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author | Tim Hosgood <thosgood@users.noreply.github.com> | 2021-01-25 03:05:01 +0000 |
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committer | GitHub <noreply@github.com> | 2021-01-25 03:05:01 +0000 |

commit | 2e6f10166cfa55a3436a58b0568936d052733d7c (patch) | |

tree | 32fd3c41faedd184195d1c4f5021011191074b30 | |

parent | 9e7b3da93f057195cbd2b5aed342475cb2b20faa (diff) | |

parent | d677ae6ee1013f3402a884b2672b56a061f8ec1b (diff) | |

download | ega-2e6f10166cfa55a3436a58b0568936d052733d7c.tar.gz ega-2e6f10166cfa55a3436a58b0568936d052733d7c.zip |

Merge pull request #184 from ryankeleti/ega2-2

more II.2

-rw-r--r-- | ega2/ega2-2.tex | 155 |

1 files changed, 128 insertions, 27 deletions

diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex index d079858..e51a6ed 100644 --- a/ega2/ega2-2.tex +++ b/ega2/ega2-2.tex @@ -685,22 +685,22 @@ There exists on $X=\Proj(S)$ exactly one quasi-coherent $\sh{O}_X$-module $\wide \end{proposition} \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}; - we thus conclude that there exists a canonical isomorphism - \[ - \theta_{g,f}\colon (M_{(f)})^\supertilde|D_+(fg) \xrightarrow{\sim} (M_{(g)})^\supertilde|D_+(fg) - \] - 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 $\theta_{g,f}=\eta_g\circ\eta_f^{-1}$. - 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, $\widetilde{S}=\sh{O}_X$. +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}; +we thus conclude that there exists a canonical isomorphism +\[ + \theta_{g,f}\colon (M_{(f)})^\supertilde|D_+(fg) \xrightarrow{\sim} (M_{(g)})^\supertilde|D_+(fg) +\] +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 $\theta_{g,f}=\eta_g\circ\eta_f^{-1}$. +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, $\widetilde{S}=\sh{O}_X$. \end{proof} \begin{definition}[2.5.3] \label{II.2.5.3} - We say that the quasi-coherent $\sh{O}_X$-module $\widetilde{M}$ defined in \sref{II.2.5.2} is \emph{associated} to the graded $S$-module $M$. +We say that the quasi-coherent $\sh{O}_X$-module $\widetilde{M}$ defined in \sref{II.2.5.2} is \emph{associated} to the graded $S$-module $M$. \end{definition} Recall that the graded $S$-modules form a category when we restrict from arbitrary homomorphisms of graded modules to homomorphisms \emph{of degree~$0$}. @@ -708,13 +708,13 @@ With this convention: \begin{proposition}[2.5.4] \label{II.2.5.4} - The functor $M\mapsto\widetilde{M}$ is an exact additive covariant functor from the category of graded $S$-modules to the category of quasi-coherent $\sh{O}_X$-modules, and it commutes with inductive limits and direct sums. +The functor $M\mapsto\widetilde{M}$ is an exact additive covariant functor from the category of graded $S$-modules to the category of quasi-coherent $\sh{O}_X$-modules, and it commutes with inductive limits and direct sums. \end{proposition} \begin{proof} - Indeed, since these properties are local, it suffices to show that they are satisfied for the sheaves of the form $\widetilde{M}|D_+(f)=(M_{(f)})^\supertilde$; - 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. +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} We denote by $\widetilde{u}$ the homomorphism $\widetilde{M}\to\widetilde{N}$ corresponding to a homomorphism $u\colon M\to N$ of degree~$0$. @@ -722,12 +722,12 @@ We immediately deduce from \sref{I.2.5.4} that the results of \sref[I]{I.1.3.9} \begin{proposition}[2.5.5] \label{II.2.5.5} - For all $\mathfrak{p}\in X=\Proj(S)$, we have $\widetilde{M}_\mathfrak{p}=M_{(\mathfrak{p})}$. +For all $\mathfrak{p}\in X=\Proj(S)$, we have $\widetilde{M}_\mathfrak{p}=M_{(\mathfrak{p})}$. \end{proposition} \begin{proof} - By definition, $\widetilde{M}_\mathfrak{p}=\varinjlim\Gamma(D_+(f),\widetilde{M})$, where $f$ runs over the set of homogeneous elements of $S_+$ such that $f\not\in\mathfrak{p}$; - since $\Gamma(D_+(f),\widetilde{M})=M_{(f)}$, the proposition follows from the definition of $M_{(\mathfrak{p})}$ \sref{II.2.2.7} +By definition, $\widetilde{M}_\mathfrak{p}=\varinjlim\Gamma(D_+(f),\widetilde{M})$, where $f$ runs over the set of homogeneous elements of $S_+$ such that $f\not\in\mathfrak{p}$; +since $\Gamma(D_+(f),\widetilde{M})=M_{(f)}$, the proposition follows from the definition of $M_{(\mathfrak{p})}$ \sref{II.2.2.7} \end{proof} \oldpage[II]{32} @@ -737,19 +737,120 @@ Even more particularly, if $S$ is \emph{essential integral}, so that $\Proj(S)=X \begin{proposition}[2.5.6] \label{II.2.5.6} - If, for all $z\in M$ and all homogeneous $f$ in $S_+$, there exists a power of $f$ that annihilates $z$, then $\widetilde{M}=0$. - This sufficient condition is also necessary if the $S_0$-algebra $S$ is generated by the set $S_1$ of homogeneous elements of degree~$1$. +If, for all $z\in M$ and all homogeneous $f$ in $S_+$, there exists a power of $f$ that annihilates $z$, then $\widetilde{M}=0$. +This sufficient condition is also necessary if the $S_0$-algebra $S$ is generated by the set $S_1$ of homogeneous elements of degree~$1$. \end{proposition} \begin{proof} - The condition $\widetilde{M}=0$ is equivalent to $M_{(f)}=0$ for all homogeneous $f$ in $S_+$. - On the other hand, if $f\in S_d$, to say that $M_{(f)}=0$ implies that, for all homogeneous $z\in M$ whose degree is some multiple of $d$, there exists a power $f^n$ such that $f^nz=0$. - If $M_{(f)}=0$ for all $f\in S_1$, then the condition of the statement is thus satisfied for all $f\in S_1$; - the condition is \emph{a fortiori} satisfied for all homogeneous $f$ in $S_+$ if $S_1$ generates $S$, since every homogeneous element of $S_+$ is then a linear combination of products of elements of $S_1$. +The condition $\widetilde{M}=0$ is equivalent to $M_{(f)}=0$ for all homogeneous $f$ in $S_+$. +On the other hand, if $f\in S_d$, to say that $M_{(f)}=0$ implies that, for all homogeneous $z\in M$ whose degree is some multiple of $d$, there exists a power $f^n$ such that $f^nz=0$. +If $M_{(f)}=0$ for all $f\in S_1$, then the condition of the statement is thus satisfied for all $f\in S_1$; +the condition is \emph{a fortiori} satisfied for all homogeneous $f$ in $S_+$ if $S_1$ generates $S$, since every homogeneous element of $S_+$ is then a linear combination of products of elements of $S_1$. \end{proof} \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 $\sh{O}_X|D_+(f)$. +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 $\sh{O}_X|D_+(f)$. \end{proposition} + +\begin{proof} +Indeed, multiplication by the invertible element $(f/1)^n$ of $S_f$ gives a bijection from $S_{(f)}=(S_f)_0$ to $(S_f)_{nd}=(S_f(nd))_0=(S(nd)_f)_0=S(nd)_{(f)}$; +in other words, the $S_{(f)}$-modules $S_{(f)}$ and $S(nd)_{(f)}$ are canonically isomorphic, whence the proposition. +\end{proof} + +\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}. +\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 $n\in\bb{Z}$. +\end{corollary} + +\begin{proof} +It suffices to remark that $X=\bigcup_{f\in S_1}D_+(f)$, by the hypothesis \sref{II.2.3.14} and to apply \sref{II.2.5.8} with $U=X$. +\end{proof} + +\begin{env}[2.5.10] +\label{II.2.5.10} +We set, for the rest of this section, +\[ +\label{II.2.5.10.1} + \sh{O}_X(n) = (S(n))^\supertilde +\tag{2.5.10.1} +\] +for all $n\in\bb{Z}$, and, for every open subset $U$ of $X$, and every $(\sh{O}_X|U)$-module $\sh{F}$, +\[ +\label{II.2.5.10.2} + \sh{F}(n) = \sh{F}\otimes_{\sh{O}_X|U}(\sh{O}_X(n)|U) +\tag{2.5.10.2} +\] +for all $n\in\bb{Z}$. +If the ideal $S_+$ is generated by $S_1$, then the functor $\sh{F}(n)$ is \emph{exact} in $\sh{F}$ for all $n\in\bb{Z}$, since $\sh{O}_X(n)$ is then an \emph{invertible} $\sh{O}_X$-module. +\end{env} + +\begin{env}[2.5.11] +\label{II.2.5.11} +Let $M$ and $N$ be graded $S$-modules. +For all $f\in S_d$ ($d>0$), we define a canonical functorial homomorphism of $S_{(f)}$-modules by +\[ +\label{II.2.5.11.1} + \lambda_f\colon M_{(f)}\otimes_{S_{(f)}}N_{(f)} \to (M\otimes_S N)_{(f)} +\tag{2.5.11.1} +\] +\oldpage[II]{33} +by composing the homomorphism $M_{(f)}\otimes_{S_{(f)}}N_{(f)}\to M_f\otimes_{S_f}N_f$ (coming from the injections $M_{(f)}\to M_f$, $N_{(f)}\to N_f$, and $S_{(f)}\to S_f$) with the canonical isomorphism $M_f\otimes_{S_f}N_f\xrightarrow{\sim}(M\otimes_S N)_f$ \sref[0]{0.1.3.4}, and by noting that, by the definition of the tensor product of two graded modules, this latter isomorphism preserves degrees; +for $x\in M_{md}$ and $y\in N_{nd}$ ($m,n\geq0$), we thus have +\[ + \lambda_f((x/f^m)\otimes(y/f^n)) = (x\otimes y)/f^{m+n}. +\] + +It immediately follows from this definition that, if $g\in S_e$ ($e>0$), then the diagram +\[ + \xymatrix{ + M_{(f)}\otimes_{S_{(f)}}N_{(f)} \ar[r]^{\lambda_f} \ar[d] + & (M\otimes_S N)_{(f)} \ar[d] + \\M_{(fg)}\otimes_{S_{(fg)}}N_{(fg)} \ar[r]_{\lambda_{fg}} + & (M\otimes_S N)_{(fg)} + } +\] +(where the vertical arrow on the right is the canonical homomorphism, and the one on the left comes from the canonical homomorphisms) commutes. +Thus $\lambda$ induces a canonical functorial homomorphism of $\sh{O}_X$-modules +\[ +\label{II.2.5.11.2} + \lambda\colon \widetilde{M}\otimes_{\sh{O}_X}\widetilde{N} \to (M\otimes_S N)^\supertilde. +\tag{2.5.11.2} +\] + +Consider, in particular, graded ideals $\fk{J}$ and $\fk{K}$ of $S$; +since $\widetilde{\fk{J}}$ and $\widetilde{\fk{K}}$ are sheaves of ideals of $\sh{O}_X$, we have a canonical homomorphism $\widetilde{\fk{J}}\otimes_{\sh{O}_X}\widetilde{\fk{K}}\to\sh{O}_X$; +the diagram +\[ +\label{II.2.5.11.3} + \xymatrix{ + \widetilde{\fk{J}}\otimes_{\sh{O}_X}\widetilde{\fk{K}} \ar[rr]^\lambda \ar[dr] + && (\widetilde{\fk{J}}\otimes_S\widetilde{\fk{K}})^\supertilde \ar[dl] + \\&\sh{O}_X& + } +\tag{2.5.11.3} +\] +then commutes. +Indeed, we can reduce to verifying this on each open subset $D_+(f)$ (for $f$ homogeneous in $S_+$), and this follows immediately from the definition \sref{2.5.11.1} of $\lambda_f$ and from \sref[I]{I.1.3.13}. + +Finally, note that, if $M$, $N$, and $P$ are graded $S$-modules, then the diagram +\[ +\label{II.2.5.11.4} + \xymatrix{ + \widetilde{M}\otimes_{\sh{O}_X}\widetilde{N}\otimes_{\sh{O}_X}\widetilde{P} \ar[r]^{\lambda\otimes1} \ar[d]_{1\otimes\lambda} + & (M\otimes_S N)^\supertilde\otimes_{\sh{O}_X}\widetilde{P} \ar[d]^\lambda + \\\widetilde{M}\otimes_{\sh{O}_X}(N\otimes_S P)^\supertilde \ar[r]_\lambda + & (M\otimes_S N\otimes_S P)^\supertilde + } +\tag{2.5.11.4} +\] +\oldpage[II]{34} +commutes. +It again suffices to verify this on each open subset $D_+(f)$, and this follows immediately from the definitions and from \sref[I]{I.1.3.13}. +\end{env} |