ZETA FUNCTIONS

YOSHIFUMI TSUCHIMOTO

Spec and Proj.

Definition 10.1. For any commutative ring A, we define its spectrum as

Spec(A) = {𝔭 ⊂ A; 𝔭 is a prime ideal of A}.

For any subset S of A we define

V (S) = V_{Spec A}(S) = {𝔭 ∈ Spec A; 𝔭 ⊃ S}

Then we may topologize Spec(A) in such a way that the closed sets are sets of the form V (S) for some S. Namely,

F : closed ⟺ ∃S ⊂ A(F = V (S))

We refer to the topology as the Zariski topology.

Exercise 10.1. Prove that Zariski topology is indeed a topolgy. That means, the collection {V (S)} satisfies the axiom of closed sets.

Exercise 10.2. Let A be a ring. Then:

(1) Show that for any f ∈ A, D(f) = {𝔭 ∈ Spec(A); f ∉ 𝔭} is an open set of Spec(A).

(2) Show that given a point 𝔭 of Spec(A) and an open set U which contains 𝔭, we may always find an element f ∈ A such that 𝔭 ∈ D(f) ⊂ U. (In other words, {D(f)} forms an open base of the Zariski topology.

Lemma 10.2. For any ring A, the following facts holds.

(1) For any subset S of A, we have

V (S) = ⋂_{s∈S} V ({s}).

(2) For any subset S of A, let us denote by ⟨S⟩ the ideal of A generated by S. then we have

V (S) = V (⟨S⟩)

Proposition 10.3. For any ring homomorphism φ : A → B, we have a map

Spec(φ) : Spec(B) ∋ 𝔭 ↦ φ^{−1}(𝔭) ∈ Spec(A).

It is continuous with respect to the Zariski topology.

Proposition 10.4. For any ring A, the following statements hold.

(1) For any ideal I of A, let us denote by π_{I} : A → A/I the canonical projection. Then Spec(π_{I}) gives a homeomorphism between Spec(A/I) and V_{Spec A}(I).

(2) For any element s of A, let us denote by ι_{s} : A → A[s^{−1}] be the canonical map. Then Spec(ι_{s}) gives a homeomorphism between Spec(A[s^{−1}]) and ∁V_{Spec_{A}}({s}).

Definition 10.5. Let X be a topological space. A closed set F of X is said to be reducible if there exist closed sets F_{1} and F_{2} such that

F = F_{1} ∪ F_{2}, F_{1} ≠ F, F_{2} ≠ F

holds. F is said to be irreducible if it is not reducible.

Definition 10.6. Let I be an ideal of a ring A. Then we define its radical to be

\sqrt{I} = \{x ∈ A; ∃N ∈ ℤ_{>0} such that x^{N} ∈ I\}.

Proposition 10.7. Let A be a ring. Then;

(1) For any ideal I of A, we have V (I) = V (\sqrt{I}).

(2) For two ideals I, J of A, V (I) = V (J) holds if and only if \sqrt{I} = \sqrt{J}.

(3) For an ideal I of A, V (I) is irreducible if and only if \sqrt{I} is a prime ideal.

It is knwon that Spec A has a structure of “locally ringed space”. A locally ringed space which locally lookes like an affine spectrum of a ring is called a scheme.

Definition 10.8. Let S = ⨁_{n∈ℕ} S_{n} be a ℕ-graded ring. We put S_{+} = ⨁_{n>0} S_{n}.

We define

Proj(S) = {𝔭 ⊂ S; 𝔭 is a homogeneous prime ideal of S, 𝔭 ⊅ S_{+}}.

It is known that Proj(S) carries a ringed space strucure on it and that it is a scheme.

Definition 10.9. Let R be a ring. Let I be an ideal of R. The scheme X̃ = Proj(S) associated to the graded ring S = ⨁_{n∈ℕ} I^{n} is called the blowing up of X with respect to I.

sheaves

10.1. sheaves. Affine spectrum Spec(A) of a ring A carries one more important structure. Namely, its structure sheaf.

We will firstly review some definitions and first properties of sheaves. To illustrate the idea, we recall an easy lemma in topology.

Lemma 10.10 (Gluing lemma). Let X, Y be a topological spaces. Let {U_{λ}}_{λ∈Λ} be an open covering of X.

