20.22 Cohomology on spectral spaces
A key result on the cohomology of spectral spaces is Lemma 20.19.2 which loosely speaking says that cohomology commutes with cofiltered limits in the category of spectral spaces as defined in Topology, Definition 5.23.1. This can be applied to give analogues of Lemmas 20.16.3 and 20.18.1 as follows.
Lemma 20.22.1. Let $X$ be a spectral space. Let $\mathcal{F}$ be an abelian sheaf on $X$. Let $E \subset X$ be a quasi-compact subset. Let $W \subset X$ be the set of points of $X$ which specialize to a point of $E$.
$H^ p(W, \mathcal{F}|_ W) = \mathop{\mathrm{colim}}\nolimits H^ p(U, \mathcal{F})$ where the colimit is over quasi-compact open neighbourhoods of $E$,
$H^ p(W \setminus E, \mathcal{F}|_{W \setminus E}) = \mathop{\mathrm{colim}}\nolimits H^ p(U \setminus E, \mathcal{F}|_{U \setminus E})$ if $E$ is a constructible subset.
Proof.
From Topology, Lemma 5.24.7 we see that $W = \mathop{\mathrm{lim}}\nolimits U$ where the limit is over the quasi-compact opens containing $E$. Each $U$ is a spectral space by Topology, Lemma 5.23.5. Thus we may apply Lemma 20.19.2 to conclude that (1) holds. The same proof works for part (2) except we use Topology, Lemma 5.24.8.
$\square$
Lemma 20.22.2. Let $f : X \to Y$ be a spectral map of spectral spaces. Let $y \in Y$. Let $E \subset Y$ be the set of points specializing to $y$. Let $\mathcal{F}$ be an abelian sheaf on $X$. Then $(R^ pf_*\mathcal{F})_ y = H^ p(f^{-1}(E), \mathcal{F}|_{f^{-1}(E)})$.
Proof.
Observe that $E = \bigcap V$ where $V$ runs over the quasi-compact open neighbourhoods of $y$ in $Y$. Hence $f^{-1}(E) = \bigcap f^{-1}(V)$. This implies that $f^{-1}(E) = \mathop{\mathrm{lim}}\nolimits f^{-1}(V)$ as topological spaces. Since $f$ is spectral, each $f^{-1}(V)$ is a spectral space too (Topology, Lemma 5.23.5). We conclude that $f^{-1}(E)$ is a spectral space and that
\[ H^ p(f^{-1}(E), \mathcal{F}|_{f^{-1}(E)}) = \mathop{\mathrm{colim}}\nolimits H^ p(f^{-1}(V), \mathcal{F}) \]
by Lemma 20.19.2. On the other hand, the stalk of $R^ pf_*\mathcal{F}$ at $y$ is given by the colimit on the right.
$\square$
Lemma 20.22.3. Let $X$ be a profinite topological space. Then $H^ q(X, \mathcal{F}) = 0$ for all $q > 0$ and all abelian sheaves $\mathcal{F}$.
Proof.
Any open covering of $X$ can be refined by a finite disjoint union decomposition with open parts, see Topology, Lemma 5.22.4. Hence if $\mathcal{F} \to \mathcal{G}$ is a surjection of abelian sheaves on $X$, then $\mathcal{F}(X) \to \mathcal{G}(X)$ is surjective. In other words, the global sections functor is an exact functor. Therefore its higher derived functors are zero, see Derived Categories, Lemma 13.16.9.
$\square$
The following result on cohomological vanishing improves Grothendieck's result (Proposition 20.20.7) and can be found in [Scheiderer].
reference
Proposition 20.22.4. Let $X$ be a spectral space of Krull dimension $d$. Let $\mathcal{F}$ be an abelian sheaf on $X$.
$H^ q(X, \mathcal{F}) = 0$ for $q > d$,
$H^ d(X, \mathcal{F}) \to H^ d(U, \mathcal{F})$ is surjective for every quasi-compact open $U \subset X$,
$H^ q_ Z(X, \mathcal{F}) = 0$ for $q > d$ and any constructible closed subset $Z \subset X$.
Proof.
We prove this result by induction on $d$.
