Lemma 20.20.1. Let $i : Z \to X$ be a closed immersion of topological spaces. For any abelian sheaf $\mathcal{F}$ on $Z$ we have $H^ p(Z, \mathcal{F}) = H^ p(X, i_*\mathcal{F})$.

## 20.20 Vanishing on Noetherian topological spaces

The aim is to prove a theorem of Grothendieck namely Proposition 20.20.7. See [Tohoku].

**Proof.**
This is true because $i_*$ is exact (see Modules, Lemma 17.6.1), and hence $R^ pi_* = 0$ as a functor (Derived Categories, Lemma 13.16.9). Thus we may apply Lemma 20.13.6.
$\square$

Lemma 20.20.2. Let $X$ be an irreducible topological space. Then $H^ p(X, \underline{A}) = 0$ for all $p > 0$ and any abelian group $A$.

**Proof.**
Recall that $\underline{A}$ is the constant sheaf as defined in Sheaves, Definition 6.7.4. It is clear that for any nonempty open $U \subset X$ we have $\underline{A}(U) = A$ as $X$ is irreducible (and hence $U$ is connected). We will show that the higher Čech cohomology groups $\check{H}^ p(\mathcal{U}, \underline{A})$ are zero for any open covering $\mathcal{U} : U = \bigcup _{i\in I} U_ i$ of an open $U \subset X$. Then the lemma will follow from Lemma 20.11.8.

Recall that the value of an abelian sheaf on the empty open set is $0$. Hence we may clearly assume $U_ i \not= \emptyset $ for all $i \in I$. In this case we see that $U_ i \cap U_{i'} \not= \emptyset $ for all $i, i' \in I$. Hence we see that the Čech complex is simply the complex

We have to see this has trivial higher cohomology groups. We can see this for example because this is the Čech complex for the covering of a $1$-point space and Čech cohomology agrees with cohomology on such a space. (You can also directly verify it by writing an explicit homotopy.) $\square$

Lemma 20.20.3. Let $X$ be a topological space such that the intersection of any two quasi-compact opens is quasi-compact. Let $\mathcal{F} \subset \underline{\mathbf{Z}}$ be a subsheaf generated by finitely many sections over quasi-compact opens. Then there exists a finite filtration

by abelian subsheaves such that for each $0 < i \leq n$ there exists a short exact sequence

with $j : U \to X$ and $j' : V \to X$ the inclusion of quasi-compact opens into $X$.

**Proof.**
Say $\mathcal{F}$ is generated by the sections $s_1, \ldots , s_ t$ over the quasi-compact opens $U_1, \ldots , U_ t$. Since $U_ i$ is quasi-compact and $s_ i$ a locally constant function to $\mathbf{Z}$ we may assume, after possibly replacing $U_ i$ by the parts of a finite decomposition into open and closed subsets, that $s_ i$ is a constant section. Say $s_ i = n_ i$ with $n_ i \in \mathbf{Z}$. Of course we can remove $(U_ i, n_ i)$ from the list if $n_ i = 0$. Flipping signs if necessary we may also assume $n_ i > 0$. Next, for any subset $I \subset \{ 1, \ldots , t\} $ we may add $\bigcup _{i \in I} U_ i$ and $\gcd (n_ i, i \in I)$ to the list. After doing this we see that our list $(U_1, n_1), \ldots , (U_ t, n_ t)$ satisfies the following property: For $x \in X$ set $I_ x = \{ i \in \{ 1, \ldots , t\} \mid x \in U_ i\} $. Then $\gcd (n_ i, i \in I_ x)$ is attained by $n_ i$ for some $i \in I_ x$.

As our filtration we take $\mathcal{F}_0 = (0)$ and $\mathcal{F}_ n$ generated by the sections $n_ i$ over $U_ i$ for those $i$ such that $n_ i \leq n$. It is clear that $\mathcal{F}_ n = \mathcal{F}$ for $n \gg 0$. Moreover, the quotient $\mathcal{F}_ n/\mathcal{F}_{n - 1}$ is generated by the section $n$ over $U = \bigcup _{n_ i \leq n} U_ i$ and the kernel of the map $j_!\underline{\mathbf{Z}}_ U \to \mathcal{F}_ n/\mathcal{F}_{n - 1}$ is generated by the section $n$ over $V = \bigcup _{n_ i \leq n - 1} U_ i$. Thus a short exact sequence as in the statement of the lemma. $\square$

Lemma 20.20.4. Let $X$ be a topological space. Let $d \geq 0$ be an integer. Assume

$X$ is quasi-compact,

the quasi-compact opens form a basis for $X$, and

the intersection of two quasi-compact opens is quasi-compact.

