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 17.26.1 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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