The Stacks project

Proof. We define a map

as follows. Let $g$ be a local section of $\mathcal{I}^{\otimes i}$ and let $\overline{s}$ be a local section of $H^ i(\mathcal{F}^\bullet )$. Then $\overline{s}$ is (locally) the class of a local section $s$ of $\mathop{\mathrm{Ker}}(d^ i : \mathcal{F}^ i \to \mathcal{F}^{i + 1})$. Then we send $g \otimes \overline{s}$ to the local section $gs$ of $(\eta _\mathcal {I}\mathcal{F})^ i \subset \mathcal{I}^ i\mathcal{F}$. Of course $gs$ is in the kernel of $d^ i$ on $\eta _\mathcal {I}\mathcal{F}^\bullet $ and hence defines a local section of $H^ i(\eta _\mathcal {I}\mathcal{F}^\bullet )$. Checking that this is well defined is without problems. We claim that this map factors through an isomorphism as given in the lemma. This we my check on stalks and hence via Lemma 20.55.4 this translates into the result of More on Algebra, Lemma 15.95.2. $\square$