NOTE ON GROTHENDIECK VANISHING THEOREM

HALVARD FAUSK

1. Introduction

A space X is called a Zariski space if all descending chains of closed subsets eventually terminates. Grothendieck proved the following vanishing theorem [1, 3.6.5].

Theorem 1.1. Let X be a Zariski space of (Krull) dimension n. Then for all sheaves of abelian groups F we have that

H^{i}(X; F) = 0

whenever i > n.

For example the underlying space of a Noetherian scheme is a Zariski space. It is natural to ask if the theorem is true for more general spaces, like the underlying space of quasi-compact schemes. The purpose of this note is to point out that the conclusion of the vanishing theorem is true for the underlying space of 0-dimensional quasi-compact schemes. The only step in Grothendieck’s original proof [1, 3.6.5] which does not immediately extend to quasi-compact schemes is the reduction to the case of irreducible spaces.

More generally we consider schemes that are paracompact. A scheme is said to be paracompact if for any open covering there is a locally finite refinement consisting of subspaces that are both (quasi) compact and open. We ask if the following is true.

Conjecture 1.2. Let X be the underlying space of a paracompact scheme of dimension n. Then for all sheaves of abelian groups F on X

H^{i}(X; F) = 0

whenever i > n.

In section 2 we recall Hochster’s characterization of the underlying spaces of schemes. We include a proof of the fact that the subspace of the prime ideal spectrum of a ring consisting of all minimal prime ideals is a Hausdorff space. This implies that the underlying space of any 0-dimensional scheme has a basis of open-closed subspaces.

In section 3 we introduce paracompact schemes. We prove that the global sections functor on the underlying space of a 0-dimensional paracompact scheme is exact on the category of sheaves of abelian groups. Hence all higher cohomology groups vanish. We also observe that any irreducible closed subscheme of a paracompact scheme is (quasi) compact.

Date: April 1, 2001.

1991 Mathematics Subject Classification. Primary ; Secondary .

In section 4 we note that the only step in Grothendieck’s proof of the vanishing theorem [1, 3.6.5], that does not extend is the reduction to the case of irreducible schemes. That is, H^{n}(X; F) = 0 if for all irreducible subsets Y in X, we have H^{m}(Y; F) = 0 for all m ≥ n.

For brevity we use the terms compact and paracompact without the Hausdorff condition (so compact instead of quasi-compact).

2. The space of minimal prime ideals

We recall some properties of the underlying space of a scheme. The following definition is from [3].

Definition 2.1. A space X is called spectral if 1) : X is T_{0}, so for any x, y ∈ X if x ∈ ȳ and y ∈ x̄ then x = y.

2) : X has a basis for the topology consisting of compact-open subspaces. In particular the intersection of two compact-open subspaces is again compact-open.

3) : Any irreducible closed subspace of X has a generic point.

4) : X is a compact space. A space is called locally spectral if it has an open covering by spectral spaces.

Remark 2.2. Since X is T_{0} it follows that the generic point in 3) is unique. Note that a basis for a topology is closed under finite intersections. Hence it is easy to see that if X has a basis of compact-open sets, then a finite intersection of compact-open sets in X is again compact-open.

It is well known that the underlying space of the prime ideal spectrum spec A of a ring A is a compact spectral space: 1) By definition of the Zariski topology. 4) The spectrum of a unital ring is compact. 2) The space spec A has an open basis consisting of spec A − ā ≅ spec A_{a}. An open subset of spec A is compact if and only if it is a finite union of open sets of the form above. Since

(spec A − ā_{1}) ∩ (spec A − ā_{2}) = spec A − \overline{a_{1}a_{2}}

it follows that finite intersections of compact-open subsets of spec A are again compact-open. 3) The prime ideal spectrum spec A is in bijection with the irreducible closed subspaces of spec A. The bijection is given by sending a prime ideal 𝔭 to its closure 𝔭̄ in spec A.

By definition the underlying space of any scheme is a locally spectral space. One can show that a locally spectral space satisfies 1), 3) and has a pre-basis for the topology consisting of compact-open sets. M. Hochster proved the following remarkable theorem [3] (we do not need to use this result here).

Theorem 2.3. A space is spectral if and only if it is the underlying space of the prime ideal spectrum of a commutative unital ring. A space is locally spectral if and only if it is the underlying space of a scheme.

We now show that the subspace min-spec A of spec A consisting of all minimal prime ideals in a commutative unital ring A is a Hausdorff space. Hence by property 2) it has a basis for the topology consisting of compactopen subspaces.

Lemma 2.4. A prime ideal 𝔭 is minimal if and only if for each a ∈ 𝔭 there is an element b ∉ 𝔭 such that ab is nilpotent.

