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authorGravatar Tim Hosgood <timhosgood@gmail.com> 2021-07-28 22:19:37 +0100
committerGravatar Tim Hosgood <timhosgood@gmail.com> 2021-07-28 22:19:37 +0100
commit58e24508592d12be0e20842033767ce31e6262df (patch)
tree653ea67a033012448c421c7e24446100dca9cb1a
parent4e1bc2e211c9ba70a3f26f46e4d289ffdc995213 (diff)
downloadega-58e24508592d12be0e20842033767ce31e6262df.tar.gz
ega-58e24508592d12be0e20842033767ce31e6262df.zip
yet another page
-rw-r--r--ega2/ega2-2.tex70
1 files changed, 69 insertions, 1 deletions
diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex
index b268667..bd60a28 100644
--- a/ega2/ega2-2.tex
+++ b/ega2/ega2-2.tex
@@ -1557,5 +1557,73 @@ This follows from \sref{II.2.8.10} and \sref{II.2.5.15}.
\begin{env}[2.8.12]
\label{II.2.8.12}
-Under
+Under the hypotheses of \sref{II.2.8.9}, for $f'\in S'_d$ ($d>0$) and $f=\varphi(f')$, the diagram
+\[
+ \xymatrix{
+ M'_{(f')} \ar[r]^\sim \ar[d]
+ & {M'}^{(d)}/(f'-1){M'}^{(d)} \ar[d]
+ \\M_{(f)} \ar[r]^\sim
+ & M^{(d)}/(f-1)M^{(d)}
+ }
+\]
+(cf. \sref{II.2.2.5}) commutes.
+\end{env}
+
+\begin{env}[2.8.13]
+\label{II.2.8.13}
+Keep the notation and hypotheses of \sref{II.2.8.9}, and let $\sh{F}'$ be an $\sh{O}_{X'}$-module;
+if we set $\sh{F}=\Phi^*(\sh{F}')$, then, for all $n\in\bb{Z}$, we have $\sh{F}(n)=\Phi^*(\sh{F}'(n))$, by \sref{II.2.8.11} and \sref[0]{0.4.3.3}.
+Then \sref[0]{0.3.7.1} we have a canonical homomorphism
+\[
+ \Gamma(\rho): \Gamma(X',\sh{F}'(n)) \to \Gamma(X,\sh{F}(n))
+\]
+which gives a canonical di-homomorphism of graded modules
+\[
+ \Gamma_\bullet(\sh{F}') \to \Gamma_\bullet(\sh{F}).
+\]
+
+Suppose that the ideal $S_+$ is generated by $S_1$, and that $\sh{F}'=\widetilde{M'}$, thus $\sh{F}=\widetilde{M}$ with $M=M'\otimes_{A'}A$.
+If $f'$ is homogeneous in $S'_+$, and $f=\varphi(f')$, then we have seen that $M_{(f)}=M'_{(f')}\otimes_{A'}A$, and the diagram
+\[
+ \xymatrix{
+ M'_0 \ar[r] \ar[d]
+ & M'_{(f')} \ar[d]
+ & =\Gamma(D_+(f'),\widetilde{M'})
+ \\M_0 \ar[r]
+ & M_{(f)}
+ & =\Gamma(D_+(f),\widetilde{M})
+ }
+\]
+thus commutes;
+we immediately conclude from this remark, and from the definition of the homomorphism $\alpha$ \sref{II.2.6.2}, that the diagram
+\[
+\label{II.2.8.13.1}
+ \xymatrix{
+ M' \ar[r]^{\alpha_{M'}} \ar[d]
+ & \Gamma_\bullet(\widetilde{M'}) \ar[d]
+ \\M \ar[r]_{\alpha_M}
+ & \Gamma_\bullet(\widetilde{M})
+ }
+\tag{2.8.13.1}
+\]
+commutes.
+Similarly, the diagram
+\[
+\label{II.2.8.13.2}
+ \xymatrix{
+ (\Gamma_\bullet(\sh{F}'))\supertilde \ar[r]^{\beta_{\sh{F}'}} \ar[d]
+ & \sh{F}' \ar[d]
+ \\(\Gamma_\bullet(\sh{F}))\supertilde \ar[r]_{\beta_{\sh{F}}}
+ & \sh{F}
+ }
+\tag{2.8.13.2}
+\]
+commutes (the vertical arrow on the right being the canonical $\Phi$-morphism $\sh{F}'\to\Phi^*(\sh{F}')=\sh{F}$).
+\end{env}
+
+\oldpage[II]{47}
+
+\begin{env}[2.8.14]
+\label{II.2.8.14}
+Still
\end{env}