $\pi'_*((\sh{O}_V/\sh{J}^{n+1})|U')$ are identical. +Since $U'$ and $U_0$ are affine, and since the $U_0$ cover $X$, we thus conclude \sref[I]{I.1.6.3} that $\pi_*(\sh{O}_V/\sh{J}^{n+1})$ is quasi-coherent, and the proof is identical for $\pi_*(\sh{J}/\sh{J}^{n+1})$. + +Finally, to prove (iii), note that $\sh{S}$ is exactly $\bb{S}_{\sh{O}_X}(\sh{L})$; +so $\varphi$ canonically induces a homomorphism of $\sh{O}_X$-algebras $\psi:\sh{S}\to\varprojlim\pi_*(\sh{O}_V/\sh{J}^{n+1})$ \sref{II.1.7.4}; +furthermore, this homomorphism sends $\sh{L}^{\otimes n}$ to $\varprojlim_m\pi_*(\sh{J}^n/\sh{J}^{n+1})$, and is thus continuous for the topologies considered, and indeed then extends to a homomorphism $\widehat{\psi}:\widehat{\sh{S}}\to\varprojlim\pi_*(\sh{O}_V/\sh{J}^{n+1})$. +To see that this is indeed an isomorphism, we can, as in the proof of (i), restrict to the case where $X=\Spec(A)$ and $V=\Spec(B)$ are affine, with $\sh{J}=\widetilde{\fk{J}}$, where $\fk{J}$ is an ideal of $B$; +there is an injection $A\to B$ corresponding to $\pi$ that identifies $A$ with a subring of $B$ that is \emph{complementary} to $B$, and $\sh{L}$ (resp. $\pi_*(\sh{O}_V/\sh{J}^{n+1})$) is the quasi-coherent $\sh{O}_X$-module associated to the $A$-module $L=\fk{J}/\fk{J}^2$ (resp. $B/\fk{J}^{n+1}$). +Since $\sh{J}$ is an \emph{invertible} $\sh{O}_V$-module, we can further assume that $\fk{J}=Bt$, where $t$ is not a zero divisor in $B$. +From the fact that $B=A\oplus Bt$, we deduce that, for all $n>0$, +\[ + B = A \oplus At \oplus At^2 \oplus \ldots \oplus At^n \oplus Bt^{n+1} +\] +and so there exists a canonical $A$-isomorphism from the ring of formal series $A[[T]]$ to $C=\varprojlim B/\fk{J}^{n+1}$ that sends $T$ to $t$. +We also have that $L=A\bar{t}$, where $\bar{t}$ is the class of $t$ modulo $Bt^2$, and the homomorphism $\varphi$ sends, by hypothesis, $\bar{t}$ to an element $t'\in C$ that is congruent to $t$ modulo $Ct^2$. +We thus deduce, by induction on $n$, that +\[ + A \oplus At' \oplus \ldots \oplus At^{'n} \oplus Ct^{n+1} + = + A \oplus At \oplus \ldots \oplus At^n \oplus Ct^{n+1} +\] +which proves that the homomorphism $\widehat{\psi}$ does indeed correspond to an isomorphism from $\prod_{n\geq0}L^{\otimes n}$ to $C$. +\end{proof} + +\begin{lemma}[8.10.2] +\label{II.8.10.2} +Under the hypotheses of Lemma~\sref{II.8.10.1}, let $g:X'\to X$ be a morphism, +\oldpage[II]{186} +write $V'=V\times_X X'$, and let $\pi':V'\to X'$ and $g:V'\to V$ be the canonical projections, so that we have the commutative diagram +\[ + \xymatrix{ + V + \ar[d]_{\pi} + & V' + \ar[l]_{g'} + \ar[d]^{\pi'} + \\X + & X' + \ar[l]^{g} + } +\] + +Then $j'=j\times1_{X'}$ is an $X'$-section of $V'$ that is also a closed immersion, and $\sh{J}'=g^{'*}(\sh{J})\sh{O}_{V'}$ is the quasi-coherent ideal of $\sh{O}_{V'}$ that defines the closed subprescheme of $V'$ associated to $j'$. +Furthermore, $\pi'_*(\sh{O}_{V'}/\sh{J}{'n+1}) = g^*(\pi_*(\sh{O}_V/\sh{J}^{n+1}))$. +Finally, $\sh{J}'$ is an $\sh{O}_{V'}$-module that is canonically isomorphic to $g^{'*}(\sh{J})$, and is, in particular, invertible if $\sh{J}$ is an invertible $\sh{O}_V$-module. \end{lemma} +\begin{proof} +\label{proof-2.8.10.2} +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; +we keep the notation from the proof of \sref{II.8.10.1}, and let $X'=\Spec(A')$. +Then $V'=\Spec(B')$, where $B'=B\otimes_A A'$, and $\sh{J}'=\widetilde{\fk{J}''}$, where $\fk{J}'=\Im(\fk{J}\otimes_A A')$. +Then $B'/\fk{J}^{'n+1}=(B/\fk{J}^{n+1})\otimes_A A'$; +furthermore, since $\fk{J}$ is a direct factor (as an $A$-module) of $B$, $\fk{J}\otimes_A A'$ is a direct factor (as an $A'$-module) of $B'$, and is thus canonically identified with $\fk{J}'$. +\end{proof} + +\begin{corollary}[8.10.3] +\label{II.8.10.3} +Assume that the hypotheses of Lemma~\sref{II.8.10.1} are satisfied, and assume further that $\pi$ is of finite type, and that $\sh{J}$ is an invertible $\sh{O}_V$-module. +Then, for all $x\in X$, the local ring at the point $j(x)$ of the fibre $\pi^{-1}(x)$ is a regular (thus integral) ring of dimension 1, whose completion is isomorphic to the formal series ring $\kres(x)[[T]]$ (where $T$ is an indeterminate); +furthermore, there exists exactly one irreducible component of $\pi^{-1}(x)$ that contains $j(x)$. +\end{corollary} + +\begin{proof} +\label{proof-2.8.10.3} +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}); +finally, since $\sh{O}_{j(x)}$ is integral, $j(x)$ belongs to exactly one of the (finitely many) irreducible components of $V$ \sref[I]{I.5.1.4}. +\end{proof} + +\begin{corollary}[8.10.4] +\label{II.8.10.4} +Suppose that the hypotheses of Lemma~\sref{II.8.10.1} are satisfied, and assume further that $\sh{J}$ is an invertible $\sh{O}_V$-module. +Let $W=V\setmin j(X)$; +for every quasi-coherent ideal $\sh{K}$ of $\sh{O}_X$, let $\sh{K}_V=\pi^*(\sh{K})\sh{O}_V$ and $\sh{K}_W=\sh{K}_V|W$. +Then $\sh{K}_V$ is the largest quasi-coherent ideal of $\sh{O}_V$ whose restriction to $W$ is $\sh{K}_W$. +\end{corollary} + +\begin{proof} +\label{proof-2.8.10.4} +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 $A$; +whence $\pi^*(\sh{K})\sh{O}_V=(\fk{K}.B)^\sim$ \sref[I]{I.1.6.9}, $\sh{K}_W=(\fk{K}.B_t)^\sim$, and the largest ideal +\oldpage[II]{187} +of $B$ whose canonical image in $B_t$ is $\fk{K}.B_t$ is the inverse image of $\fk{K}.B_t$, that is, the set of $s\in B$ such that, for some integer $n>0$, we have $t^ns\in\fk{K}.B$. +We have to show that this last relation implies that $s\in\fk{K}.B$, or again that the canonical image of $t$ is not a zero divisor in $B/\fk{K}B=(A/\fk{K})\otimes_AB$, which follows from \sref{II.8.10.2} applied to $X'=\Spec(A/\fk{K})$. +\end{proof} + +\begin{corollary}[8.10.5] +\label{II.8.10.5} +Suppose that the hypotheses of \sref{II.8.10.3} are satisfied; +let $W=V\setmin j(X)$, $x$ be a point of $X$, $\sh{K}$ a quasi-coherent ideal of $\sh{O}_X$, and $z$ the generic point of the irreducible component of $\pi^{-1}(x)$ that contains $j(x)$ \sref{II.8.10.3}. +\begin{enumerate} + \item[\rm{(i)}] Let $g$ be a section of $\sh{O}_V$ over $V$ such that $g|W$ is a section of $\sh{K}_W$ over $W$ (using the notation from \sref{II.8.10.4}). + Then $g$ is a section of $\sh{K}_V$; + if further $g(z)\neq0$, and if, for every integer $m>0$, we denote by $g_m^x$ the germ at the point $x$ of the canonical image $g_m$ of $g$ in $\Gamma(X,\pi_*(\sh{O}_V/\sh{J}^{m+1}))$, then there exists an integer $m>0$ such that the image of $g_m^x$ in + \[ + (\pi_*(\sh{O}_V/\sh{J}^{m+1}))_x \otimes_{\sh{O}_x} \kres(x) + \] + is $\neq0$. + \item[\rm{(ii)}] Suppose further that the conditions of \sref{II.8.10.1}[iii] are fulfilled. + Then, if there exists a section $g$ of $\sh{K}_V$ over $V$ such that $g(z)\neq0$, then there exists an integer $n\geq0$ and a section $f$ of $\sh{K}.\sh{L}^{\otimes n}=\sh{K}\otimes\sh{L}^{\otimes n}\subset\sh{L}^{\otimes n}$ such that $f(x)\neq0$. + If $g$ is a section of $\sh{J}$, we can take $n>0$. +\end{enumerate} +\end{corollary} + +\begin{proof} +\label{proof-2.8.10.5} +\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}; + the fibre $\pi^{-1}(x)$ is then affine of ring $B'=B\otimes_A\kres(x)$, and there exists in $B'$ an element $t'$ which is not a zero divisor and is such that $B'=\kres(x)\oplus B't'$. + Since $j(x)$ is a specialisation of $z$ and since $g(z)\neq0$, we necessarily have that $g_{(j)x}\neq0$. + But $\sh{O}_{j(x)}$ is a separated local ring \sref{II.8.10.3}, and thus embeds into its completion, and the image of $g$ in this completion is thus not null. + But this completion is isomorphic to $\varprojlim_n(B'/B't^{'n+1})$ \sref{II.8.10.3}; + if $g'=g\otimes1\in B'$, there then exists an integer $m$ such that $g'\not\in B't^{'m+1}$, or, again, the image $g'_m$ of $g'$ in $B'/B't^{'m+1}$ is not null. + But since $g'_m$ is exactly the image of $g_m^x$, our claim is proved. + \item[\rm{(ii)}] By \sref{II.8.10.1}[iii], $\pi_*(\sh{O}_V/\sh{J}^{m+1})$ is isomorphic to the direct sum of the $\sh{L}^{\otimes k}$ for $0\leq k\leq m$; + we denote by $f_k$ the section of $\sh{L}^{\otimes k}$ over $X$ that is the component of the element of $\bigoplus_{k=0}^m\Gamma(X,\sh{L}^{\otimes k})$ which corresponds to $g_m$ by this isomorphism. + Choosing $m$ as in (i), there is thus an index $k$ such that $f_k(x)\neq0$, by (i). + To see that $f_k$ is a section of $\sh{K}\sh{L}^{\otimes k}$, it suffices to consider, as above, the case where $X$ and $V$ are affine, and this follows immediately from the fact that $g\in\fk{K}.B$ (with the notation from \sref{II.8.10.4}). + The final claim follows from the fact that the hypothesis $g\in\Gamma(V,\sh{J})$ implies that $f_0=0$. +\end{enumerate} +\end{proof} + +\begin{env}[8.10.6] +\label{II.8.10.6} +\emph{Proof of \sref{II.8.9.4}.