Lemma 30.20.7. Given a morphism of schemes $f : X \to Y$ and a quasi-coherent sheaf $\mathcal{F}$ on $X$. Assume

1. $Y$ locally Noetherian,

2. $f$ proper, and

3. $\mathcal{F}$ coherent.

Let $y \in Y$ be a point. Consider the infinitesimal neighbourhoods

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

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

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

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

Proof. This is just a reformulation of a special case of the theorem on formal functions, Theorem 30.20.5. Let us spell it out. Note that $\mathcal{O}_{Y, y}$ is a Noetherian local ring. Consider the canonical morphism $c : \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) \to Y$, see Schemes, Equation (26.13.1.1). This is a flat morphism as it identifies local rings. Denote momentarily $f' : X' \to \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y})$ the base change of $f$ to this local ring. We see that $c^*R^ pf_*\mathcal{F} = R^ pf'_*\mathcal{F}'$ by Lemma 30.5.2. Moreover, the infinitesimal neighbourhoods of the fibre $X_ y$ and $X'_ y$ are identified (verification omitted; hint: the morphisms $c_ n$ factor through $c$).

Hence we may assume that $Y = \mathop{\mathrm{Spec}}(A)$ is the spectrum of a Noetherian local ring $A$ with maximal ideal $\mathfrak m$ and that $y \in Y$ corresponds to the closed point (i.e., to $\mathfrak m$). In particular it follows that

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

In this case also, 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 30.9.9 we see that

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

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

Comment #6822 by Matthew Hase-Liu on

Minor typo: I think the last $i_{n,\*}\mathcal{F}$ (at the end of the proof) should be $i_{n,\*}\mathcal{F}_n$.

Comment #6823 by Matthew Hase-Liu on

sorry, should be i_{n,*}\mathcal{F_n}

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