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
of abelian groups.
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
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
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. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$
). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)