The Stacks project

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

\[ \xymatrix{ 0 \ar[r] & H^0(X, \mathcal{I}) \ar[r] & H^0(X', \mathcal{O}_{X'}^*) \ar[r] & H^0(X, \mathcal{O}^*_ X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^1(X, \mathcal{I}) \ar[r] & \mathop{\mathrm{Pic}}\nolimits (X') \ar[r] & \mathop{\mathrm{Pic}}\nolimits (X) \ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\ & H^2(X, \mathcal{I}) \ar[r] & \ldots \ar[r] & \ldots } \]

of abelian groups.

Proof. This is the long exact cohomology sequence associated to the short exact sequence of sheaves of abelian groups

\[ 0 \to \mathcal{I} \to \mathcal{O}_{X'}^* \to \mathcal{O}_ X^* \to 0 \]

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. $\square$

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