(1) If we are given a collection of continuous maps {f_{λ} : U_{λ} → Y }_{λ∈Λ} such that

f_{λ}|_{U_{λ}∩U_{μ}} = f_{μ}|_{U_{λ}∩U_{μ}}

holds for any pair (λ, μ) ∈ Λ^{2}, then we have a unique continuous map f : X → Y such that

f|_{U_{λ}} = f_{λ}

holds for any λ ∈ Λ.

(2) Conversely, if we are given a continuous map f : X → Y, then we obtain a collection of maps {f_{λ} : U_{λ} → Y }_{λ∈Λ} by restriction.

Proof. (1) It is easy to verify that we have a well-defined map

f : X → Y

with

f|_{U_{λ}} = f_{λ}.

The continuity of f is proved by verifying that the inverse image of any open set V ⊂ Y by f is open in X. □

10.1.1. A convention. Before proceeding further, we employ the following convention.

For an open covering {U_{λ}}_{λ∈Λ} of a topological space X, we write

U_{λμ} = U_{λ} ∩ U_{μ}, U_{λμν} = U_{λ} ∩ U_{μ} ∩ U_{ν},

and so on.

10.1.2. presheaves. We first define presheaves.

Definition 10.11. Let X be a topological space. We say “a presheaf ℱ of rings over X is given” if we are given the following data.

(1) For each open set U ⊂ X, a ring denoted by ℱ(U). (which is called the ring of sections of ℱ on U.)

(2) For each pair U, V of open subsets of X such that V ⊂ U, a ring homomorphism (called restriction)

ρ_{V U} : ℱ(U) → ℱ(V).

with the properties (1) ℱ(∅) = 0.

(2) We have ρ_{U,U} = identity for any open subset U ⊂ X.

(3) We have

ρ_{W V} ρ_{V U} = ρ_{W V}

for any open sets U, V, W ⊂ X such that W ⊂ V ⊂ U.

10.1.3. sheaves.

Definition 10.12. Let X be a topological space. A presheaf ℱ of rings over X is called a sheaf if for any open set U ⊂ X and for any open covering {U_{λ}}_{λ∈Λ} of U, it satisfies the following conditions.

(1) (“Locality”) If there is given a local section f, g ∈ ℱ(U) such

that

ρ_{U_{λ}U}(f) = ρ_{U_{λ}U}(g)

holds for all λ ∈ Λ, then we have f = g

(2) (“Gluing lemma”). If there is given a collection of sections {f_{λ}}_{λ∈Λ} such that

ρ_{U_{λμ}U_{λ}}(f_{λ}) = ρ_{U_{λμ}U_{μ}}(f_{μ})

holds for any pair (λ, μ) ∈ Λ^{2}, then we have a section f ∈ ℱ(U) such that

ρ_{U_{λ}U}(f) = f_{λ}

holds for all λ ∈ Λ.

We may similarly define sheaf of sets, sheaf of modules, etc.

Lemma 10.13. Let X be a topological set with an open base 𝔘. To define a sheaf ℱ over X we only need to define ℱ(U) for every member U of 𝔘 and check the sheaf axiom for open bases. In precise, given such data, we may always find a unique sheaf 𝒢 on X such that G(U) ≅ F (U) holds in a natural way. (That means, the isomorphism commutes with restrictions wherever they are defined.)

Proof. Let ℱ be such. For any open set U ⊂ X, we define a presheaf 𝒢 by the following formula.

𝒢(U) = \{(s_{V}) ∈ ∏_{V ∈𝔘,V ⊂U} ℱ(V); \begin{matrix}ρ_{W V} (s_{V}) = s_{W} for any V, W ∈ 𝔘 \\ with the property W ⊂ V ⊂ U. \end{matrix} \}

Restriction map of 𝒢 is defined in an obvious manner.

Then it is easy to see that 𝒢 satisfies the sheaf axiom and that

𝒢(U) ≅ ℱ(U)

holds for any U ∈ 𝔘 in a natural way.

□

Lemma 10.14. Let A be a ring.

(1) We have a sheaf 𝒪 of rings on Spec(A) which is defined uniquely by the property

𝒪(O_{f}) = A_{f} (∀f ∈ A)

(2) For any A-module M we have a sheaf M̃ of modules on Spec(A) which is defined uniquely by the property

M̃(O_{f}) = M_{f} (∀f ∈ A)

(3) For any A-module M, the sheaf M̃ is a sheaf of 𝒪-modules on Spec(A). That means, it is a sheaf of modules over Spec(A) with an additional 𝒪-action (which is defined in an obvious way.)

