The Stacks project

Lemma 31.17.2. Let $\pi : X \to Y$ be a finite morphism of schemes. If there exists a norm of degree $d$ for $\pi $, then there exists a homomorphism of abelian groups

\[ \text{Norm}_\pi : \mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (Y) \]

such that $\text{Norm}_\pi (\pi ^*\mathcal{N}) \cong \mathcal{N}^{\otimes d}$ for all invertible $\mathcal{O}_ Y$-modules $\mathcal{N}$.

Proof. We will use the correspondence between isomorphism classes of invertible $\mathcal{O}_ X$-modules and elements of $H^1(X, \mathcal{O}_ X^*)$ given in Cohomology, Lemma 20.6.1 without further mention. We explain how to take the norm of an invertible $\mathcal{O}_ X$-module $\mathcal{L}$. Namely, by Lemma 31.17.1 there exists an open covering $Y = \bigcup V_ j$ such that $\mathcal{L}|_{\pi ^{-1}V_ j}$ is trivial. Choose a generating section $s_ j \in \mathcal{L}(\pi ^{-1}V_ j)$ for each $j$. On the overlaps $\pi ^{-1}V_ j \cap \pi ^{-1}V_{j'}$ we can write

\[ s_ j = u_{jj'} s_{j'} \]

for a unique $u_{jj'} \in \mathcal{O}^*_ X(\pi ^{-1}V_ j \cap \pi ^{-1}V_{j'})$. Thus we can consider the elements

\[ v_{jj'} = \text{Norm}_\pi (u_{jj'}) \in \mathcal{O}_ Y^*(V_ j \cap V_{j'}) \]

These elements satisfy the cocycle condition (because the $u_{jj'}$ do and $\text{Norm}_\pi $ is multiplicative) and therefore define an invertible $\mathcal{O}_ Y$-module. We omit the verification that: this is well defined, additive on Picard groups, and satisfies the property $\text{Norm}_\pi (\pi ^*\mathcal{N}) \cong \mathcal{N}^{\otimes d}$ for all invertible $\mathcal{O}_ Y$-modules $\mathcal{N}$. $\square$


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