The Stacks project

Lemma 68.22.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ and let $\mathcal{F}$ be a quasi-coherent sheaf on $Y$. Assume

  1. $Y$ locally Noetherian,

  2. $f$ proper, and

  3. $\mathcal{F}$ coherent.

Let $\overline{y}$ be a geometric point of $Y$. Consider the “infinitesimal neighbourhoods”

\[ \xymatrix{ X_ n = \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}/\mathfrak m_{\overline{y}}^ n) \times _ Y X \ar[r]_-{i_ n} \ar[d]_{f_ n} & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}/\mathfrak m_{\overline{y}}^ n) \ar[r]^-{c_ n} & Y } \]

of the fibre $X_1 = X_{\overline{y}}$ and set $\mathcal{F}_ n = i_ n^*\mathcal{F}$. Then we have

\[ \left(R^ pf_*\mathcal{F}\right)_{\overline{y}}^\wedge \cong \mathop{\mathrm{lim}}\nolimits _ n H^ p(X_ n, \mathcal{F}_ n) \]

as $\mathcal{O}_{Y, \overline{y}}^\wedge $-modules.

Proof. This is just a reformulation of a special case of the theorem on formal functions, Theorem 68.22.5. Let us spell it out. Note that $\mathcal{O}_{Y, \overline{y}}$ is a Noetherian local ring, see Properties of Spaces, Lemma 65.24.4. 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 68.11.2. 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

\[ \left(R^ pf_*\mathcal{F}\right)_{\overline{y}} = \Gamma (Y, R^ pf_*\mathcal{F}) = H^ p(X, \mathcal{F}) \]

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 68.12.9 we see that

\[ H^ p(X_ n, \mathcal{F}_ n) = H^ p(X, i_{n, *}\mathcal{F}) = H^ p(X, \mathcal{F}/\mathfrak m^ n\mathcal{F}) \]

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


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