Proof. ⇒ If the property is not satisfied then there exists an element a ∈ 𝔭 such that for all s ∈ S = A − 𝔭 the product sa is not nilpotent. Hence a and S generates a multiplicative system S̃ (not containing 0). The ring S̃^{−1}A is nonzero, hence has a maximal ideal. This gives a prime ideal in A that does not meet S̃, hence is strictly contained in 𝔭. So 𝔭 is not a minimal prime ideal. ⇐ Let 𝔮 ⊂ 𝔭 be a minimal prime ideal contained in 𝔭 (which exists by Zorn’s lemma). For any element a ∈ 𝔭 there is by assumption an element b ∉ 𝔭 such that ab is nilpotent. Since 𝔮 is a prime ideal a ∈ 𝔮, hence 𝔮 = 𝔭 and we conclude that 𝔭 must be a minimal prime ideal. □

For the next lemma let ann (a) denote the ideal {x ∈ A | ax is nilpotent}.

Lemma 2.5. For any element a ∈ A we have that

ā ∪ \overline{ann (a)} = spec A

and

ā ∩ \overline{ann (a)} ∩ min-spec A = ∅.

Proof. If a ∉ 𝔭 then for all b ∈ ann (a) we have b ∈ 𝔭 since 𝔭 is a prime ideal, hence ann (a) ⊂ 𝔭. So if 𝔭 ∉ ā then 𝔭 ∈ \overline{ann (a)}.

Let a ∈ 𝔭 where 𝔭 is a minimal prime. By Lemma 2.4 there is an element b ∈ A − 𝔭 such that ab is nilpotent, so ann (a) ⊄ 𝔭. This shows that min-spec A ∩ ā ∩ \overline{ann (a)} = ∅. □

Since the two closed sets ā ∩ min-spec A and \overline{ann (a)} ∩ min-spec A are complementary in min-spec A they are also open in min-spec A. Let 𝔭 ≠ 𝔮 be minimal prime ideals. Then we can assume that there is an element a ∈ 𝔭−𝔮. So 𝔭 ∈ ā and 𝔮 ∉ ā. Hence min-spec A is a totally disconnected space, in particular a Hausdorff space. Since compact subsets of Hausdorff spaces are closed we get that min-spec A has a basis for the topology consisting of open-closed sets.

Any open subspace U of a space X has dimension less then or equal to the dimension of X. So 0-dimensional schemes are locally isomorphic to the spectrum of 0-dimensional rings. Hence 0-dimensional schemes are Hausdorff.

Proposition 2.6. Let X be the underlying space of a 0-dimensional scheme. Then X has a basis for the topology given by open and closed sets.

For more information about the underlying space of min-spec A see [2, 4].

3. paracompact schemes

Definition 3.1. A scheme X is said to be paracompact if for any open covering {U_{α}} of X there is a locally finite refinement {V_{γ}} of {U_{α}} (so every V_{γ} is contained in some U_{α}) such that each V_{γ} is compact-open in X.

Locally finite means that for any point x in X there is an open neighborhood W_{x} of x such that W_{x} has nonempty intersection with only finitely many of the sets V_{γ}.

Remark 3.2. All compact schemes are paracompact. A closed subspace of a paracompact scheme is again paracompact.

A scheme is quasi-separated if the intersection of any two compact-open subspaces is again compact-open. Note that this is equivalent to require the intersection of any two open affine subschemes to be a finite union of open affine subschemes. All affine schemes are quasi-separated. All schemes of dimension 0 are quasi-separated. The underlying spaces of a quasi-separated schemes are exactly spaces satisfying properties 1), 2) and 3) of definition 2.1

Lemma 3.3. Let X be a paracompact quasi-separated scheme. Then any covering {U_{α}} of X has a compact-open refinement {V_{γ}}, such that for any given γ there are only finitely many γ′ such that V_{γ} ∩ V_{γ′} is nonempty.

Proof. Since the space X has a basis of compact-open sets and is paracompact, we can assume that each U_{α} in the covering of X is compact-open, and moreover that for any point x in X there is a compact-open neighborhood W_{x} of x which only meets finitely many of the U_{α}. Hence W_{x} = ∪_{α}(W_{x}∩U_{α}) is a finite union of compact-open sets. Each U_{α} is compact so there exists finitely many x_{1}^{α}, x_{2}^{α}, … x_{n_{α}}^{α} such that

U_{α} = ∪_{i=1}^{n_{α}}W_{x_{i}^{α}} ∩ U_{α}.