Proof. We prove (2).

From the previous Lemma, we only need to prove locality and gluing lemma for open sets of the form O_{f}. That means, in proving the properties (1) and (2) of Definition 10.12, we may assume that U_{λ} = O_{f_{λ}}, U = O_{f} for some elements f_{λ}, f ∈ A.

Furthermore, in doing so we may use the identification O_{f} ≈ Spec A_{f}. By replacing A by A_{f}, this means that we may assume that O_{f} = Spec(A).

To sum up, we may assume

U = Spec(A), U_{λ} = O_{f_{λ}}.

To simplify the notation, in the rest of the proof, we shall denote by

i_{λ} : M → M_{f_{λ}}

the canonical map which we have formerly written i_{f_{λ}}. Furthermore, for any pair λ, μ of the index set, we shall denote by i_{λμ} the canonical map

i_{λμ} : M → M_{f_{λ}f_{μ}}.

Locality: Compactness of Spec(A) (Theorem ??) implies that there exist finitely many open sets {O_{f_{j}}}_{j=1}^{k} among U_{λ} such that ∪_{j=1}^{k}O_{f_{j}} = Spec(A). In particular there exit elements {c_{j}}_{j=1}^{k} of A such that

(PU)  c_{1}f_{1} + c_{2}f_{2} + ⋯ + c_{k}f_{k} = 1

holds.

Let m, n ∈ M be elements such that

i_{j}(m) = i_{j}(n) ( in M_{f_{j}}.)

With the help of the “module version” of Lemma ??, we see that for each j, there exist positive integers N_{j} such that

f_{j}^{N_{j}}(m − n) = 0

holds for all j ∈ {1, 2, 3, …, k}. Let us take the maximum N of {N_{j}}. It is easy to see that

f_{j}^{N}(m − n) = 0

holds for any j. On the other hand, taking (kN)-th power of the equation (PU) above, we may find elements {a_{j}} ⊂ A such that

a_{1}f_{1}^{N} + a_{2}f_{2}^{N} + ⋯ + a_{k}f_{k}^{N} = 1

holds. Then we compute

m − n = (a_{1}f_{1}^{N} + a_{2}f_{2}^{N} + ⋯ + a_{k}f_{k}^{N})(m − n) = 0

to conclude that m = n.

Gluing lemma:

Let {m_{λ} ∈ M_{f_{λ}}} be given such that they satisfy

i_{λμ}(m_{λ}) = i_{λμ}(m_{μ})

for any λ, μ. We fist choose a finite subcovering {O_{f_{j}} = U_{λ_{j}}}_{j=1}^{k} of {U_{λ}}. Then we may choose a positive integer N_{1} such that

m_{λ_{j}} = x_{j}/f_{j}^{N_{1}} (∃x_{j} ∈ M)

holds for all j ∈ {1, 2, 3, …, k}.

i_{jl}(x_{j}f_{l}^{N_{1}}) = i_{jl}(x_{l}f_{j}^{N})

Then by the same argument which appears in the “locality” part, there exists a positive integer N_{2} such that

(f_{i}f_{j})^{N_{2}}(x_{j}f_{l}^{N_{1}} − x_{l}f_{j}^{N_{1}}) = 0

holds for all j, l ∈ {1, 2, 3, …, k}. We rewrite the above equation as follows.

(f_{j}^{N_{2}}x_{j})f_{l}^{N_{2}+N_{1}} − (f_{l}^{N_{2}}x_{l})f_{j}^{N_{2}+N_{1}} = 0.

On the other hand, by taking k(N_{1} + N_{2})-th power of the equation (PU), we may see that there exist elements {b_{j}} ∈ A such that

∑_{j=1}^{k} b_{j}f_{j}^{N_{1}+N_{2}} = 1

holds.

Now we put

n = ∑_{j} b_{j}(f_{j}^{N_{2}}x_{j}).

Then since for any l

(f_{j}^{N_{2}}x_{j}) = (f_{l}^{N_{2}}x_{l})f_{j}^{N_{2}+N_{1}}/f_{l}^{N_{2}+N_{1}} = f_{j}^{N_{2}+N_{1}}m_{λ_{l}}

holds on O_{l}, we have i_{l}(n) = m_{λ_{l}}.

