Lemma 37.4.2. Let $X \subset X'$ be a thickening. Let $n$ be an integer invertible in $\mathcal{O}_ X$. Then the map $\mathop{\mathrm{Pic}}\nolimits (X')[n] \to \mathop{\mathrm{Pic}}\nolimits (X)[n]$ is bijective.

Proof for a finite order thickening. By the general principle explained following Definition 37.2.1 this reduces to the case of a first order thickening. Then may use Lemma 37.4.1 to see that it suffices to show that $H^1(X, \mathcal{I})[n]$, $H^1(X, \mathcal{I})/n$, and $H^2(X, \mathcal{I})[n]$ are zero. This follows as multiplication by $n$ on $\mathcal{I}$ is an isomorphism as it is an $\mathcal{O}_ X$-module. $\square$

Proof in general. Let $\mathcal{I} \subset \mathcal{O}_{X'}$ be the quasi-coherent ideal sheaf cutting out $X$. Then we have a short exact sequence of abelian groups

$0 \to (1 + \mathcal{I})^* \to \mathcal{O}_{X'}^* \to \mathcal{O}_ X^* \to 0$

We obtain a long exact cohomology sequence as in the statement of Lemma 37.4.1 with $H^ i(X, \mathcal{I})$ replaced by $H^ i(X, (1 + \mathcal{I})^*)$. Thus it suffices to show that raising to the $n$th power is an isomorphism $(1 + \mathcal{I})^* \to (1 + \mathcal{I})^*$. Taking sections over affine opens this follows from Algebra, Lemma 10.32.8. $\square$

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