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. $\square$

Comments (3)

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}

Post a comment

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02OD. Beware of the difference between the letter 'O' and the digit '0'.