Now, take any other open set O_{f_{μ}} = U_{μ} from the covering {U_{λ}}. {O_{f_{j}}}_{j=1}^{k} ∪ {O_{f_{μ}}} is again a finite open covering of Spec(A). We thus know from the argument above that there exists an element n_{1} of M such that

i_{j}(n_{1}) = m_{f_{j}}, i_{μ}(n_{1}) = m_{μ}.

From the locality, n_{1} coincides with n. In particular, i_{μ}(n) = m_{μ} holds. This means n satisfies the requirement for the “glued object”.

□

Corollary 10.15. Let A be a commutative ring. Let B be a noncommutative ring which contains A as a central subalgebra (that means, Z(B) ⊃ A). Then there exists a sheaf B̃ of 𝒪-algebras over Spec(A)

10.2. Benefit of being a sheaf. By saying that 𝒪 is a sheaf on Spec(A), we may easily use the arguments we have used to proved the locality and the gluing lemma.

For example, the proof we gave in Theorem ??, especially the part where we chose the idempotent p_{1}, was a bit complicated.

Let us give another proof using the sheaf arguments. There exists a unique element p ∈ A = 𝒪(Spec(A)) which coincides with 1 on U_{1} = V (J) and with 0 on U_{2} = V (I). From the uniqueness we see that

p^{2} = p

holds since p^{2} satisfies the same properties as p. The rest of the proof is the same.

As the second easier example, we consider the following undergraduate problem. Problem: Find the inverse of the matrix

(\begin{matrix}3 & 5 \\ 1 & 2 \end{matrix}) .

A student may compute (using “operations on rows”) as follows.

(\begin{matrix}3 & 5 & | & 1 & 0 \\ 1 & 2 & | & 0 & 1 \end{matrix}) → (\begin{matrix}1 & 5/3 & | & 1/3 & 0 \\ 1 & 2 & | & 0 & 1 \end{matrix})

→ (\begin{matrix}1 & 5/3 & | & 1/3 & 0 \\ 0 & 1/3 & | & −1/3 & 1 \end{matrix}) → (\begin{matrix}1 & 5/3 & | & 1/3 & 0 \\ 0 & 1 & | & −1 & 3 \end{matrix})
→ (\begin{matrix}1 & 0 & | & 2 & −5 \\ 0 & 1 & | & −1 & 3 \end{matrix})

The calculation is valid on Spec(ℤ[1/3]).

Another student may calculate (using “operations on columns”) as follows.

(\begin{matrix}3 & 5 & | & 1 & 0 \\ 1 & 2 & | & 0 & 1 \end{matrix}) → (\begin{matrix}3 & 5/2 & | & 1 & 0 \\ 1 & 1 & | & 0 & 1/2 \end{matrix})

→ (\begin{matrix}1/2 & 5/2 & | & 1 & 0 \\ 0 & 1 & | & −1/2 & 1/2 \end{matrix}) → (\begin{matrix}1 & 5/2 & | & 2 & 0 \\ 0 & 1 & | & −1 & 1/2 \end{matrix})
→ (\begin{matrix}1 & 1 & | & 2 & −5 \\ 0 & 1 & | & −1 & 3 \end{matrix})

The calculation is valid on Spec(ℤ[1/2]). Of course, both calculations are valid on the intersection Spec(ℤ[1/2]) ∩ Spec(ℤ[1/3]) = Spec(ℤ[1/6]).

The gluing lemma asserts that the answer obtained individually is automatically an answer on the whole of Spec(ℤ). Of course, in this special case, there are lots of easier ways to tell that. But one may imagine this kind of thing is helpful when we deal with more complicated objects.

10.3. homomorphisms of (pre)sheaves.

Definition 10.16. Let ℱ_{1}, ℱ_{2} be presheaves of modules on a topological space X. Then we say that a sheaf homomorphism

φ : ℱ_{1} → ℱ_{2}

is given if we are given a module homomorphism

φ_{U} : ℱ_{1}(U) → ℱ_{2}(U)

for each open set U ⊂ X with the following property hold.

