The Stacks project

Lemma 33.44.8. Let $k$ be a field, let $X$ be a proper scheme of dimension $\leq 1$ over $k$, and let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $n$. Then

\[ \deg (\mathcal{E}) = \deg (\wedge ^ n(\mathcal{E})) = \deg (\det (\mathcal{E})) \]

Proof. By Lemma 33.44.6 and elementary arithmetic, we reduce to the case of a proper curve. Then there exists a modification $f : X' \to X$ such that $f^*\mathcal{E}$ has a filtration whose successive quotients are invertible modules, see Divisors, Lemma 31.36.1. By Lemma 33.44.4 we may work on $X'$. Thus we may assume we have a filtration

\[ 0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_ n = \mathcal{E} \]

by locally free $\mathcal{O}_ X$-modules with $\mathcal{L}_ i = \mathcal{E}_ i/\mathcal{E}_{i - 1}$ is invertible. By Modules, Lemma 17.25.1 and induction we find $\det (\mathcal{E}) = \mathcal{L}_1 \otimes \ldots \otimes \mathcal{L}_ n$. Thus the equality follows from Lemma 33.44.7 and additivity (Lemma 33.44.3). $\square$

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