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# stacks / stacks-project

Replace throughout

Thanks to Zhaodong Cai
https://stacks.math.columbia.edu/tag/0AYU#comment-4732
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aisejohan committed Dec 10, 2019
1 parent 2cfc834 commit d339aeffc7d214941e4776242046564ca64134cb
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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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