If $d = 0$, then $X$ is a profinite space, see Topology, Lemma 5.23.8. Thus (1) holds by Lemma 20.22.3. If $U \subset X$ is quasi-compact open, then $U$ is also closed as a quasi-compact subset of a Hausdorff space. Hence $X = U \amalg (X \setminus U)$ as a topological space and we see that (2) holds. Given $Z$ as in (3) we consider the long exact sequence
\[ H^{q - 1}(X, \mathcal{F}) \to H^{q - 1}(X \setminus Z, \mathcal{F}) \to H^ q_ Z(X, \mathcal{F}) \to H^ q(X, \mathcal{F}) \]
Since $X$ and $U = X \setminus Z$ are profinite (namely $U$ is quasi-compact because $Z$ is constructible) and since we have (2) and (1) we obtain the desired vanishing of the cohomology groups with support in $Z$.
Induction step. Assume $d \geq 1$ and assume the proposition is valid for all spectral spaces of dimension $< d$. We first prove part (2) for $X$. Let $U$ be a quasi-compact open. Let $\xi \in H^ d(U, \mathcal{F})$. Set $Z = X \setminus U$. Let $W \subset X$ be the set of points specializing to $Z$. By Lemma 20.22.1 we have
\[ H^ d(W \setminus Z, \mathcal{F}|_{W \setminus Z}) = \mathop{\mathrm{colim}}\nolimits _{Z \subset V} H^ d(V \setminus Z, \mathcal{F}) \]
where the colimit is over the quasi-compact open neighbourhoods $V$ of $Z$ in $X$. By Topology, Lemma 5.24.7 we see that $W \setminus Z$ is a spectral space. Since every point of $W$ specializes to a point of $Z$, we see that $W \setminus Z$ is a spectral space of Krull dimension $< d$. By induction hypothesis we see that the image of $\xi $ in $H^ d(W \setminus Z, \mathcal{F}|_{W \setminus Z})$ is zero. By the displayed formula, there exists a $Z \subset V \subset X$ quasi-compact open such that $\xi |_{V \setminus Z} = 0$. Since $V \setminus Z = V \cap U$ we conclude by the Mayer-Vietoris (Lemma 20.8.2) for the covering $X = U \cup V$ that there exists a $\tilde\xi \in H^ d(X, \mathcal{F})$ which restricts to $\xi $ on $U$ and to zero on $V$. In other words, part (2) is true.
Proof of part (1) assuming (2). Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $. Set
\[ \mathcal{G} = \mathop{\mathrm{Im}}(\mathcal{I}^{d - 1} \to \mathcal{I}^ d) = \mathop{\mathrm{Ker}}(\mathcal{I}^ d \to \mathcal{I}^{d + 1}) \]
For $U \subset X$ quasi-compact open we have a map of exact sequences as follows
\[ \xymatrix{ \mathcal{I}^{d - 1}(X) \ar[r] \ar[d] & \mathcal{G}(X) \ar[r] \ar[d] & H^ d(X, \mathcal{F}) \ar[d] \ar[r] & 0 \\ \mathcal{I}^{d - 1}(U) \ar[r] & \mathcal{G}(U) \ar[r] & H^ d(U, \mathcal{F}) \ar[r] & 0 } \]
The sheaf $\mathcal{I}^{d - 1}$ is flasque by Lemma 20.12.2 and the fact that $d \geq 1$. By part (2) we see that the right vertical arrow is surjective. We conclude by a diagram chase that the map $\mathcal{G}(X) \to \mathcal{G}(U)$ is surjective. By Lemma 20.12.6 we conclude that $\check{H}^ q(\mathcal{U}, \mathcal{G}) = 0$ for $q > 0$ and any finite covering $\mathcal{U} : U = U_1 \cup \ldots \cup U_ n$ of a quasi-compact open by quasi-compact opens. Applying Lemma 20.11.9 we find that $H^ q(U, \mathcal{G}) = 0$ for all $q > 0$ and all quasi-compact opens $U$ of $X$. By Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) we conclude that
\[ H^ q(X, \mathcal{F}) = H^ q\left( \Gamma (X, \mathcal{I}^0) \to \ldots \to \Gamma (X, \mathcal{I}^{d - 1}) \to \Gamma (X, \mathcal{G}) \right) \]
In particular the cohomology group vanishes if $q > d$.
Proof of (3). Given $Z$ as in (3) we consider the long exact sequence
\[ H^{q - 1}(X, \mathcal{F}) \to H^{q - 1}(X \setminus Z, \mathcal{F}) \to H^ q_ Z(X, \mathcal{F}) \to H^ q(X, \mathcal{F}) \]
Since $X$ and $U = X \setminus Z$ are spectral spaces (Topology, Lemma 5.23.5) of dimension $\leq d$ and since we have (2) and (1) we obtain the desired vanishing.
$\square$
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