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MAPPING F1-LAND:
AN OVERVIEW OF GEOMETRIES
OVER THE FIELD WITH ONE ELEMENT
JAVIER L ´OPEZ PE ˜NA AND OLIVER LORSCHEID
ABSTRACT. This paper gives an overview of the various approaches towards F1-geometry.
In a first part, we review all known theories in literature so far, which are: Deitmar’s
F1-schemes, Toe¨n and Vaquie´’s F1-schemes, Haran’s F-schemes, Durov’s generalized
schemes, Soule´’s varieties over F1 as well as his and Connes-Consani’s variations of this
theory, Connes and Consani’s F1-schemes, the author’s torified varieties and Borger’s Λ-
schemes. In a second part, we will tie up these different theories by describing functors
between the different F1-geometries, which partly rely on the work of others, partly de-
scribe work in progress and partly gain new insights in the field. This leads to a commu-
tative diagram of F1-geometries and functors between them that connects all the reviewed
theories. We conclude the paper by reviewing the second author’s constructions that lead
to realization of Tits’ idea about Chevalley groups over F1.
CONTENTS
Introduction 2
1. The building bricks of F1-geometries 4
1.1. M-schemes after Kato and Deitmar 4
1.2. Schemes over F1 in the sense of Toe¨n and Vaquie´ 5
1.3. Haran’s non-additive geometry 6
1.4. Durov’s generalized schemes 7
1.5. Varieties over F1 in the sense of Soule´ and its variations 8
1.6. Schemes over F1 in the sense of Connes and Consani 10
1.7. Torified varieties 11
1.8. Λ-schemes after Borger 12
2. Paths and bridges 12
2.1. Toric varieties as M-schemes 13
2.2. Comparison of M-schemes with TV-schemes 14
2.3. M-schemes as M0- and CC-schemes 14
2.4. M- and M0-schemes as F-schemes 15
2.5. M0-schemes as generalized schemes with zero 15
2.6. Relation between Durov’s generalized rings with zero and Haran’s F-rings 15
2.7. M-schemes as Λ-schemes 16
2.8. Toric varieties and affinely torified varieties 16
2.9. Relation between affinely torified varieties, S-varieties and its variations 16
2.10. Generalized torified schemes and CC-schemes 17
2.11. The map of F1-land 18
2.12. Algebraic groups over F1 18
References 20
J. Lo´pez Pen˜a was supported by the EU Marie-Curie fellowship PIEF-GA-2008-221519 at Queen Mary Uni-
versity of London.
O. Lorscheid was supported by the Max-Planck-Institut fu¨r Mathematik in Bonn.
1

2 JAVIER L ´OPEZ PE ˜NA AND OLIVER LORSCHEID
INTRODUCTION
The hour of birth to the field with one element was given in Jacques Tits’ paper [40]
from 1956, in which he indicated that the analogy between the symmetric group Sn and
the Chevalley group GLn(Fq) (as observed by Robert Steinberg in [38] in 1951) should
find an explanation by interpreting Sn as a Chevalley group over the “field of characteristic
one”. Though Tits’ idea was not taken serious at that time, it is one of the guiding thoughts
in the development of F1-geometry today. For a recent treatment of Steinberg’s paper, see
[37], and for an overview of what geometric objects over Fq should become if q = 1, see
[5].
It took more than 35 years that the field of one element reoccurred in mathematical liter-
ature. In an unpublished note ([21]), Mikhail Kapranov and Alexander Smirnov developed
the philosophy that the set of the n-th roots should be interpreted as F1n , a field extension
of F1 in analogy to the field extension Fpn of Fp. A scheme that contains the n-th roots of
unity should be thought of as a scheme over F1n . The tensor product F1n ⊗F1 Z should be
the group ring Z[Z/nZ].
