Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 49 (2008), No. 2, 517-525.

𝔽_{1}-schemes and Toric Varieties

Anton Deitmar

Mathematisches Institut Auf der Morgenstelle 10, 72076 Tübingen, Germany e-mail: deitmar@uni-tuebingen.de

Abstract. In this paper it is shown that integral 𝔽_{1}-schemes of finite type are essentially the same as toric varieties. A description of the 𝔽_{1}-zeta function in terms of toric geometry is given. Etale morphisms and universal coverings are introduced. MSC 2000: 14A15, 11G25, 14L32, 14M32 Keywords: F1-schemes, toric varieties, field of one element

Introduction

There are by now several attempts to make the theory of the field of one element 𝔽_{1} rigorous. In [10] the authors formalize the transition from rings to schemes on a categorial level and apply this machinery to the category of sets to obtain the category of 𝔽_{1}-schemes as in [1]. In [3] and [5] the authors extend the definition of rings in order to capture a structure that deserves to be called 𝔽_{1}. In [1] the author tried instead to fix the minimum properties any of these theories must share. The current paper extends this line of thought. We use terminology of [1] and [2].

In this paper, a ring will always be commutative with unit and a monoid will always be commutative. An ideal 𝔞 of a monoid A is a subset with A𝔞 ⊂ 𝔞. A prime ideal is an ideal 𝔭 such that S_{𝔭} = A ^{}𝔭 is a submonoid of A. For a prime ideal 𝔭 let A_{𝔭} = S_{𝔭}^{−1}A be the localization at 𝔭. The spectrum of a monoid A is the set of all prime ideals with the obvious Zariski-topology (see [1]). Similar to the theory of rings, one defines a structure sheaf O_{X} on X = spec (A), and one defines a scheme over 𝔽_{1} to be a topological space together with a sheaf of monoids, locally isomorphic to spectra of monoids.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2008 Heldermann Verlag

518 A. Deitmar: 𝔽_{1}-schemes and Toric Varieties

A 𝔽_{1}-scheme X is of finite type, if it has a finite covering by affine schemes U_{i} = spec (A_{i}) such that each A_{i} is a finitely generated monoid. For a ring R, we write X_{R} for the R-base-change of X, so X_{R} = X_{ℤ} ⊗_{ℤ} R.

For a monoid A we let A⊗ℤ be the monoidal ring ℤ[A]. This defines a functor from monoids to rings which is left adjoint to the forgetful functor that sends a ring R to the multiplicative monoid (R, ×). This construction is compatible with gluing, so one gets a functor X ↦ X_{ℤ} from 𝔽_{1}-schemes to ℤ-schemes. In [2] we have shown that X is of finite type if and only if X_{ℤ} is a ℤ-scheme of finite type.

We say that the monoid A is integral, if it has the cancellation property, i.e., if ab = ac implies b = c in A. This is equivalent to saying that A injects into its quotient group or A is a submonoid of a group.

By a module of a monoid A we mean a set M together with a map A×M → M; (a, m) ↦ am with 1m = m and (ab)m = a(bm). A stationary point of a module is a point m ∈ M with am = m for every a ∈ A. A pointed module is a pair (M, m_{0}) consisting of an A-module M and a stationary point m_{0} ∈ M.

1. Flatness

Recall the tensor product of two modules M, N of A:

M ⊗ N = M ⊗_{A} N = M × N/ ∼,

where ∼ is the equivalence relation generated by (am, n) ∼ (m, an) for every a ∈ A, m ∈ M, n ∈ N. The class of (m, n) is written as m ⊗ n. The tensor product M ⊗ N becomes a module via a(m ⊗ n) = (am) ⊗ n. For example, the module A ⊗ M is isomorphic to M.

Let now (M, m_{0}) and (N, n_{0}) be two pointed modules of A, then (M ⊗N, m_{0}⊗ n_{0}) is a pointed module, called the pointed tensor product.

The category Mod_{0}(A) of pointed modules and pointed morphisms has a terminal and initial object 0, so it makes sense to speak of kernels and cokernels. It is easy to see that every morphism f in Mod_{0}(A) possesses both. One defines the image of f as im(f) = ker(coker(f)) and the coimage as coim(f) = coker(ker(f)).

