The Stacks project

Proof. This is just a reformulation of a special case of the theorem on formal functions, Theorem 69.22.5. Let us spell it out. Note that $\mathcal{O}_{Y, \overline{y}}$ is a Noetherian local ring, see Properties of Spaces, Lemma 66.24.4. Consider the canonical morphism $c : \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}) \to Y$. This is a flat morphism as it identifies local rings. Denote $f' : X' \to \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}})$ the base change of $f$ to this local ring. We see that $c^*R^ pf_*\mathcal{F} = R^ pf'_*\mathcal{F}'$ by Lemma 69.11.2. Moreover, we have canonical identifications $X_ n = X'_ n$ for all $n \geq 1$.

Hence we may assume that $Y = \mathop{\mathrm{Spec}}(A)$ is the spectrum of a strictly henselian Noetherian local ring $A$ with maximal ideal $\mathfrak m$ and that $\overline{y} \to Y$ is equal to $\mathop{\mathrm{Spec}}(A/\mathfrak m) \to Y$. It follows that

because $(Y, \overline{y})$ is an initial object in the category of étale neighbourhoods of $\overline{y}$. The morphisms $c_ n$ are each closed immersions. Hence their base changes $i_ n$ are closed immersions as well. Note that $i_{n, *}\mathcal{F}_ n = i_{n, *}i_ n^*\mathcal{F} = \mathcal{F}/\mathfrak m^ n\mathcal{F}$. By the Leray spectral sequence for $i_ n$, and Lemma 69.12.9 we see that

Hence we may indeed apply the theorem on formal functions to compute the limit in the statement of the lemma and we win. $\square$