(1) For any open subsets V, U ⊂ X such that V ⊂ U, we have

ρ_{V,U} ∘ φ_{U} = φ_{V} ∘ ρ_{V,U}.

(The property is also commonly referred to as “φ commutes with restrictions”.)

Definition 10.17. A homomorphism of sheaves is defined as a homomorphism of presheaves.

10.4. example of presheaves and sheafification. To proceed our theory further, we need to study a bit more about presheaves. Unfortunately, a sheaf of modules M̃ on an affine schemes are “too good”. Namely, in terms of cohomology (which we study later,) we have always

H^{i}(Spec(A), M̃) = 0 (if i > 0).

So to study some important problems on sheaf theory (which we will sure to encounter when we deal with non-affine schemes,) we need to study some examples from other mathematical areas.

A first example is a presheaf which satisfies the “locality” of sheaf axiom, but which fails to obey “gluing lemma”.

Example 10.18. Let X = ℝ be the (usual) real line with the usual Lebesgue measure. Then we have a presheaf of L^{1}-functions given by

L^{1}(U) = {f : U → ℂ; |f| is integrable}.

L^{1} is a presheaf which satisfies the “locality” of sheaf axiom, but which fails to obey “gluing lemma”. Indeed, Let {U_{n} = (−n, n)} be an open covering of ℝ and define a section f_{n} on U_{n} by

f_{n}(x) = 1 (x ∈ U_{n}).

Then we see immediately that {f_{n}} is a family of sections which satisfies the assumption of “gluing lemma”. The function which should appear as the “glued function” is the constant function 1, which fails to be integrable on the whole of ℝ.

We may “sheafificate” the presheaf L^{1} above. Instead of L^{1}-functions we consider functions which are locally L^{1}. Namely, for any open subset U ⊂ ℝ, we consider

L_{loc}^{1}(U) = \{f : U → ℂ; \begin{matrix}∀x ∈ U, ∃V (open in U) ∋ x \\ such that |f| is integrable on V \end{matrix} \}

The presheaf so defined is a sheaf, which we may call “the sheaf of locally L^{1}-functions”.

Example 10.19. Similarly, we may consider a presheaf U ↦ Bdd(U) of bounded functions on a topological space X. We may sheafificate this example and the sheaf so created is the sheaf of locally bounded functions.

Example 10.20. It is psychologically a bit difficult to give an example of a presheaf which does not satisfy the locality axiom of a sheaf. But there are in fact a lot of them.

For any differentiable (C^{∞}) manifold X (students which are not familiar with the manifolds may take X as an open subset of ℝ^{n} for an example.), we define a presheaf 𝒢 on X defined as follows

𝒢(U) = C^{∞}(U × U) = {complex valued C^{∞}-functions on U × U}.

The restriction is defined in an obvious manner. It is an easy exercise to see that the presheaf does not satisfy the locality axiom of a sheaf.

To sheafificate this, we first need to introduce an equivalence relation on 𝒢(U).

f ∼ g ⟺ ( \begin{matrix}there exists an open covering \{U_{λ}\} of U \\ such that ρ_{U_{λ},U}f = ρ_{U_{λ},U}g \\ for any λ. \end{matrix} )

Then we may easily see that

f ∼ g ⟺ ( \begin{matrix}there exists an open neighborhood V of \\ the diagonal Δ_{U} ⊂ U × U \\ such that f = g on V \end{matrix})

holds.

Then we define

ℱ(U) = 𝒢(U)/ ∼ .

It is now an easy exercise again to verify that ℱ so defined is a sheaf. (Readers who are familiar with the theory of jets may notice that the sheaf is related to the sheaf of jets. In other words, there is a sheaf homomorphism from this sheaf to the sheaf of jets.)

10.5. sheafification of a sheaf. In the preceding subsection, we have not been explained what “sheafification” really means. Here is the definition.

Lemma 10.21. Let 𝒢 be a presheaf on a topological space X. Then there exists a sheaf sheaf(𝒢) and a presheaf morphism

ι_{𝒢} : 𝒢 → sheaf(𝒢)

such that the following property holds.

(1) If there is another sheaf ℱ with a presheaf morphism

α : 𝒢 → ℱ,

then there exists a unique sheaf homomorphism

α̃ : sheaf(𝒢) → ℱ

such that

α = α̃ ∘ ι_{𝒢}

holds. Furthermore, such sheaf(𝒢), ι_{𝒢} is unique.