A morphism is called strong, if the natural map from coim(f) to im(f) is an isomorphism. Kernels and cokernels are strong. If A →^{f} B →^{g} C is given with g being strong and gf = 0, then the induced map coker(f) → C is strong. Likewise, if f is strong and gf = 0, then the induced map A → ker g is strong. A map is strong if and only if it can be written as a cokernel followed by a kernel.

The usual notion of exact sequences applies, and we say that a sequence of morphisms is strong exact if it is exact and all morphisms in the sequence are strong.

A module F ∈ Mod_{0}(A) is called flat, if the functor X ↦ F ⊗ X is strongexact, i.e., if for every strong exact sequence

0 → M → N → P → 0

the induced sequence

0 → F ⊗ M → F ⊗ N → F ⊗ P → 0

is strong exact as well.

It is easy to see that a pointed module F is flat if and only if for every injection M ↪ N of pointed modules the map F ⊗ M → F ⊗ N is an injection.

Examples. If A is a group, then every module is flat. Let S be a submonoid of A. Then the localization S^{−1}A is a flat A-module. The direct sum G ⊕ F of two flat modules is flat. Finally, consider the free monoid in one generator C_{+} = {1, τ, τ^{2}, …}, then an A-module M is flat if and only if τm = τm′ implies m = m′ for all m, m′ ∈ M. This is equivalent to saying that M is a C_{+}-submodule of a module of the quotient group C_{∞} = τ^{ℤ} of C_{+}. The same characterization holds for every integral monoid.

A morphism φ : A → B of monoids is called flat if B is flat as an A-module. A morphism of 𝔽_{1}-schemes f : X → Y is called flat if for every x ∈ X the morphism of monoids f^{#} : O_{Y,f(x)} → O_{X,x} is flat.

The following is straightforward.

• A morphism of monoids φ : A → B is flat if and only if the induced morphism of 𝔽_{1}-schemes spec B → spec A is flat.

• The composition of flat morphisms is flat.

• The base change of a flat morphism by an arbitrary morphism is flat.

Remark. It is easy to see that if ℤ[F] is flat as ℤ[A]-module, then F is flat as A-module. The converse is already false if A is a group. As an example let k be

a field and let A be the group of all matrices of the form ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) where x ∈ k.

Let A act on k^{2} in the usual way and trivially on k. Consider the exact sequence of ℤ[A]-modules,

[figure: 0; k; k^{2}; k; 0,; α; β]

where α(x) = ( \begin{matrix}x \\ 0 \end{matrix} ), β ( \begin{matrix}x \\ y \end{matrix} ) = y. Let F = \{1\} the trivial A-module, then for

every ℤ[A]-module M one has M ⊗_{ℤ[A]}ℤ[F] = H_{0}(A, M). Note that H_{0}(A, k) = k and that H_{0}(α) = 0, so it is not injective, hence ℤ[F] is not flat.

2. Algebraic extensions

Let A be a submonoid of B. An element b ∈ B is called algebraic over A, if there exists n ∈ ℕ with b^{n} ∈ A. The extension B/A is called algebraic, if every b ∈ B is algebraic over A. An algebraic extension B/A is called strictly algebraic, if for every a ∈ A the equation x^{n} = a has at most n solutions in B.

If B/A is algebraic, then ℤ[B]/ℤ[A] is an algebraic ring extension, but the converse is wrong in general, as the following example shows: Let A = 𝔽_{1} and B be the set of two elements, 1 and b with b^{2} = b.

A monoid A is called algebraically closed, if every equation of the form x^{n} = a with a ∈ A has a solution in A. Every monoid A can be embedded into an algebraically closed one, and if A is a group, then there exists a smallest such embedding, called the algebraic closure of A. For example, the algebraic closure 𝔽̄_{1} of 𝔽_{1} is the group μ_{∞} of all roots of unity, which is isomorphic to ℚ/ℤ.

3. Etale morphisms

Recall that a homomorphism φ: A → B of monoids is called a local homomorphism, if φ^{−1}(B^{×}) = A^{×} (every φ satisfies “⊃”). For a monoid A let m_{A} = A ^{}A^{×} be its maximal ideal. It is easy to see that a homomorphism φ: A → B is local if and only if φ(m_{A}) ⊂ m_{B}.

