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authorGravatar Tim Hosgood <timhosgood@gmail.com> 2021-07-28 15:42:17 +0100
committerGravatar Tim Hosgood <timhosgood@gmail.com> 2021-07-28 15:42:17 +0100
commitd8777945e8ca3fbfcd49615b4494d2dfbe76cef5 (patch)
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parentc39a3169cdf1dcd816bbfcd6f0153db5f379a9d3 (diff)
downloadega-d8777945e8ca3fbfcd49615b4494d2dfbe76cef5.tar.gz
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diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex
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@@ -1472,5 +1472,61 @@ With the notation of \sref{II.2.8.4}, if $M''$ is a graded $S''$-module, then, t
\begin{proposition}[2.8.8]
\label{II.2.8.8}
-Under
-\end{proposition} \ No newline at end of file
+Under the hypotheses of \sref{II.2.8.1}, let $M'$ be a graded $S'$-module.
+Then there exists a canonical functorial homomorphism $\nu$ from the $(\sh{O}_X|G(\varphi))$-module $\Phi^*(\widetilde{M'})$ to the $(\sh{O}_X|G(\varphi))$-module $(M'\otimes_{S'}S)\supertilde|G(\varphi)$.
+If the ideal $S'_+$ is generated by $S'_1$, then $\nu$ is an isomorphism.
+\end{proposition}
+
+\begin{proof}
+Indeed, for $f'\in S'_d$ ($d>0$), we define a canonical functorial homomorphism of $S_{(f)}$-modules (where $f=\varphi(f')$)
+\[
+\label{II.2.8.8.1}
+ \nu_f: M'_{(f')}\otimes_{S'_{(f')}}S_{(f)} \to (M'\otimes_{S'}S)_{(f)}
+\tag{2.8.8.1}
+\]
+by composing the homomorphism $M'_{(f')}\otimes_{S'_{(f')}}S_{(f)}\to M'_{f'}\otimes_{S'_{f'}}S_f$ and the canonical isomorphism $M'_{f'}\otimes_{S'_{f'}}S_f\xrightarrow{\sim}(M'\otimes_{S'}S)_f$ \sref[0]{0.1.5.4}, and noting that the latter preserves degrees.
+We can immediately verify the compatibility of $\nu_f$ with the restriction operators from $D_+(f)$ to $D_+(fg)$ (for any $g'\in S'_+$ and $g=\varphi(g')$), whence the definition of the homomorphism
+\[
+ \nu: \Phi^*(\widetilde{M'}) \to (M'\otimes_{S'}S)\supertilde|G(\varphi)
+\]
+taking \sref[I]{I.1.6.5} into account.
+To prove the second claim, it suffices to show that $\nu_f$ is an isomorphism for all $f'\in S_1$, since $G(\varphi)$ is then a union of the $D_+(\varphi(f'))$.
+We first define a $\bb{Z}$-bilinear $M'_m\times S_n\to M'_{(f')}\otimes_{S'_{(f')}}S_{(f)}$ by sending $(x',s)$ to the element $(x'/{f'}^m)\otimes(s/f^n)$ (with the convention that $x'/{f'}^m$ is ${f'}^{-m}x'/1$ when $m<0$).
+\oldpage[II]{45}
+We claim that, in the proof of \sref{II.2.5.13}, this map gives rise to a di-homomorphism of modules
+\[
+ \eta_f: M'\otimes_{S'}S \to M'_{(f')}\otimes_{S_{(f')}}S_{(f)}.
+\]
+Furthermore, if, for $r>0$, we have $f^r\sum_i(x'_i\otimes s_i)=0$, then this can also be written as $\sum_i({f'}^rx'_i\otimes s_i)=0$, whence, by \sref[0]{0.1.5.4}, $\sum_i({f'}^rx_i/{f'}^{m_i+r})\otimes(s_i/f^{n_i})=0$, i.e. $\eta_f(\sum_i x_i\otimes y_i)0=$, which proves that $\eta_f$ factors as $M'\otimes_{S'}S\to(M'\otimes_{S'}S)_f\xrightarrow{\eta'_f}M'_{(f')}\otimes_{S'_{(f')}}S_{(f)}$;
+we finally can prove that $\eta'_f$ and $\nu_f$ are inverse isomorphisms to one another.
+
+In particular, it follows from \sref{II.2.1.2.1} that we have a canonical homomorphism
+\[
+\label{II.2.8.8.2}
+ \Phi^*(\sh{O}_{X'}(n)) \xrightarrow{\sim} \sh{O}_X(n)|G(\varphi)
+\tag{2.8.8.2}
+\]
+for all $n\in\bb{Z}$.
+\end{proof}
+
+\begin{env}[2.8.9]
+\label{II.2.8.9}
+Let $A$ and $A'$ be rings, and $\psi:A'\to A$ a ring homomorphism, defining a morphism $\Psi:\Spec(A)\to\Spec(A')$.
+Let $S'$ be a positively-graded $A'$-algebra, and set $S=S'\otimes_{A'}A$, which is evidently an $A$-algebra graded by the $S'_n\otimes_{A'}A$;
+the map $\varphi:s'\to s'\otimes1$ is then a graded ring homomorphism that makes the diagram \sref{II.2.8.5.1} commute.
+Since $S_+$ is here the $A$-module generated by $\varphi(S'_+)$, we have $G(\varphi)=\Proj(S)=X$;
+whence, setting $X'=\Proj(S')$, we have the commutative diagram
+\[
+\label{II.2.8.9.1}
+ \xymatrix{
+ X \ar[r]^{\Phi} \ar[d]_p
+ & X' \ar[d]
+ \\Y \ar[r]_{\Psi}
+ & Y'
+ }
+\tag{2.8.9.1}
+\]
+
+Now let $M'$ be a graded $S'$-module, and set $M=M'\otimes_{A'}A=M'\otimes_{S'}S$.
+Under these conditions:
+\end{env}