} +The question is local on $Y$ \sref{II.4.6.4}; +since $\varepsilon$ is a $Y$-section, we can thus replace $C$ by an affine open neighbourhood $U$ of a point of $\varepsilon(Y)$ such that $\varepsilon(Y)\cap U$ is closed in $U$. +In other words, we can assume that $C$ is affine, and that $Y$ is a closed subprescheme of $C$ (and thus also affine) defined by a quasi-coherent sheaf +\oldpage[II]{188} +$\sh{I}$ of ideals of $\sh{O}_C$. +Since $p$ is separated and quasi-compact, $X$ is thus a quasi-compact scheme, and we are reduced to proving that $\sh{L}$ is \emph{ample} \sref{II.4.6.4}. +By criterion~\sref{II.4.5.2}[a)], we must thus prove the following: +for every quasi-coherent ideal $\sh{K}$ of $\sh{O}_X$ and every point $x\in X$ not belonging to the support of $\sh{O}_X/\sh{K}$, there exists an integer $n>0$ and a section $f$ of $\sh{K}\otimes\sh{L}^{\otimes n}$ over $X$ such that $f(x)\neq0$. + +For this, set +\begin{align*} + \sh{K}_V &= \pi^*(\sh{K})\sh{O}_V +\\ \sh{K}_W &= \sh{K}_V|W +\end{align*} +where $W=V\setmin j(X)$; +since the restriction of $q$ to $W$ is a quasi-compact immersion to $C$, it follows from \sref[I]{I.9.4.2} that $\sh{K}_W$ is the restriction to $W$ of a quasi-coherent ideal $\sh{K}'_V$ of $\sh{O}_V$ of the form +\[ + \sh{K}'_V = q^*(\sh{K}_C)\sh{O}_V +\] +where $\sh{K}_C$ is a quasi-coherent ideal of $\sh{O}_C$. +Furthermore, since, by hypotheses, $q^{-1}(Y)\subset j(X)$, and since $Y$ is defined by the ideal $\sh{I}$, the restriction to $W$ of $q^*(\sh{I})\sh{O}_V$ is identical to that of $\sh{O}_V$, and so $\sh{K}_W$ is also the restriction to $W$ of $q^*(\sh{I}\sh{K}_C)\sh{O}_V$, and we can thus suppose that $\sh{K}_C\subset\sh{I}$, whence +\[ +\label{II.8.10.6.1} + \sh{K}'_V \subset q^*(\sh{I})\sh{O}_V \subset \sh{J} +\tag{8.10.6.1} +\] +taking into account \sref[I]{I.4.4.6} and the commutativity of \sref{II.8.9.4.1}. +Furthermore, we deduce from \sref{II.8.10.4} that +\[ +\label{II.8.10.6.2} + \sh{K}'_V \subset \sh{K}_V. +\tag{8.10.6.2} +\] +With this in mind, it follows from \sref{II.8.10.3} that $j(x)$ belongs to exactly one irreducible component of $\pi^{-1}(x)$; +let $z$ be the generic point of this component, and let $z'=q(z)$. +By \sref{II.8.10.5}, the proof will be finished (taking \sref{II.8.10.6.1} and \sref{II.8.10.6.2} into account) if we show the existence of a section $g$ of $\sh{K}'_V$ over $V$ such that $g(z)\neq0$. +But, by hypothesis, $\sh{K}$ has a restriction equal to that of $\sh{O}_X$ in an open neighbourhood of $x$; +also, it follows from \sref{II.8.10.3} that $z\neq j(x)$, and so $z\in W$, and thus $(\sh{K}_W)_z=\sh{O}_{V,z}$, whence, by definition, $(\sh{K}_C)_{z'}=\sh{O}_{C,z}$. +Since $C$ is affine, there is thus a section $g'$ of $\sh{K}_C$ over $C$ such that $g'(z')\neq0$, and by taking $g$ to be the section of $\sh{K}'_V$ corresponding canonically to $g'$, we indeed have $g(z)\neq0$, which finishes the proof. +\end{env} + +\begin{remark}[8.10.7] +\label{II.8.10.7} +We ignore the question of whether or not condition (ii) in \sref{II.8.9.4} is superfluous or not. +In any case, the conclusion does not hold if we do not assume the existence of a $Y$-morphism $\pi:V\to X$ such that $\pi\circ j=1_X$; +we briefly point out how we can indeed construct a counterexample, whose details will not be developed until later on. +We take $Y=\Spec(k)$, where $k$ is a field, and $C=\Spec(A)$, where $A=k[T_1,T_2]$, and the $Y$-section $\varepsilon$ corresponding to the augmentation homomorphism $A\to k$. +We denote by $C'$ the scheme induced by $C$ by blowing up the closed point $a=\varepsilon(Y)$ of $C$; +if $D$ is the inverse image of $a$ in $C'$, we consider in $D$ a closed point $b$, +\oldpage[II]{189} +and we denote by $V$ the scheme induced by $C'$ by blowing up $b$; +$X$ is the closed subprescheme of $V$ given by the inverse image of $a$ by the structure morphism $q:V\to C$. +We now show that $X$ is the union of two irreducible components, $X_1$ and $X_2$, where $X_1$ is the inverse image of $b$ in $V$. +It is immediate that the ideal $\sh{J}$ of $\sh{O}_V$ that defines $X$ is again invertible, we we can show that $j^*(\sh{J})=\sh{L}$ (where $j$ is the canonical injection $X\to V$) is not ample, by considering the ``degree'' of the inverse image of $\sh{L}$ in $X_1$, which would be $>0$ if $\sh{L}$ were ample, but we can show (by an elementary intersection calculation) that it is in fact equal to $0$. +\end{remark} + \subsection{Uniqueness of contractions} -\label{subsection:II.8.11} +\label{subsection:2.8.11} + +\begin{lemma}[8.11.1] +\label{II.8.11.1} +Let $U$ and $V$ be preschemes, and $h=(h_0,\lambda):U\to V$ a surjective morphism. +Suppose that +\begin{enumerate} + \item $\lambda:\sh{O}_V\to h_*(\sh{O}_U) = (h_0)_*(\sh{O}_U)$ is an isomorphism; + \item the underlying space of $V$ can be identified with the quotient of the underlying space of $U$ by the relation $h_0(x)=h_0(y)$ (\emph{a condition which always holds whenever the morphism $h$ is \emph{open} or \emph{closed}, or, \emph{a fortiori} when $h$ is \emph{proper}.}) +\end{enumerate} +Then, for every prescheme $W$, the map +\[ +\label{II.8.11.1.1} + \Hom(V,W) \to \Hom(U,W) +\tag{8.11.1.1} +\] +that, to each morphism $v=(v_0,\nu)$ from $V$ to $W$, associates the morphism $u=v\circ h=(u_0,\mu)$, is a bijection from $\Hom(V,W)$ to the set of $u$ such that $u_0$ is constant on every fibre $h_0^{-1}(x)$. +\end{lemma} + +\begin{proof} +\label{proof-2.8.11.1} +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 $U/R$. +We can also, replacing $V$ by some isomorphic prescheme if necessary, suppose that $\lambda$ is the identity; +by hypothesis, $\mu$ is then a homomorphism $\mu:\sh{O}_W\to(u_0)_*(\sh{O}_U) = (v_0)_*((h_0)_*(\sh{O}_U))$ such that the corresponding homomorphism $\mu^\sharp:u_0^*(\sh{O}_W)\to\sh{O}_U$ is \emph{local} on every fibre. +Since $(v_0)_*((h_0)_*(\sh{O}_U))=(v_0)_*(\sh{O}_V)$, we necessarily have that $\nu=u$, and everything then reduces to showing that the corresponding homomorphism $\nu^\sharp:v_0^*(\sh{O}_W)\to\sh{O}_V$ is local on every fibre. +But every $y\in V$ is of the form $h_0(x)$ for some $x\in U$; +let $z=v_0(y)=u_0(x)$. +Then \sref[0]{0.3.5.5} the homomorphism $\mu_x^\sharp$ factors as +\[ + \mu_x^\sharp: \sh{O}_z \xrightarrow{\nu_y^\sharp} \sh{O}_y \xrightarrow{\lambda_x^\sharp} \sh{O}_x. +\] +By hypothesis, $\lambda_x^\sharp$ and $\mu_x^\sharp$ are local homomorphisms; +thus $\lambda_x^\sharp$ sends every invertible element of $\sh{O}_y$ to an invertible element of $\sh{O}_x$; +if $\nu_y^\sharp$ sent a non-invertible element of $\sh{O}_z$ to an invertible element of $\sh{O}_y$, then $\mu_x^\sharp$ would send this element of $\sh{O}_z$ to an invertible element of $\sh{O}_x$, contradicting the hypothesis, whence the lemma. +\end{proof} + +\begin{corollary}[8.11.2] +\label{II.8.11.2} +Let $U$ be an integral prescheme, and $V$ a normal prescheme; +then every morphism $h:U\to V$ that is universally closed, birational, and radicial, is also an isomorphism. +\end{corollary} + +\begin{proof} +\label{proof-2.8.11.2} +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 $V$. +To prove the corollary, it will suffice to show that $\lambda:\sh{O}_V\to(h_0)_*(\sh{O}_U)$ is an isomorphism: we can then apply \sref{II.8.11.1}, which proves that the map \sref{II.8.11.1.1} is bijective (the fibres $h_0^{-1}(x)$ each consisting of a single point); +thus $h$ will be an isomorphism. +The question clearly being local on $V$, we can suppose that $V=\Spec(A)$ is affine, of an integral and integrally closed ring \sref{II.8.8.6.1}; +$h$ then corresponds \sref[I]{I.2.2.4} to a homomorphism $\varphi:A\to\Gamma(U,\sh{O}_U)$, and everything reduces to showing that $\varphi$ is an isomorphism. +But, if $K$ is the field of fractions of $A$, then $\Gamma(U,\sh{O}_U)$ has, by hypothesis, $K$ as its field of fractions, and $A$ is a subring of $\Gamma(U,\sh{O}_U)$, with $\varphi$ being the canonical injection \sref[I]{I.8.2.7}. +Since the morphism $h$ satisfies the hypotheses of \sref{II.7.3.11}, $\Gamma(U,\sh{O}_U)$ is a subring of the integral closure of $A$ in $K$, and is thus identical to $A$ by hypothesis. +\end{proof} + +\begin{remark}[8.11.3] +\label{II.8.11.3} +We will see in chapter~III \sref[III]{III.4.4.11} that, whenever $V$ is a \emph{locally Noetherian} prescheme, every morphism $h:U\to V$ that is proper and quasi-finite (in particular, every morphism satisfying the hypotheses of \sref{II.8.11.2}) is necessarily \emph{finite}. +The conclusion of \sref{II.8.11.2} then follows in this case from \sref{II.6.1.15}. +\end{remark} + +\begin{env}[8.11.4] +\label{II.8.11.4} +We will now see that, in Grauert's criterion, we can often prove that the prescheme $C$ and the ``contraction'' $q$ are determined in an \emph{essentially unique} manner. +\end{env} + +\begin{lemma}[8.11.5] +\label{II.8.11.5} +Let $Y$ be a prescheme, $p:X\to Y$ a proper morphism, $\sh{L}$ a $p$-ample invertible $\sh{O}_X$-module, $C$ a $Y$-prescheme, $\varepsilon:Y\to C$ a $Y$-section, and $q:V=\bb{V}(\sh{L})\to C$ a $Y$-morphism, all such that the diagram in \sref{II.8.9.1.1} commutes. +Suppose further that, if $p=(p_0,\theta)$, then $\theta:\sh{O}_Y\to p_*(\sh{O}_X)$ is an isomorphism. +Let $\sh{S}'=\bigoplus_{n\geq0}p_*(\sh{L}^{\otimes n})$ and $C'=\Spec(\sh{S}')$, and let $q':\bb{V}(\sh{L})\to C'$ be the canonical $Y$-morphism \sref{II.8.8.5}. +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} +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. +We are thus in a situation where we can apply \sref{II.8.11.1}, and we will have proven the lemma if we show that $q$ is constant on every fibre $q^{'-1}(z')$, where $z'\in C'$. +But this condition is trivially satisfied for $z'\not\in\varepsilon'(Y)$. +If $z'\in\varepsilon'(Y)$, then there exists exactly one $y\in Y$ such that $z'=\varepsilon'(y)$, and, by commutativity of \sref{II.8.8.5.2} and the fact that $q'$ sends $\bb{V}(\sh{L})\setmin j(X)$ to $C'\setmin\varepsilon'(Y)$, $q^{'-1}(z')=j(p^{-1}(y))$; +the commutativity of the diagram in \sref{II.8.9.1.1} then proves our claim. +\end{proof} + +\begin{corollary}[8.11.6] +\label{II.8.11.6} +Under the hypotheses of \sref{II.8.11.5}, suppose further that $q$ is proper, and that the restriction of $q$ to $\bb{V}(\sh{L})\setmin j(X)$ is an isomorphism to $C\setmin\varepsilon(Y)$. +Then the morphism $u$ is universally closed, surjective, and radicial, and its restriction to $C'\setmin\varepsilon'(Y)$ is an isomorphism to $C\setmin\varepsilon(Y)$. +\end{corollary} + +\begin{proof} +\label{proof-2.8.11.6} +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} +in \sref{II.8.8.5.2} and \sref{II.8.9.1.1} shows that the restriction of $u$ to the closed subprescheme $\varepsilon'(Y)$ of $C'$ is an isomorphism to the closed subprescheme $\varepsilon(Y)$ of $C$, from which we immediately deduce that, for all $z'\in\varepsilon'(Y)$, if $z=u(z')$, then $u$ defines an isomorphism from $\kres(z)$ to $\kres(z')$. +These remarks prove that $u$ is bijective and radicial; +furthermore, if $\psi:C\to Y$ and $\psi':C'\to Y$ are the structure morphisms, then $\psi'=\psi\circ u$, and, since $\psi'$ is separated \sref{II.1.2.4}, so too is $u$ \sref[I]{I.5.5.1}[v]. +We have already seen, in the proof of \sref{II.8.11.5}, that $q'$ is surjective; +since $q=u\circ q'$ is proper, we finally conclude, from \sref{II.5.4.3} and \sref{II.5.4.9}, that $u$ is universally closed. +\end{proof} + +\begin{proposition}[8.11.7] +\label{II.8.11.7} +Let $Y$ be a prescheme, $X$ an \emph{integral} prescheme, $p:X\to Y$ a proper morphism, $\sh{L}$ a $p$-ample invertible $\sh{O}_X$-module, $C$ a \emph{normal} $Y$-prescheme, $\varepsilon:Y\to C$ a $Y$-section, and $q:V=\bb{V}(\sh{L})\to C$ a $Y$ morphism, all such that the diagram in \sref{II.8.9.1.1} commutes. +Suppose further that, if $p=(p_0,\theta)$, then $\theta:\sh{O}_Y\to p_*(\sh{O}_X)$ is an isomorphism. +Let $\sh{S}'=\bigoplus_{n\geq0}p_*(\sh{L}^{\otimes n})$ and $C'=\Spec(\sh{S}')$, and let $q':\bb{V}(\sh{L})\to C'$ be the canonical $Y$-morphism \sref{II.8.8.5}. +Then the unique $Y$-morphism $u:C'\to C$ such that $q=u\circ q'$ is an \emph{isomorphism}. +\end{proposition} + +\begin{proof} +\label{proof-2.8.11.7} +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. +\end{proof} + +\begin{remark}[8.11.8] +\label{II.8.11.8} +\begin{itemize} + \item[\rm{(i)}] The hypothesis that $C$ is normal implies that $X$ is also normal. + Indeed, $C'=\Spec(\sh{S}')$ is then normal, being isomorphic to $C$, and integral, by \sref{II.8.8.6}; + we thus conclude that $\Proj(\sh{S}')$ is \emph{normal}. + Indeed, the question is local on $Y$; + if $Y$ is affine, with $\sh{S}'=\widetilde{S'}$, then the ring $S'=\Gamma(C',\sh{S}')$ is integral and integrally closed \sref{II.8.8.6.1}, and so, for every homogeneous element $f\in S'_+$, the graded ring $S'_f$ is integral and integrally closed \cite[t.~I, p.