$H^ p(X, j_!\underline{\mathbf{Z}}_ U) = 0$ for all $p > d$ and any quasi-compact open $j : U \to X$.

Then $H^ p(X, \mathcal{F}) = 0$ for all $p > d$ and any abelian sheaf $\mathcal{F}$ on $X$.

**Proof.**
Let $S = \coprod _{U \subset X} \mathcal{F}(U)$ where $U$ runs over the quasi-compact opens of $X$. For any finite subset $A = \{ s_1, \ldots , s_ n\} \subset S$, let $\mathcal{F}_ A$ be the subsheaf of $\mathcal{F}$ generated by all $s_ i$ (see Modules, Definition 17.4.5). Note that if $A \subset A'$, then $\mathcal{F}_ A \subset \mathcal{F}_{A'}$. Hence $\{ \mathcal{F}_ A\} $ forms a system over the directed partially ordered set of finite subsets of $S$. By Modules, Lemma 17.4.6 it is clear that

by looking at stalks. By Lemma 20.19.1 we have

Hence it suffices to prove the vanishing for the abelian sheaves $\mathcal{F}_ A$. In other words, it suffices to prove the result when $\mathcal{F}$ is generated by finitely many local sections over quasi-compact opens of $X$.

Suppose that $\mathcal{F}$ is generated by the local sections $s_1, \ldots , s_ n$. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf generated by $s_1, \ldots , s_{n - 1}$. Then we have a short exact sequence

From the long exact sequence of cohomology we see that it suffices to prove the vanishing for the abelian sheaves $\mathcal{F}'$ and $\mathcal{F}/\mathcal{F}'$ which are generated by fewer than $n$ local sections. Hence it suffices to prove the vanishing for sheaves generated by at most one local section. These sheaves are exactly the quotients of the sheaves $j_!\underline{\mathbf{Z}}_ U$ where $U$ is a quasi-compact open of $X$.

Assume now that we have a short exact sequence

with $U$ quasi-compact open in $X$. It suffices to show that $H^ q(X, \mathcal{K})$ is zero for $q \geq d + 1$. As above we can write $\mathcal{K}$ as the filtered colimit of subsheaves $\mathcal{K}'$ generated by finitely many sections over quasi-compact opens. Then $\mathcal{F}$ is the filtered colimit of the sheaves $j_!\underline{\mathbf{Z}}_ U/\mathcal{K}'$. In this way we reduce to the case that $\mathcal{K}$ is generated by finitely many sections over quasi-compact opens. Note that $\mathcal{K}$ is a subsheaf of $\underline{\mathbf{Z}}_ X$. Thus by Lemma 20.20.3 there exists a finite filtration of $\mathcal{K}$ whose successive quotients $\mathcal{Q}$ fit into a short exact sequence

with $j'' : W \to X$ and $j' : V \to X$ the inclusions of quasi-compact opens. Hence the vanishing of $H^ p(X, \mathcal{Q})$ for $p > d$ follows from our assumption (in the lemma) on the vanishing of the cohomology groups of $j''_!\underline{\mathbf{Z}}_ W$ and $j'_!\underline{\mathbf{Z}}_ V$. Returning to $\mathcal{K}$ this, via an induction argument using the long exact cohomology sequence, implies the desired vanishing for it as well. $\square$

Example 20.20.5. Let $X = \mathbf{N}$ endowed with the topology whose opens are $\emptyset $, $X$, and $U_ n = \{ i \mid i \leq n\} $ for $n \geq 1$. An abelian sheaf $\mathcal{F}$ on $X$ is the same as an inverse system of abelian groups $A_ n = \mathcal{F}(U_ n)$ and $\Gamma (X, \mathcal{F}) = \mathop{\mathrm{lim}}\nolimits A_ n$. Since the inverse limit functor is not an exact functor on the category of inverse systems, we see that there is an abelian sheaf with nonzero $H^1$. Finally, the reader can check that $H^ p(X, j_!\mathbf{Z}_ U) = 0$, $p \geq 1$ if $j : U = U_ n \to X$ is the inclusion. Thus we see that $X$ is an example of a space satisfying conditions (2), (3), and (4) of Lemma 20.20.4 for $d = 0$ but not the conclusion.