Definition 10.22. The sheaf sheaf(𝒢) (together with ι_{𝒢}) is called the sheafification of 𝒢.

The proof of Lemma 10.21 is divided in steps.

The first step is to know the uniqueness of such sheafification. It is most easily done by using universality arguments. ([?] has a short explanation on this topic.)

Then we divide the sheafification process in two steps.

Lemma 10.23. (First step of sheafification) Let 𝒢 be a presheaf on a topological space X. Then for each open set U ⊂ X, we may define a equivalence relation on 𝒢(U) by

f ∼ g ⟺ ( \begin{matrix}there exists an open covering \{U_{λ}\} of U \\ such that ρ_{U_{λ},U}f = ρ_{U_{λ},U}g \\ for any λ. \end{matrix} )

Then we define

𝒢^{(1)}(U) = 𝒢(U)/ ∼ .

Then 𝒢^{(1)} is a presheaf that satisfies the locality axiom of a sheaf. There is also a presheaf homomorphism from 𝒢 to 𝒢^{(1)}. Furthermore, 𝒢^{(1)} is universal among such.

Lemma 10.24. (Second step of sheafification) Let 𝒢 be a presheaf on a topological space X which satisfies the locality axiom of a sheaf. Then we define a presheaf 𝒢^{(2)} in the following manner. First for any open covering {U_{λ}} of an open set U ⊂ X, we define

𝒢^{(2)}(U; \{U_{λ}\}) = \{\{r_{λ}\} ∈ ∏_{λ∈Λ} G(U_{λ}); \begin{matrix}ρ_{U_{λμ},U_{μ}}f_{μ} = ρ_{U_{λμ},U_{λ}}f_{λ} \\ for any λ, μ ∈ Λ. \end{matrix} \}

Then we define

𝒢^{(2)}(U) = \underrightarrow{lim}_{\{U_{λ}\}} 𝒢^{(2)}(U; \{U_{λ}\})

Then we may see that G^{(2)} is a sheaf and that there exists a homomorphism from G to G^{(2)}. Furthermore, G^{(2)} is universal among such.

Proofs of the above two lemma are routine work and are left to the reader.

Finish of the proof of Lemma 10.21: We put

sheaf(𝒢) = ((𝒢)^{(1)})^{(2)}
□

10.6. stalk of a presheaf.

Definition 10.25. Let 𝒢 be a presheaf on a topological space X. Let P ∈ X be a point. We define the stalk of 𝒢 on P as

𝒢_{P} = \underrightarrow{lim}_{U∋P} 𝒢(U)

It should be noted at this stage that

Lemma 10.26. Let 𝒢 be a presheaf on a topological space X. The natural map

𝒢 → sheaf(𝒢)

induces an isomorphism of stalk at each point x ∈ X.

10.7. kernels, cokernels, etc. on sheaves of modules. In this subsection we restrict ourselves to deal with sheaves of modules.

To shorten our statements, we call a presheaf which satisfies (only) the sheaf axiom (1) (locality) a “(1)-presheaf”.

Lemma 10.27. Let φ : ℱ → 𝒢 be a homomorphism between sheaves of modules. Then we have

(1) The presheaf kernel of φ is a sheaf. We call it the sheaf kernel Ker(φ) of φ.

(2) The presheaf image of φ is not necessarily a sheaf, but it is a (1)-presheaf. We call the sheafification of the presheaf image as the sheaf image Image(φ) of φ.

(3) The presheaf cokernel of φ is not necessarily a sheaf. We call the sheafification of the cokernel as the sheaf cokernel Coker(φ) of φ.

Definition 10.28. A sequence of homomorphisms of sheaves of modules

ℱ_{1} →^{f_{1}} ℱ_{2} →^{f_{2}} ℱ_{3}

is said to be exact if Image(f_{1}) = Ker(f_{2}) holds.

Lemma 10.29. A sequence of homomorphisms of sheaves of modules

ℱ_{1} →^{f_{1}} ℱ_{2} →^{f_{2}} ℱ_{3}

is exact if and only if it is exact stalk wise, that means, if and only if the sequence

(ℱ_{1})_{P} →^{f_{1}} (ℱ_{2})_{P} →^{f_{2}} (ℱ_{3})_{P}

is exact for all point P.