Lemma 37.4.1. Let X \subset X' be a first order thickening with ideal sheaf \mathcal{I}. Then there is a canonical exact sequence
of abelian groups.
Some material on Picard groups of thickenings.
Lemma 37.4.1. Let X \subset X' be a first order thickening with ideal sheaf \mathcal{I}. Then there is a canonical exact sequence
of abelian groups.
Proof. This is the long exact cohomology sequence associated to the short exact sequence of sheaves of abelian groups
where the first map sends a local section f of \mathcal{I} to the invertible section 1 + f of \mathcal{O}_{X'}. We also use the identification of the Picard group of a ringed space with the first cohomology group of the sheaf of invertible functions, see Cohomology, Lemma 20.6.1. \square
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
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 nth 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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Comment #9740 by Fiasco on