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authorGravatar Tim Hosgood <timhosgood@gmail.com> 2021-01-25 05:58:36 +0000
committerGravatar Tim Hosgood <timhosgood@gmail.com> 2021-01-25 05:58:36 +0000
commit9f9c067009ca9eed621de38a31f4730c6461cbef (patch)
tree222d8c93275eae919257115331b39c155c3589e4
parent2e6f10166cfa55a3436a58b0568936d052733d7c (diff)
downloadega-9f9c067009ca9eed621de38a31f4730c6461cbef.tar.gz
ega-9f9c067009ca9eed621de38a31f4730c6461cbef.zip
another env
-rw-r--r--ega2/ega2-2.tex38
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diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex
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--- a/ega2/ega2-2.tex
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@@ -854,3 +854,41 @@ Finally, note that, if $M$, $N$, and $P$ are graded $S$-modules, then the diagra
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}
+
+\begin{env}[2.5.12]
+\label{II.2.5.12}
+Under the hypotheses of \sref{II.2.5.11}, we define a functorial canonical homomorphism of $S_{(f)}$-modules
+\[
+\label{II.2.5.12.1}
+ \mu_f\colon (\Hom_S(M,N))_{(f)} \to \Hom_{S_{(f)}}(M_{(f)},N_{(f)})
+\tag{2.5.12.1}
+\]
+by sending $u/f^n$, where $u$ is a homomorphism of degree~$nd$, to the homomorphism $M_{(f)}\to N_{(f)}$ that sends $x/f^m$ ($x\in M_{md}$) to $u(x)/f^{m+n}$.
+For $g\in S_e$ ($e>0$), we again have a commutative diagram:
+\[
+ \xymatrix{
+ (\Hom_S(M,N))_{(f)} \ar[r]^{\mu_f} \ar[d]
+ & \Hom_{S_{(f)}}(M_{(f)},N_{(f)}) \ar[d]
+ \\(\Hom_S(M,N))_{(fg)} \ar[r]_{\mu_{fg}}
+ & \Hom_{S_{(fg)}}(M_{(fg)},N_{(fg)})
+ }
+\]
+(where the vertical arrow on the left is the canonical homomorphism, and the one on the right comes from the canonical homomorphisms).
+We thus again conclude (taking \sref[I]{I.1.3.8} into account) that the $\mu_f$ define a functorial canonical homomorphism of $\sh{O}_X$-modules
+\[
+\label{II.2.5.12.2}
+ \mu\colon (\Hom_S(M,N))^\supertilde \to \shHom_{\sh{O}_X}(\widetilde{M},\widetilde{N})
+\tag{2.5.12.2}
+\]
+\end{env}
+
+\begin{proposition}[2.5.13]
+\label{II.2.5.13}
+Suppose that the ideal $S_+$ is generated by $S_1$.
+Then the homomorphism $\lambda$ \sref{II.2.5.11.2} is an isomorphism;
+so too is the homomorphism $\mu$ \sref{II.2.5.12.2} if the graded $S$-module $M$ admits a finite presentation \sref{II.2.1.1}
+\end{proposition}
+
+\begin{proof}
+
+\end{proof}