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authorGravatar Tim Hosgood <thosgood@users.noreply.github.com> 2021-01-26 03:19:42 +0000
committerGravatar GitHub <noreply@github.com> 2021-01-26 03:19:42 +0000
commit4a44e8d65347f2cfd46f4f18cf94c3100d6173ee (patch)
tree963a1e51cd7bd6c9e209714ae67dd2eb8c39169d
parentc74eed364cfbf2c5fcdc19c0cb47a1f85e4b3ec0 (diff)
parent6cda5c5c18e93e4689d7d5065cba5b465b9abc9a (diff)
downloadega-4a44e8d65347f2cfd46f4f18cf94c3100d6173ee.tar.gz
ega-4a44e8d65347f2cfd46f4f18cf94c3100d6173ee.zip
Merge pull request #186 from ryankeleti/ega2-2
fixed some errors
-rw-r--r--ega2/ega2-2.tex20
1 files changed, 10 insertions, 10 deletions
diff --git a/ega2/ega2-2.tex b/ega2/ega2-2.tex
index f6be303..89a77d7 100644
--- a/ega2/ega2-2.tex
+++ b/ega2/ega2-2.tex
@@ -824,14 +824,14 @@ Thus $\lambda$ induces a canonical functorial homomorphism of $\sh{O}_X$-modules
\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 $\widetilde{\fk{J}}\otimes_{\sh{O}_X}\widetilde{\fk{K}}\to\sh{O}_X$;
+Consider, in particular, graded ideals $\mathfrak{J}$ and $\mathfrak{K}$ of $S$;
+since $\widetilde{\mathfrak{J}}$ and $\widetilde{\mathfrak{K}}$ are sheaves of ideals of $\sh{O}_X$, we have a canonical homomorphism $\widetilde{\mathfrak{J}}\otimes_{\sh{O}_X}\widetilde{\mathfrak{K}}\to\sh{O}_X$;
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]
+ \widetilde{\mathfrak{J}}\otimes_{\sh{O}_X}\widetilde{\mathfrak{K}} \ar[rr]^\lambda \ar[dr]
+ && (\widetilde{\mathfrak{J}}\otimes_S\widetilde{\mathfrak{K}})^\supertilde \ar[dl]
\\&\sh{O}_X&
}
\tag{2.5.11.3}
@@ -891,8 +891,8 @@ so too is the homomorphism $\mu$ \sref{II.2.5.12.2} if the graded $S$-module $M$
\begin{proof}
Since $X$ is the union of the $D_+(f)$ for $f\in S_1$ \sref{II.2.3.14}, we are led to proving that $\lambda_f$ and $\mu_f$ are isomorphisms, under the given hypotheses, whenever $f$ is homogeneous and \emph{of degree~$1$}.
-But we can then define a $\ZZ$-bilinear map $M_m\times N_n\to M_{(f)}\otimes_{S_{(f)}}N_{(f)}$ by sending $(x,y)$ to the element $(x/f^m)\otimes(y/f^n)$ (if $m<0$, we write $x/f^m$ to mean $f^{-m}x/1$);
-these maps define a $\ZZ$-linear map $M\otimes_{\bb{Z}}N\to M_{(f)}\otimes_{S_{(f)}}N_{(f)}$, and, if $s\in S_q$, this map sends $(sx)\otimes y$ to $(s/f^q)((x/f^m)\otimes(y/f^n))$ (for $x\in M_m$ and $y\in N_n$).
+But we can then define a $\bb{Z}$-bilinear map $M_m\times N_n\to M_{(f)}\otimes_{S_{(f)}}N_{(f)}$ by sending $(x,y)$ to the element $(x/f^m)\otimes(y/f^n)$ (if $m<0$, we write $x/f^m$ to mean $f^{-m}x/1$);
+these maps define a $\bb{Z}$-linear map $M\otimes_{\bb{Z}}N\to M_{(f)}\otimes_{S_{(f)}}N_{(f)}$, and, if $s\in S_q$, this map sends $(sx)\otimes y$ to $(s/f^q)((x/f^m)\otimes(y/f^n))$ (for $x\in M_m$ and $y\in N_n$).
We thus obtain a di-homomorphism of modules $\gamma_f: M\otimes_S N\to M_{(f)}\otimes_{S_{(f)}}N_{(f)}$, with respect to the canonical homomorphism $S\to S_{(f)}$ (sending $s\in S_q$ to $s/f^q$).
Suppose furthermore that, for an element $\sum_i(x_i\otimes y_i)$ of $M\otimes_S N$ (with $x_i$ and $y_i$ homogeneous of degree $m_i$ and $n_i$, respectively), we have that $f^r\sum_i(x_i\otimes y_i)=0$, or, in other words, that $\sum_i(f^rx_i\otimes y_i)=0$.
We thus deduce, by \sref[0]{0.1.3.4}, that $\sum_i(f^rx_i/f^{m_i+r})\otimes(y_i/f^{n_i})=0$, i.e. $\gamma_f(\sum_i(x_i\otimes y_i))=0$.
@@ -921,18 +921,18 @@ We easily note that the composed map
is the isomorphism $z/f^h\mapsto z/f^{h-n}$ from $(N(-n))_{(f)}$ to $N_{(f)}$, and thus $\mu_f$ is an isomorphism.
\end{proof}
-If the ideal $S_+$ is generated by $S_1$, then we deduce from \sref{II.2.5.13} that, for every graded ideal $\fk{J}$ of $S$, and for every graded $S$-module $M$, we have
+If the ideal $S_+$ is generated by $S_1$, then we deduce from \sref{II.2.5.13} that, for every graded ideal $\mathfrak{J}$ of $S$, and for every graded $S$-module $M$, we have
\[
\label{II.2.5.13.1}
- \widetilde{\fk{J}}\cdot\widetilde{M} = (\fk{J}\cdot M)^\supertilde
+ \widetilde{\mathfrak{J}}\cdot\widetilde{M} = (\mathfrak{J}\cdot M)^\supertilde
\tag{2.5.13.1}
\]
up to canonical isomorphism;
this follows from the commutativity of the diagram
\[
\xymatrix{
- \widetilde{\fk{J}}\otimes_{\sh{O}_X}\widetilde{M} \ar[rr]^\lambda \ar[dr]
- && (\fk{J}\otimes_S M)^\supertilde
+ \widetilde{\mathfrak{J}}\otimes_{\sh{O}_X}\widetilde{M} \ar[rr]^\lambda \ar[dr]
+ && (\mathfrak{J}\otimes_S M)^\supertilde
\\&\widetilde{M}
}
\]