The Stacks project

59.53 Stalks of higher direct images

The stalks of higher direct images can often be computed as follows.

Theorem 59.53.1. Let $f: X \to S$ be a quasi-compact and quasi-separated morphism of schemes, $\mathcal{F}$ an abelian sheaf on $X_{\acute{e}tale}$, and $\overline{s}$ a geometric point of $S$ lying over $s \in S$. Then

\[ \left(R^ nf_* \mathcal{F}\right)_{\overline{s}} = H_{\acute{e}tale}^ n( X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}\mathcal{F}) \]

where $p : X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) \to X$ is the projection. For $K \in D^+(X_{\acute{e}tale})$ and $n \in \mathbf{Z}$ we have

\[ \left(R^ nf_*K\right)_{\overline{s}} = H_{\acute{e}tale}^ n(X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K) \]

In fact, we have

\[ \left(Rf_*K\right)_{\overline{s}} = R\Gamma _{\acute{e}tale}(X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K) \]

in $D^+(\textit{Ab})$.

Proof. Let $\mathcal{I}$ be the category of étale neighborhoods of $\overline{s}$ on $S$. By Lemma 59.51.6 we have

\[ (R^ nf_*\mathcal{F})_{\overline{s}} = \mathop{\mathrm{colim}}\nolimits _{(V, \overline{v}) \in \mathcal{I}^{opp}} H_{\acute{e}tale}^ n(X \times _ S V, \mathcal{F}|_{X \times _ S V}). \]

We may replace $\mathcal{I}$ by the initial subcategory consisting of affine étale neighbourhoods of $\overline{s}$. Observe that

\[ \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) = \mathop{\mathrm{lim}}\nolimits _{(V, \overline{v}) \in \mathcal{I}} V \]

by Lemma 59.33.1 and Limits, Lemma 32.2.1. Since fibre products commute with limits we also obtain

\[ X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) = \mathop{\mathrm{lim}}\nolimits _{(V, \overline{v}) \in \mathcal{I}} X \times _ S V \]

We conclude by Lemma 59.51.5. For the second variant, use the same argument using Lemma 59.52.3 instead of Lemma 59.51.5.

To see that the last statement is true, it suffices to produce a map $\left(Rf_*K\right)_{\overline{s}} \to R\Gamma _{\acute{e}tale}(X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K)$ in $D^+(\textit{Ab})$ which realizes the ismorphisms on cohomology groups in degree $n$ above for all $n$. To do this, choose a bounded below complex $\mathcal{J}^\bullet $ of injective abelian sheaves on $X_{\acute{e}tale}$ representing $K$. The complex $f_*\mathcal{J}^\bullet $ represents $Rf_*K$. Thus the complex

\[ (f_*\mathcal{J}^\bullet )_{\overline{s}} = \mathop{\mathrm{colim}}\nolimits _{(V, \overline{v}) \in \mathcal{I}^{opp}} (f_*\mathcal{J}^\bullet )(V) \]

represents $(Rf_*K)_{\overline{s}}$. For each $V$ we have maps

\[ (f_*\mathcal{J}^\bullet )(V) = \Gamma (X \times _ S V, \mathcal{J}^\bullet ) \longrightarrow \Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}\mathcal{J}^\bullet ) \]

and the target complex represents $R\Gamma _{\acute{e}tale}(X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K)$ in $D^+(\textit{Ab})$. Taking the colimit of these maps we obtain the result. $\square$

Remark 59.53.2. Let $f : X \to S$ be a morphism of schemes. Let $K \in D(X_{\acute{e}tale})$. Let $\overline{s}$ be a geometric point of $S$. There are always canonical maps

\[ (Rf_*K)_{\overline{s}} \longrightarrow R\Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K) \longrightarrow R\Gamma (X_{\overline{s}}, K|_{X_{\overline{s}}}) \]

where $p : X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) \to X$ is the projection. Namely, consider the commutative diagram

\[ \xymatrix{ X_{\overline{s}} \ar[r] \ar[d]^{f_{\overline{s}}} & X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) \ar[r]_-p \ar[d]^{f'} & X \ar[d]^ f \\ \overline{s} \ar[r]^-i & \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) \ar[r]^-j & S } \]

We have the base change maps

\[ i^{-1}Rf'_*(p^{-1}K) \to Rf_{\overline{s}, *}(K|_{X_{\overline{s}}}) \quad \text{and}\quad j^{-1}Rf_*K \to Rf'_*(p^{-1}K) \]

(Cohomology on Sites, Remark 21.19.3) for the two squares in this diagram. Taking global sections we obtain the desired maps. By Cohomology on Sites, Remark 21.19.5 the composition of these two maps is the usual (base change) map $(Rf_*K)_{\overline{s}} \to R\Gamma (X_{\overline{s}}, K|_{X_{\overline{s}}})$.

Comments (2)

Comment #2356 by Yu-Liang Huang on

Just a small suggestion: the projection and the index on cohomology are both denoted by , perhaps it's better to change one of them.

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