~257 and 261]{I-13}, and thus so too is the ring $S'_{(f)}$ of its degree-zero terms, because the intersection of $S'_f$ with the field of fractions of $S'_{(f)}$ is equal to $S'_{(f)}$; + this proves our claim \sref{II.6.3.4}. + Finally, since $X$ is isomorphic to an open subprescheme of $\Proj(\sh{S}')$ \sref{II.8.8.1}, $X$ is indeed normal. + We can thus express \sref{II.8.11.7} in the following form: + \emph{If $X$ is integral and normal, and $p=(p_0,\theta):X\to Y$ is a proper morphism such that $\theta:\sh{O}_Y\to p_*(\sh{O}_X)$ is an isomorphism, then, for every $p$-ample $\sh{O}_X$-module $\sh{L}$, there exists exactly one way of contracting the null section of $V=\bb{V}(\sh{L})$ to obtain a normal $Y$-scheme $C$ and a proper $Y$-morphism $q:V\to C$.} + \item[\rm{(ii)}] When $p$ is proper, the hypothesis $p_*(\sh{O}_X)=\sh{O}_Y$ can be considered as an auxiliary hypothesis, not really restricting the generality of the result. + Indeed, if it is not satisfied, then it suffices to replace $Y$ with the $Y$-scheme $Y'=\Spec(p_*(\sh{O}_X))$, and to consider $X$ as a $Y'$-scheme. + We will return to this general method in chapter~III, \textsection~4. +\end{itemize} +\end{remark} \subsection{Quasi-coherent sheaves on based cones} -\label{subsection:II.8.12} +\label{subsection:2.8.12} \begin{env}[8.12.1] \label{II.8.12.1} -Let us take the hypotheses and notation of \sref{II.8.3.1}. +Let us use the hypotheses and notation of \sref{II.8.3.1}. Let $\sh{M}$ be a \emph{quasi-coherent graded $\sh{S}$-module}; to avoid any confusion, we denote by $\widetilde{\sh{M}}$ the quasi-coherent $\sh{O}_C$-module \oldpage[II]{192} -associated to $\sh{M}$ \sref{II.1.4.3} when $\sh{M}$ is considered as a \emph{nongraded} $\sh{S}$-module, and by $\shProj_0(\sh{M})$ the quasi-coherent $\sh{O}_X$-module associated to $\sh{M}$, $\sh{M}$ being considered this time as a graded $\sh{S}$-module (in other words, the $\sh{O}_X$-module denoted by $\widetilde{\sh{M}}$ in \sref{II.3.2.2}). +associated to $\sh{M}$ \sref{II.1.4.3} when $\sh{M}$ is considered as a \emph{non-graded} $\sh{S}$-module, and by $\shProj_0(\sh{M})$ the quasi-coherent $\sh{O}_X$-module associated to $\sh{M}$, $\sh{M}$ being considered this time as a graded $\sh{S}$-module (in other words, the $\sh{O}_X$-module denoted by $\widetilde{\sh{M}}$ in \sref{II.3.2.2}). In addition, we set \[ \label{II.8.12.1.1} \sh{M}_X=\shProj_0(\sh{M})=\bigoplus_{n\in\bb{Z}}\shProj_0(\sh{M}(n)); - \tag{8.12.1.1} +\tag{8.12.1.1} \] the quasi-coherent graded $\sh{O}_X$-algebra $\sh{S}_X$ being defined by \sref{II.8.6.1.1}, $\shProj(\sh{M})$ is equipped with a structure of a \emph{(quasi-coherent) graded $\sh{S}_X$-module}, by means of the canonical homomorphisms \sref{II.3.2.6.1} \[ \label{II.8.12.1.2} \sh{O}_X(m)\otimes_{\sh{O}_X}\shProj_0(\sh{M}(n))\to\shProj_0(\sh{S}(m)\otimes_\sh{S}\sh{M}(n))\to\shProj_0(\sh{M}(m+n)), - \tag{8.12.1.2} +\tag{8.12.1.2} \] the verification of the axioms of sheaves of modules being done using the commutative diagram in \sref{II.2.5.11.4}. @@ -2149,7 +2501,7 @@ If $Y=\Spec(A)$ is affine, $\sh{S}=\widetilde{S}$, and $\sh{M}=\widetilde{M}$, w \[ \label{II.8.12.1.3} \Gamma(X_f,\shProj(\widetilde{M}))=M_f - \tag{8.12.1.3} +\tag{8.12.1.3} \] by the definitions and \sref{II.8.2.9.1}. @@ -2157,14 +2509,205 @@ Now consider the quasi-coherent graded $\widehat{\sh{S}}$-module \[ \label{II.8.12.1.4} \widehat{\sh{M}}=\sh{M}\otimes_\sh{S}\widehat{\sh{S}} - \tag{8.12.1.4} +\tag{8.12.1.4} \] -($\widehat{\sh{S}}$ defined by \sref{II.8.3.1.1}); we deduce a quasi-coherent graded $\sh{O}_{\widehat{C}}$-module $\shProj_0(\widehat{\sh{M}})$, which we will also denote by +($\widehat{\sh{S}}$ being defined by \sref{II.8.3.1.1}); this induces a quasi-coherent graded $\sh{O}_{\widehat{C}}$-module $\shProj_0(\widehat{\sh{M}})$, which we will also denote by \[ \label{II.8.12.1.5} \sh{M}^\square=\shProj_0(\widehat{\sh{M}}). - \tag{8.12.1.5} +\tag{8.12.1.5} \] It is clear \sref{II.3.2.4} that $\sh{M}^\square$ is an additive functor which is \emph{exact} in $\sh{M}$, commuting with direct sums and with inductive limits. \end{env} + +\begin{proposition}[8.12.2] +\label{II.8.12.2} +With the notation of \sref{II.8.3.2}, we have canonical functorial isomorphisms +\[ +\label{II.8.12.2.1} + i^*(\sh{M}^\square)\xrightarrow{\sim}\widetilde{\sh{M}}, + \quad + j^*(\sh{M}^\square)\xrightarrow{\sim}\shProj_0(\sh{M}). +\tag{8.12.2.1} +\] +Indeed, $i^*(\sh{M}^\square)$ is canonically identified with $(\widehat{\sh{M}}/(\bb{z}-1)\widehat{\sh{M}})^\sim$ on $\Spec(\widehat{\sh{S}}/(\bb{z}-1)\widehat{\sh{S}})$ by \sref{II.3.2.3}; +the first of the canonical isomorphisms \sref{II.8.12.2.1} is then immediately induced \sref{II.1.4.1} by the canonical isomorphism $\widehat{\sh{M}}/(\bb{z}-1)\widehat{\sh{M}}\xrightarrow{\sim}\sh{M}$. +The canonical immersion $j:X\to C$ corresponds to the canonical homomorphism $\widehat{\sh{S}}\to\sh{S}$ with kernel $\bb{z}\widehat{\sh{S}}$ \sref{II.8.3.2}; +the second homomorphism \sref{II.8.12.2.1} is the particular case of the canonical homomorphism \sref{II.3.5.2}[ii], since here we have $\widehat{\sh{M}}\otimes_{\widehat{\sh{S}}}\sh{S}=\sh{M}$; +to verify that this is an isomorphism, we can restrict to the case where $Y=\Spec(A)$ is affine, $\sh{S}=\widetilde{S}$, and $\sh{M}=\widetilde{M}$; +by appealing to \sref{II.2.8.8}, the proof that, for all homogeneous $f$ in $S_+$, the preceding homomorphism, restricted to $X_f$, restricts to an isomorphism, is then immediate. +\end{proposition} + +\oldpage[II]{193} + +By an abuse of language, we again say, thanks to the existence of the first isomorphism \sref{II.8.12.2.1}, that $\sh{M}^\square$ is the \emph{projective closure} of the $\sh{O}_X$-module $\widetilde{\sh{M}}$ (it being implicit that the data of the $\sh{O}_C$-module $\widetilde{\sh{M}}$ includes the grading of the $\sh{S}$-module $\sh{M}$). + +\begin{env}[8.12.3] +\label{II.8.12.3} +With the notation of \sref{II.8.3.5}, we have a canonical functorial homomorphism +\[ +\label{II.8.12.3.1} + p^*(\shProj(\sh{M}))\to\sh{M}^\square|\widehat{E}. +\tag{8.12.3.1} +\] + +Indeed, this is a particular case of the homomorphism $\nu^\sharp$ defined more generally in \sref{II.3.5.6}. +If $Y=\Spec(A)$ is affine, $\sh{S}=\widetilde{S}$, and $\sh{M}=\widetilde{M}$, then, by appealing to \sref{II.2.8.8}, the restriction of \sref{II.8.12.3.1} to $p^{-1}(X_f)=\widehat{C}_f$ (for some homogeneous $f$ in $S_+$) corresponds to the canonical homomorphism +\[ +\label{II.8.12.3.2} + M_{(f)}\otimes_{S_{(f)}}S_f^\leq \to M_f^\leq +\tag{8.12.3.2} +\] +taking into account \sref{II.8.2.3.2} and \sref{II.8.2.5.2}. +\end{env} + +\begin{env}[8.12.4] +\label{II.8.12.4} +Let us place ourselves in the settings of \sref{II.8.5.1}, and assume its hypotheses and keep its notation. +It follows from \sref{II.1.5.6} that, for every quasi-coherent graded $\sh{S}$-module $\sh{S}$, we have, on one hand, a canonical isomorphism +\[ +\label{II.8.12.4.1} + \Phi^*(\widetilde{\sh{M}}) \xrightarrow{\sim} (q^*(\sh{M})\otimes_{q^*(\sh{S})}\sh{S}')^\sim +\tag{8.12.4.1} +\] +of $\sh{O}_{C'}$-modules; +on the other hand, \sref{II.3.5.6} implies the existence of a canonical $\Proj(\varphi)$-morphism +\[ +\label{II.8.12.4.2} + \shProj_0\sh{M} \to (\shProj_0(q^*(\sh{M}))\otimes_{q^*(\sh{S})}\sh{S}')|G(\varphi) +\tag{8.12.4.2} +\] +and also of a canonical $\widehat{\Phi}$-morphism +\[ +\label{II.8.12.4.3} + \shProj_0\widehat{\sh{M}} \to (\shProj_0(q^*(\widehat{\sh{M}}))\otimes_{q^*(\widehat{\sh{S}})}\widehat{\sh{S}}')|G(\widehat{\varphi}). +\tag{8.12.4.3} +\] +\end{env} + +\begin{env}[8.12.5] +\label{II.8.12.5} +Consider now the setting of \sref{II.8.6.1}, with the same notation; +we thus take $Y'=X$, the morphism $q:X\to Y$ being the structure morphism, and $\varphi$ the canonical $q$-morphism \sref{II.8.6.1.2}. +We then have a canonical isomorphism +\[ +\label{II.8.12.5.1} + q^*(\sh{M})\otimes_{q^*(\sh{S})}\sh{S}_X^\geq \xrightarrow{\sim} \sh{M}_X^\geq +\tag{8.12.5.1} +\] +by setting $\sh{M}_X^\geq=\bigoplus_{n\geq0}\shProj_0(\sh{M}(n))$. +We can indeed restrict to the case where $Y=\Spec(A)$ is affine, $\sh{S}=\widetilde{S}$, and $\sh{M}=\widetilde{M}$, and define the isomorphism \sref{II.8.12.5.1} on each of the affine open subsets $X_f$ (where $f$ is homogeneous in $S_+$), by verifying the compatibility with taking a homogeneous multiple of $f$. +But the restriction to $X_f$ of the left-hand side of \sref{II.8.12.5.1} is $\widetilde{M}'=((M\otimes_A S_{(f)})\otimes_{S\otimes_A S_{(f)}}S_f^\geq)^\sim$ by \sref{II.8.6.2.1}; +since we have a canonical isomorphism from $M\otimes_A S_{(f)}$ to $M\otimes_S(S\otimes_A S_{(f)})$, we have an induced isomorphism from $\widetilde{M}'$ to $(M\otimes_S S_f^\geq)^\sim$, and the latter is canonically isomorphic, by \sref{II.8.2.9.1}, to the restriction to $X_f$ of the right-hand side of \sref{II.8.12.5.1}, and satisfies the required