A local homomorphism φ: A → B is called unramified if

• φ(m_{A})B = m_{B} and

• φ injects A^{×} into B^{×} and B/φ(A) is a finite strictly algebraic extension. Note that if φ is unramified, then so are all localizations φ_{𝔭} : A_{φ^{−1}(𝔭)} → B_{𝔭} for 𝔭 ∈ spec B.

A morphism f : X → Y of 𝔽_{1}-schemes is called unramified, if for every x ∈ X the local morphism f^{#} : O_{Y,f(x)} → O_{X,x} is unramified.

A morphism f : X → Y of 𝔽_{1}-schemes is called locally of finite type, if every point in Y has an open affine neighborhood V = spec A such that f^{−1}(V) is a union of open affines spec B_{i} with B_{i} finitely generated as a monoid over A. The morphism is of finite type if for every point in Y the number of B_{i} can be chosen finite. The morphism is called finite, if every y ∈ Y has an open affine neighborhood V = spec A such that f^{−1}(V) is affine, equal to spec B, where B is finitely generated as A-module.

A morphism f : X → Y of finite type is called étale, if f is flat and unramified. It is called an étale covering, if it is also finite.

Proposition 3.1. The étale coverings of spec 𝔽_{1} are the morphisms of the form spec A → spec 𝔽_{1}, where A is a finite cyclic group. The scheme spec 𝔽̄_{1} has no non-trivial étale coverings.

Proof. Clear. □

A connected scheme over 𝔽_{1}, which has only the trivial étale covering, is called simply connected.

Proposition 3.2. The schemes spec 𝔽̄_{1}, spec C_{+} ×_{𝔽_{1}} 𝔽̄_{1} and ℙ1 ; 𝔽̄_{1} are simply connected.

Proof. The first has been dealt with. For the second, let A = μ_{∞} × C_{+}. Then spec A = spec C_{+} ×_{𝔽_{1}} 𝔽̄_{1}. Let f : X → spec A be an étale covering. As f is finite, X is affine, say X = spec B. Let φ : A → B denote the corresponding morphism of monoids. The space spec A consists of two points, the generic point η_{A} and the closed point c_{A}. Likewise, let η_{B}, c_{B} denote the generic and closed points of spec B. One has f(η_{B}) = η_{A}. We will show that f(c_{B}) = c_{A}. Assume the contrary. Then φ^{−1}(m_{B}) is empty, hence φ maps A to the unit group B^{×}. The localization at the closed point c_{B} then maps μ_{∞} × C_{∞} to B^{×} and is unramified, hence injective. But as C_{+} → C_{∞} is not finite, neither can φ be finite, a contradiction. So we conclude f(c_{B}) = c_{A}, and so the corresponding localization, which is φ itself, is unramified. Let s = φ(τ), where τ is the generator of C_{+}. Then φ(m_{A})B = m_{B} implies m_{B} = sB, and so B = B^{×} ·∪sB (disjoint union). Also, B^{×} is an algebraic extension of A^{×} ≅ μ_{∞}, hence equals φ(A^{×}). As B is finitely generated and flat as A-module, there are b_{1}, …, b_{r} ∈ B with

sB = B^{×}s^{ℕ} ·∪B^{×}s^{ℕ}b_{1} ·∪ ⋯∪B^{×}s^{ℕ}b_{r}.

If we assume r > 0, then b_{1} is algebraic over φ(A) = B^{×} ·∪B^{×}s^{ℕ}, so let N be the smallest number in ℕ such that b_{1}^{N} ∈ φ(A). Then b_{1}^{N} ∉ B^{×} ≅ μ_{∞}, because, as the extension is strictly algebraic, then b_{1} would be in B^{×} already. So b_{1}^{N} ∈ B^{×}s^{ℕ}. As the group B^{×} is divisible, we can replace b_{1} with a B^{×} multiple to get b_{1}^{N} = s^{M} for some M ∈ ℕ. Then b_{1} ∉ B^{×}s^{ℕ}b_{1}, as b_{1} = b^{∗}s^{k}b_{1} leads to s^{M} = b_{1}^{N} = (b^{∗})^{N}s^{kN+M} which contradicts the injectivity of φ. But then b_{1} must be in one of the other B^{×}s^{ℕ}-orbits,which contradicts the disjointness of these orbits. We conclude r = 0, i.e. B = B^{×} ·∪B^{×}s^{ℕ} ≅ A as claimed. The assertion for ℙ1 ; 𝔽̄_{1} is an easy consequence. □

4. Toric varieties

Recall a toric variety is an irreducible variety V over ℂ together with an algebraic action of the r-dimensional torus GL_{1}^{r}, such that V contains an open orbit.

