76.11 Picard groups of thickenings
Some material on Picard groups of thickenings.
Lemma 76.11.1. Let $S$ be a scheme. Let $X \subset X'$ be a first order thickening of algebraic spaces over $S$ with ideal sheaf $\mathcal{I}$. Then there is a canonical exact sequence
\[ \xymatrix{ 0 \ar[r] & H^0(X, \mathcal{I}) \ar[r] & H^0(X', \mathcal{O}_{X'}^*) \ar[r] & H^0(X, \mathcal{O}^*_ X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^1(X, \mathcal{I}) \ar[r] & \mathop{\mathrm{Pic}}\nolimits (X') \ar[r] & \mathop{\mathrm{Pic}}\nolimits (X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^2(X, \mathcal{I}) \ar[r] & \ldots \ar[r] & \ldots } \]
of abelian groups.
Proof.
Recall that $X_{\acute{e}tale}= X'_{\acute{e}tale}$, see Lemma 76.9.6 and more generally the discussion in Section 76.9. The sequence of the lemma is the long exact cohomology sequence associated to the short exact sequence of sheaves of abelian groups
\[ 0 \to \mathcal{I} \to \mathcal{O}_{X'}^* \to \mathcal{O}_ X^* \to 0 \]
on $X_{\acute{e}tale}$ 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 site with the first cohomology group of the sheaf of invertible functions, see Cohomology on Sites, Lemma 21.6.1.
$\square$
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