It is now easy to see that {W_{x_{i}^{α}} ∩ U_{α}} is a covering by compact-open subspaces such that any one of the W_{x_{i}^{α}} ∩U_{α} only intersects finitely many other subspaces of the form W_{x_{j}^{β}} ∩ U_{β}. □

Proposition 3.4. Let X be a paracompact scheme of dimension 0. Then the global sections functor Γ(X; −) is exact from the category of sheaves of abelian groups to the category of abelian groups. In particular

H^{n}(X; F) = 0

for all sheaves of abelian groups and n > 0.

Proof. Let ϵ : F → G be epic. For every element g ∈ G(X) there is a covering {U_{α}} of X and f_{α} ∈ F(U_{α}) such that ϵ(f_{α}) = g|_{U_{α}}. Since X is Hausdorff we can by the previous lemma assume that each U_{α} is an open-closed set and that each U_{α} intersects only finitely many other U_{β}.

We decompose U_{α} into a disjoint union of open-closed sets. Let the indexes of all the U_{β} which intersects U_{α} nonempty be β_{1}, β_{2}, …, β_{n}. The decomposition consist of the 2^{n} open-closed spaces obtained by forming all combinations of intersections

U_{α} ∩ ⋂_{i=1}^{n} \{ \begin{matrix}U_{β_{i}} \\ X − U_{β_{i}} \end{matrix}

(some of them possibly empty). This gives a decomposition of X into a disjoint union of open-closed sets X = ⨿_{γ} V_{γ}, such that each U_{α} is a finite disjoint union of some of the V_{γ}. Since F is a sheaf we have that

F(X) ≅ ∏_{γ} F(V_{γ}).

For each γ choose an α_{γ} such that V_{γ} ⊂ U_{α}. Denote by f the element in F(X) which corresponds under the isomorphism to

∏_{γ} f_{α_{γ}}|_{V_{γ}}.

Since for all γ

ϕ(f)|_{V_{γ}} = ϕ(f_{α_{γ}})|_{V_{γ}} = g|_{V_{γ}}

we get that ϕ(f) = g so

ϵ : Γ(X; F) → Γ(X; G)

is a surjective map. Hence the global sections functor Γ(X; −) is exact. □

Lemma 3.5. Let X be a paracompact scheme. Then all closed irreducible subsets of X are compact.

Proof. Any irreducible set is the closure x̄ of a unique generic point x. Let {U_{α}} be a locally finite covering of X by compact open sets. So the point x meets only finitely many of the U_{α}. Let

U = ∪_{U_{α}∌x}U_{α}.

This is an open set not containing the point x. Hence we have that

x̄ ⊂ X − U ⊂ ∪_{U_{α}∋x}U_{α}.

Since only finitely many U_{α} contains x the rightmost space is a finite union of compact spaces, hence is itself compact. So the closed subspace x̄ is also compact. □

4. Grothendieck vanishing theorem

A space is called a Zariski space if all descending chains of closed subsets eventually terminates. Zariski spaces can be decomposed into finitely many maximal irreducible closed subsets. Grothendieck proved the following vanishing theorem [1, 3.6.5].

Theorem 4.1. Let X be a Zariski space of (Krull) dimension n. Then for all sheaves of abelian groups F we have that

H^{i}(X; F) = 0

whenever i > n.

When X is a compact space it is easy to see that Γ(X; −) commutes with directed colimits, and that directed colimits of flasque sheaves are flasque. Since injective sheaves are flasque, and the sheaf cohomology can be computed using a flaque resolutions, we get that H^{∗}(X; −) commutes with directed colimits. It follows from [1, 3.6.1] that when X is compact H^{i}(X; F) = 0 for all sheaves of abelian groups F and all i > n if and only if

H^{i}(X; ℤ_{U}) = 0

for all open subsets U of X and all i > n.

Following the proof of theorem [1, 3.6.5], using lemma 3.3 and proposition 3.4 we observe that the conclusion of the vanishing theorem holds for paracompact schemes, if we can prove the following.

Conjecture 4.2. Let X be the underlying space of a paracompact scheme of dimension n. Given that the vanishing theorem is true for paracompact schemes of dimension less then n. If for all closed irreducible subspaces Y of X and for all abelian sheaves G on Y the cohomology groups H^{i}(Y; G) = 0 whenever i > n, then H^{i}(X; F) = 0 for all abelian sheaves F on X and all i > n

References

[1] A. Grothendieck, Sur quelques points d’algebre homologique, Chapitre V, Tohoku Mathematical Journal, second series, Vol. 9 (1957).

[2] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Transactions of AMS 115 (1965) 110-130.

[3] M. Hochster, Prime ideal structure in commutative rings, Transactions of AMS, 142 (1969) 43-60.

[4] M. Hochster, The minimal prime spectrum of a commutative ring, Canadian J. Math. 23 (1971) 749-758.

Department of Mathematics, The University of Chicago, Chicago IL 60637

   E-mail address: fausk@math.uchicago.edu