As toric varieties can be constructed via lattices it follows that every toric variety is the lift X_{ℂ} of an 𝔽_{1}-scheme X. For integral schemes of finite type there is a converse direction given in the following theorem, which shows that integral 𝔽_{1}-schemes of finite type are essentially the same as toric varieties.

Theorem 4.1. Let X be a connected integral 𝔽_{1}-scheme of finite type. Then every irreducible component of X_{ℂ} is a toric variety. The components of X_{ℂ} are mutually isomorphic as toric varieties.

Proof. Let U = spec A be an open affine subset of X. Let η be the generic point of X, then the localization G = A_{η} is the quotient group of A. At the same time, G is the stalk O_{X,η}, so G does not depend on the choice of U up to canonical isomorphism. Let φ : A → G be the quotient map, which is injective as X is integral. The ℂ-algebra homomorphism,

ℂ[A] → ℂ[G] ⊗ ℂ[A]
a ↦ φ(a) ⊗ a

defines an action of the algebraic group G = spec ℂ[G] on spec ℂ[A]. Since this is compatible with the restriction maps of the structure sheaf, we get an algebraic action of the group scheme G on X_{ℂ}. As X is integral, G = spec ℂ[G] = spec ℂ[A_{η}] also is an open subset V_{ℂ} of X_{ℂ}, and for U_{ℂ} = spec ℂ[A] the map

O(U_{ℂ}) = ℂ[A] \underrightarrow{φ} ℂ[G] = O(V_{ℂ})

is the restriction map of the structure sheaf O of X_{ℂ}. The map ℂ[A] → ℂ[G] is injective and ℂ[G] has zero Jacobson radical, so it follows that V_{ℂ} is dense in X_{ℂ}, so in particular it meets every irreducible component. The group G is a finitely generated abelian group, so G ≡ ℤ^{r} × F for a finite abelian group F. Hence G ≡ GL_{1}^{r} × F as a group-scheme. As G meets every component of X_{ℂ}, the latter are permuted by F. Whence the claim. □

To formulate the next result, we will briefly recall the standard construction of toric varieties, see [4]. Let N be a lattice, i.e., a group isomorphic to ℤ^{n} for some n. A fan Δ in N is a finite collection of proper convex rational polyhedral cones σ in the real vector space N_{ℝ} = N ⊗ ℝ, such that every face of a cone in Δ is in Δ and the intersection of two cones in Δ is a face of each. (Here zero is considered a face of every cone.) We explain the notation further: A convex cone is a convex subset σ of N_{ℝ} with ℝ_{≥0}σ = σ, it is polyhedral, if it is finitely generated and rational, if the generators lie in the lattice N. Finally, a cone is called proper if it does not contain a non-zero sub vector space of N_{ℝ}.

Let a fan Δ be given. Let M = Hom(N, ℤ) be the dual lattice. For a cone σ ∈ Δ the dual cone σ̆ is the cone in the dual space M_{ℝ} consisting of all α ∈ M_{ℝ} such that α(σ) ≥ 0. This defines a monoid A_{σ} = σ̆ ∩M. Set U_{σ} = spec (ℂ[A_{σ}]). If τ is a face of σ, then A_{τ} ⊃ A_{σ}, and this inclusion gives rise to an open embedding U_{τ} ↪ U_{σ}. Along these embeddings we glue the affine varieties U_{σ} to obtain a variety X_{Δ} over ℂ, which has a given 𝔽_{1}-structure. Then X_{Δ} is a toric variety, the torus being U_{0} ≅ GL_{1}^{n}. Every toric variety is given in this way.