compatibility conditions. +\oldpage[II]{194} + +Replacing $\sh{M}$ by $\widehat{\sh{M}}$, $\sh{S}$ by $\widehat{\sh{S}}$, and $\sh{S}_X$ by $(\sh{S}_X^\geq)^\wedge$ in the previous argument, we similarly have a canonical isomorphism +\[ +\label{II.8.12.5.2} + q^*(\widehat{\sh{M}})\otimes_{q^*(\widehat{\sh{S}})}(\sh{S}_X^\geq)^\wedge \xrightarrow{\sim} (\sh{M}_X^\geq)^\wedge. +\tag{8.12.5.2} +\] + +If we recall \sref{II.8.6.2} that the structure morphism $u:\Proj(\sh{S}_X^\geq)\to X$ is an isomorphism, then we deduce, first of all, from the above, that we have a canonical $u$-isomorphism +\[ +\label{II.8.12.5.3} + \shProj_0\sh{M} \xrightarrow{\sim} \shProj_0(\sh{M}_X^\geq) +\tag{8.12.5.3} +\] +as a particular case of \sref{II.8.12.4.2}. +We note that, with the notation from the proof of \sref{II.8.6.2}, this reduces to seeing that the canonical homomorphism $M_{(f)}\otimes_{S_{(f)}}(S_f^\geq)^{(d)}\to(M_f^\geq)^{(d)}$ is an isomorphism whenever $f\in S_d$, which is immediate. + +Secondly, the isomorphism \sref{II.8.12.5.2} gives us, by this time applying \sref{II.8.12.4.3} to the canonical morphism $r=\Proj(\widehat{\alpha}):\widehat{C}_X\to\widehat{C}$, a canonical $r$-morphism +\[ +\label{II.8.12.5.4} + \sh{M}^\square \to (\sh{M}_X^\geq)^\square. +\tag{8.12.5.4} +\] + +Recall now \sref{II.8.6.2} that the restrictions of $r$ to the pointed cones $\widehat{E}_X$ and $E_X$ are \emph{isomorphisms} to $\widehat{E}$ and $E$ (respectively). +Furthermore: +\end{env} + +\begin{proposition}[8.12.6] +\label{II.8.12.6} +The restrictions to $\widehat{E}_X$ and $E_X$ of the canonical $r$-morphism \sref{II.8.12.5.4} are isomorphisms +\[ +\label{II.8.12.6.1} + \sh{M}^\square|\widehat{E} \xrightarrow{\sim} (\sh{M}_X^\geq)^\square|\widehat{E}_X +\tag{8.12.6.1} +\] +\[ +\label{II.8.12.6.2} + \sh{M}^\sim|\widehat{E} \xrightarrow{\sim} (\sh{M}_X^\geq)^\sim|\widehat{E}_X. +\tag{8.12.6.2} +\] +\end{proposition} + +\begin{proof} +\label{proof-2.8.12.6} +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 +\[ + \widehat{M}_{(f)}\otimes_{\widehat{S}_{(f)}}(S_f^\geq)_{(f/1)}^\wedge \to (M\otimes_S S_f^\geq)_{(f/1)}^\wedge +\] +is an isomorphism; +but, by \sref{II.8.2.3.2} and \sref{II.8.2.5.2}, the left-hand side is canonically identified with $M_f^\leq\otimes_{S_f^\leq}(S_f^\geq)_{f/1}^\leq$, and thus with $M_f^\leq$, by \sref{II.8.2.7.2}, and the right-hand side with $(M_f^\geq)_{f/1}^\leq$, and thus also with $M_f^\leq$, by \sref{II.8.2.9.2}, whence the conclusion concerning \sref{II.8.12.6.1}; +\sref{II.8.12.6.2} then follows from \sref{II.8.12.6.1} and \sref{II.8.12.2.1}. +\end{proof} + +\begin{corollary}[8.12.7] +\label{II.8.12.7} +With the identifications of \sref{II.8.6.3}, the restriction of $(\sh{M}_X^\geq)^\square$ to $\widehat{E}_X$ can be identified with $(\sh{M}_X^\leq)^\sim$, and the restriction of $(\sh{M}_X^\geq)^\square$ to $E_x$ with $\widetilde{\sh{M}}_X$. +\end{corollary} + +\begin{proof} +\label{proof-2.8.12.7} +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} + +\begin{proposition}[8.12.8] +\label{II.8.12.8} +Under the hypotheses of \sref{II.8.6.4}, the canonical homomorphism \sref{II.8.12.3.1} is an isomorphism. +\end{proposition} + +\begin{proof} +\label{proof-2.8.12.8} +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 $\sh{S}_1$. +With the notation of \sref{II.8.12.3}, we then have, for some $f\in S_1$, that $S_f^\leq=S_{(f)}[1/f]$, and the canonical homomorphism $M_{(f)}\otimes_{S_{(f)}}S_f^\leq\to M_f^\leq$ is an isomorphism, by the definition of $M_f^\leq$. +\end{proof} + +\begin{env}[8.12.9] +\label{II.8.12.9} +Consider now the quasi-coherent $\sh{S}$-modules +\[ + \sh{M}_{[n]}=\bigoplus_{m\geq n}\sh{M}_m +\] +and (with the notation of \sref{II.8.7.2}) the graded quasi-coherent $\sh{S}^\natural$-module +\[ +\label{II.8.12.9.1} + \sh{M}^\natural=\left(\bigoplus_{n\geq0}\sh{M}_{[n]}\right)^\sim. +\tag{8.12.9.1} +\] + +We have seen \sref{II.8.7.3} that there exists a canonical $C$-isomorphism $h:C_X\xrightarrow{\sim}\Proj(\sh{S}^\natural)$. +Furthermore: +\end{env} + +\begin{proposition}[8.12.10] +\label{II.8.12.10} +There exists a canonical $h$-isomorphism +\[ +\label{II.8.12.10.1} + \shProj_0(\sh{M}^\natural) \xrightarrow{\sim} \widetilde{\sh{M}}_X. +\tag{8.12.10.1} +\] +\end{proposition} + +\begin{proof} +\label{proof-2.8.12.10} +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} |