Lemma 20.20.6. Let $X$ be an irreducible topological space. Let $\mathcal{H} \subset \underline{\mathbf{Z}}$ be an abelian subsheaf of the constant sheaf. Then there exists a nonempty open $U \subset X$ such that $\mathcal{H}|_ U = \underline{d\mathbf{Z}}_ U$ for some $d \in \mathbf{Z}$.

**Proof.**
Recall that $\underline{\mathbf{Z}}(V) = \mathbf{Z}$ for any nonempty open $V$ of $X$ (see proof of Lemma 20.20.2). If $\mathcal{H} = 0$, then the lemma holds with $d = 0$. If $\mathcal{H} \not= 0$, then there exists a nonempty open $U \subset X$ such that $\mathcal{H}(U) \not= 0$. Say $\mathcal{H}(U) = n\mathbf{Z}$ for some $n \geq 1$. Hence we see that $\underline{n\mathbf{Z}}_ U \subset \mathcal{H}|_ U \subset \underline{\mathbf{Z}}_ U$. If the first inclusion is strict we can find a nonempty $U' \subset U$ and an integer $1 \leq n' < n$ such that $\underline{n'\mathbf{Z}}_{U'} \subset \mathcal{H}|_{U'} \subset \underline{\mathbf{Z}}_{U'}$. This process has to stop after a finite number of steps, and hence we get the lemma.
$\square$

Proposition 20.20.7 (Grothendieck). Let $X$ be a Noetherian topological space. If $\dim (X) \leq d$, then $H^ p(X, \mathcal{F}) = 0$ for all $p > d$ and any abelian sheaf $\mathcal{F}$ on $X$.

**Proof.**
We prove this lemma by induction on $d$. So fix $d$ and assume the lemma holds for all Noetherian topological spaces of dimension $< d$.

Let $\mathcal{F}$ be an abelian sheaf on $X$. Suppose $U \subset X$ is an open. Let $Z \subset X$ denote the closed complement. Denote $j : U \to X$ and $i : Z \to X$ the inclusion maps. Then there is a short exact sequence

see Modules, Lemma 17.7.1. Note that $j_!j^*\mathcal{F}$ is supported on the topological closure $Z'$ of $U$, i.e., it is of the form $i'_*\mathcal{F}'$ for some abelian sheaf $\mathcal{F}'$ on $Z'$, where $i' : Z' \to X$ is the inclusion.

We can use this to reduce to the case where $X$ is irreducible. Namely, according to Topology, Lemma 5.9.2 $X$ has finitely many irreducible components. If $X$ has more than one irreducible component, then let $Z \subset X$ be an irreducible component of $X$ and set $U = X \setminus Z$. By the above, and the long exact sequence of cohomology, it suffices to prove the vanishing of $H^ p(X, i_*i^*\mathcal{F})$ and $H^ p(X, i'_*\mathcal{F}')$ for $p > d$. By Lemma 20.20.1 it suffices to prove $H^ p(Z, i^*\mathcal{F})$ and $H^ p(Z', \mathcal{F}')$ vanish for $p > d$. Since $Z'$ and $Z$ have fewer irreducible components we indeed reduce to the case of an irreducible $X$.

If $d = 0$ and $X$ is irreducible, then $X$ is the only nonempty open subset of $X$. Hence every sheaf is constant and higher cohomology groups vanish (for example by Lemma 20.20.2).

Suppose $X$ is irreducible of dimension $d > 0$. By Lemma 20.20.4 we reduce to the case where $\mathcal{F} = j_!\underline{\mathbf{Z}}_ U$ for some open $U \subset X$. In this case we look at the short exact sequence

where $Z = X \setminus U$. By Lemma 20.20.2 we have the vanishing of $H^ p(X, \underline{\mathbf{Z}}_ X)$ for all $p \geq 1$. By induction we have $H^ p(X, i_*\underline{\mathbf{Z}}_ Z) = H^ p(Z, \underline{\mathbf{Z}}_ Z) = 0$ for $p \geq d$. Hence we win by the long exact cohomology sequence. $\square$

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