Lemma 4.2. Let B be a submonoid of the monoid A of finite index. Then the map ψ : spec A → spec B defined by ψ(𝔭) = 𝔭 ∩ B is a bijection.

Proof. Let N ∈ ℕ be such that a^{N} ∈ B for every a ∈ A. To see injectivity, let ψ(𝔭) = ψ(𝔮) and let a ∈ 𝔭. Then a^{N} ∈ 𝔮 and so a ∈ 𝔮 as 𝔮 is a prime ideal. This shows 𝔭 ⊂ 𝔮 and by symmetry we get equality. For surjectivity, let 𝔭_{B} ∈ spec B and let 𝔭 = {a ∈ A : a^{N} ∈ 𝔭_{B}}. Then ψ(𝔭) = 𝔭_{B}. □

Proposition 4.3. Suppose that Δ is a fan in a lattice of dimension n. For j = 0, …, n let f_{j} be the number of cones in Δ of dimension j. Set

c_{j} = ∑_{k=j}^{n} f_{n−k}(−1)^{k+j} ( \begin{matrix}k \\ j \end{matrix} ) .

Let X be the corresponding toric variety, then the 𝔽_{1}-zeta function of X equals

ζ_{X}(s) = s^{c_{0}}(s − 1)^{c_{1}} ⋯ (s − n)^{c_{n}}.

Proof. Let σ ∈ Δ be a cone of dimension k. Let F be a face of σ̆. Let 𝔭_{F} = A_{σ} ^{}F. Then 𝔭_{F} is a non-empty prime ideal in A_{σ}. The map F ↦ 𝔭_{F} is a bijection between the set of all faces of σ̆ and the set of non-empty prime ideals of A_{σ}. The set S_{𝔭} = A ^{}𝔭 equals M ∩F. The quotient group Quot(S_{𝔭}) is isomorphic to ℤ^{f}, where f is the dimension of F. There is a bijection between the set of faces of σ and the set of faces of σ̆ mapping a face τ to the face F of all α ∈ σ̆ with α(τ) = 0. The dimension of F then equals n − dim(τ). So let f_{j}^{σ} denote the number of faces of σ of dimension j. Then the zeta polynomial of X_{σ} equals

N_{σ}(x) = ∑_{k=0}^{n} f_{k}^{σ}(x − 1)^{n−k}.

Let N_{Δ} be the zeta polynomial of X_{Δ}. We get

N_{Δ}(x) = ∑_{k=0}^{n} f_{k}(x − 1)^{n−k}

= ∑_{k=0}^{n} f_{k} ∑_{j=0}^{n−k} ( \begin{matrix}n − k \\ j \end{matrix} ) x^{j}(−1)^{n−k−j}

= ∑_{k=0}^{n} f_{n−k} ∑_{j=0}^{k} ( \begin{matrix}k \\ j \end{matrix} ) x^{j}(−1)^{k−j}

= ∑_{j=0}^{n} x^{j} ∑_{k=j}^{n} f_{n−k} ( \begin{matrix}k \\ j \end{matrix} ) (−1)^{k−j}.

This implies the claim. □

5. Valuations

On the infinite cyclic monoid C_{+} = {1, τ, τ^{2}, …} we have a natural linear order given by τ^{k} ≤ τ^{l} ⇔ k ≤ l. Let φ, ψ be two monoid morphisms from a monoid A to C_{+}. Then define φ ≤ ψ ⇔ φ(a) ≤ ψ(a) ∀a ∈ A. A valuation on A is a non-trivial homomorphism v : A → C_{+} which is minimal with respect to the order ≤ among all non-trivial homomorphisms from A to C_{+}. Let V (A) denote the set of valuations on A.

Lemma 5.1. Let

[figure: 1; A; B; F; 1; φ]

be an exact sequence of monoids, where F is a finite abelian group. Then for every valuation v ∈ V (A) there exists a unique valuation w on B and k ∈ ℕ such that

w|_{A} = v^{k}.

Mapping v to w sets up a bijection from V (A) to V (B).

Proof. Let F′ be a subgroup of F and let B′ be the preimage of F′ under φ. We get two exact sequences

[figure: 1; A; B′; F′; 1,]

and

[figure: 1; B′; B; F/F′; 1.]

