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Replace throughout

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aisejohan committed Dec 10, 2019
1 parent 2cfc834 commit d339aeffc7d214941e4776242046564ca64134cb
Showing with 4 additions and 4 deletions.
  1. +4 −4 varieties.tex
@@ -9212,13 +9212,13 @@ \section{Degrees on curves}
Thus
$$
Rf_*f^*\mathcal{E} = f_*f^*\mathcal{E} =
\mathcal{E} \otimes_{\mathcal{O}_X} f_*\mathcal{O}_X
\mathcal{E} \otimes_{\mathcal{O}_X} f_*\mathcal{O}_{X'}
$$
Since $f$ induces an isomorphism on local rings at generic points of
all irreducible components of dimension $1$ we see that the kernel
and cokernel
$$
0 \to \mathcal{K} \to \mathcal{O}_X \to f_*\mathcal{O}_X \to \mathcal{Q} \to 0
0 \to \mathcal{K} \to \mathcal{O}_X \to f_*\mathcal{O}_{X'} \to \mathcal{Q} \to 0
$$
have supports of dimension $\leq 0$. Note that tensoring this with
$\mathcal{E}$ is still an exact sequence as $\mathcal{E}$ is locally free.
@@ -9228,15 +9228,15 @@ \section{Degrees on curves}
& =
\chi(X, \mathcal{E}) - \chi(X, f_*f^*\mathcal{E}) \\
& =
\chi(X, \mathcal{E}) - \chi(X, \mathcal{E} \otimes f_*\mathcal{O}_X) \\
\chi(X, \mathcal{E}) - \chi(X, \mathcal{E} \otimes f_*\mathcal{O}_{X'}) \\
& =
\chi(X, \mathcal{K} \otimes \mathcal{E}) -
\chi(X, \mathcal{Q} \otimes \mathcal{E}) \\
& =
n\chi(X, \mathcal{K}) -
n\chi(X, \mathcal{Q}) \\
& =
n\chi(X, \mathcal{O}_X) - n\chi(X, f_*\mathcal{O}_X) \\
n\chi(X, \mathcal{O}_X) - n\chi(X, f_*\mathcal{O}_{X'}) \\
& =
n\chi(X, \mathcal{O}_X) - n\chi(X', \mathcal{O}_{X'})
\end{align*}

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