Assume we have proven the lemma for each of these two sequences, then it follows for the original one. In this way we reduce the proof to the case when F is a finite cyclic group. We first show existence of w for given v. For this let f_{0} be a generator of F and let l be its order. Choose a b_{0} in the preimage φ^{−1}(f_{0}). Then b_{0}^{l} ∈ A, and v(b_{0}^{l}) = τ^{n} for some n ≥ 0. If n = 0, then set k = 1 and define w : B → C_{+} by w(b_{0}^{j}a) = v(a) for a ∈ A and j ≥ 0. If n > 0, then set k = l/gcd(l, n) and let w : B → C_{+} be defined by w(b_{0}^{j}a) = τ^{j}v(a)^{k}. This shows existence of the extension w. □

6. Cohomology

Cohomology is not defined over 𝔽_{1}. I am grateful to Ofer Gabber for bringing the following example to my attention. Let X be the topological space consisting of three points η, X_{+}, x_{−}. The open sets besides the trivial ones are U = {η}, U_{+} = {η, x_{+}}, U_{−} = {η, x_{−}}. Let A be a subgroup of the abelian group B and let C = B/A. Let F be the sheaf of abelian groups on X with F(U_{±}) = A and F(U) = B and the restriction being the inclusion. Let G be the constant sheaf B and let H be the quotient sheaf G/F. As G is flabby, the long cohomology sequence terminates and looks like this:

0 → H^{0}(F) → H^{0}(G) → H^{0}(H) → H^{1}(F) → 0.

In concrete terms this is

0 → A → B → C × C → (C × C)/Δ → 0,

where Δ means the diagonal in C × C. Let f : X → X be the homeomorphism with f(x_{+}) = x_{−}, f(x_{−}) = x_{+}, and f(η) = η. There is a natural isomorphism f_{∗}F ≅ F and for the other sheaves as well. On the global sections of F and G this induces the trivial map, whereas on H^{0}(H) it induces the flip (a, b) ↦ (b, a), which on H^{1}(F) amounts to the same as the inversion a ↦ −a. The naturality of these isomorphisms means that if the sheaves and the cohomology groups are defined over 𝔽_{1}, then so must be the flip. This, however, is not the case, as for a set S the inversion on the abelian group ℤ[S] is not induced by a self-map of S.

Even more convincing is the fact that in this example there are different injective resolutions which produce different cohomology groups.

References

[1] Deitmar, A.: Schemes over 𝔽_{1}. In: Geer, G. van der; Moonen, B. J. J.; Schoof, R. (eds.), Number Fields and Function Fields – Two Parallel Worlds. Progress in Mathematics 239 (2005), 87–100. Zbl 1098.14003

[2] Deitmar, A.: Remarks on zeta functions and K-theory over 𝔽_{1}. Proc. Japan Acad., Ser. A, 82(8) (2006), 141–146. Zbl pre05161657

[3] Durov, N.: New Approach to Arakelov Geometry.

http://arxiv.org/abs/0704.2030

[4] Fulton, W.: Introduction to toric varieties. The 1989 William H. Roever Lectures in Geometry. Annals of Mathematics Studies 131. Princeton University Press, Princeton, NJ, 1993 Zbl 0813.14039

[5] Haran, S.: Non-additive geometry . Compos. Math. 143 (2007), 618–688. Zbl pre05166038

[6] Kurokawa, B.; Ochiai, H.; Wakayama, A.: Absolute Derivations and Zeta Functions. Doc. Math., J. DMV Extra Vol. 2003, 565–584. Zbl 1101.11325

[7] Kurokawa, N.: Zeta functions over 𝔽_{1}. Proc. Japan Acad. Ser. A, Math. Sci. 81(10) (2005), 180–184 .

[8] Soulé, C.: Les variétés sur le corps à un élément. Mosc. Math. J. 4(1) (2004), 217–244. Zbl 1103.14003

[9] Tits, J.: Sur les analogues algébriques des groupes semi-simples complexes. Centre Belge Rech. math., Colloque d’Algèbre supérieure, Bruxelles du 19 au 22 déc. 1956, (1957), 261–289. Zbl 0084.15902

[10] Toen, B.; Vaquie, M: Under Spec Z. http://arxiv.org/abs/math/0509684

Received July 8, 2007