In the early nineties, Christoph Deninger published his studies ([13], [14], [15], et
cetera) on motives and regularized determinants. In his paper [14], Deninger gave a de-
scription of conditions on a category of motives that would admit a translation of Weil’s
proof of the Riemann hypothesis for function fields of projective curves over finite fields
Fq to the hypothetical curve SpecZ. In particular, he showed that the following formula
would hold:
2−1/2π−s/2Γ(s2)ζ(s) =
det∞
(
1
2π (s−Θ)
∣∣∣H1(SpecZ,OT )
)
det∞
(
1
2π (s−Θ)
∣∣∣H0(SpecZ,OT )
)
det∞
(
1
2π (s−Θ)
∣∣∣H2(SpecZ,OT )
)
where det∞ denotes the regularized determinant, Θ is an endofunctor that comes with the
category of motives and Hi(SpecZ,OT ) are certain proposed cohomology groups. This
description combines with Nobushige Kurokawa’s work on multiple zeta functions ([23])
from 1992 to the hope that there are motives h0 (“the absolute point”), h1 and h2 (“the
absolute Tate motive”) with zeta functions
ζhw(s) = det∞
( 1
2π (s−Θ)
∣∣∣Hw(SpecZ,OT )
)
for w = 0, 1, 2. Deninger computed that ζh0(s) = s/2π and that ζh2(s) = (s− 1)/2π. It
was Yuri Manin who proposed in [30] the interpretation of h0 as SpecF1 and the interpre-
tation of h2 as the affine line over F1. The quest for a proof of the Riemann hypothesis was
from now on a main motivation to look for a geometry over F1. Kurokawa continued his
work on zeta functions in the spirit of F1-geometry in the collaboration [24] with Hiroyuki
Ochiai and Masato Wakayama and in [25].
In 2004, Christophe Soule´ proposed a first definition of an algebraic variety over F1 in
[36], based on the observation that the extension of the base field of a scheme can be char-
acterized by a universal property. His suggestion for a variety over F1 is an object involving
a functor, a complex algebra, a scheme and certain morphisms and natural transformations
such that a corresponding universal property is satisfied. Shortly after that, many different
approaches to F1-geometry arose.
Anton Deitmar reinterpreted in [9] the notion of a fan as given by Kazuya Kato in
[22] as a scheme over F1. He calculated zeta functions for his F1-schemes ([10]) and
showed that the F1-schemes whose base extension to C are complex varieties correspond
to toric varieties ([11]). Bertrand To¨en and Michel Vaquie´ associated to any symmetric
monoidal category with certain additional properties a category of geometric objects. In
the case of the category of sets together the cartesian product, the geometric objects are

AN OVERVIEW OF GEOMETRIES OVER THE FIELD WITH ONE ELEMENT 3
locally representable functors on monoids, which are F1-schemes in the sense of Toe¨n
and Vaquie´. Florian Marty developed theory on Zariski open objects ([33]) and smooth
morphisms ([34]) in this context.
Shai Haran (cf. [20]) proposed using certain categories modeled over finite sets as a
replacement for rings, and actually produced a candidate for the compactification SpecZ
of SpecZ in his framework. Nikolai Durov developed in [16] an extension of classical
algebraic geometry within a categorical framework that essentially implied replacing rings
by a certain type of monads. As a byproduct of his theory he obtained a definition of F1
and an algebraic geometry attached to it. See [17] for a summary.
In 2008, Alain Connes, Katia Consani and Matilde Marcolli showed ([6]) that the Bost-
Connes system defined in [4] admits a realization as a geometric object in the sense of
Soule´. The suggestion of Soule´ to consider a variation of the functor in his approach and
other ideas, led to Connes and Consani the variation of Soule´’s approach as presented in
[7]. That paper contains a first contribution to Tits’ idea of Chevalley groups over F1,
namely, Connes and Consani define them as varieties over F12 . Soule´ himself wrote only
later a text with his originally suggested modification. It can be found in this volume ([37]).
Yuri Manin proposed in [31] a notion of analytic geometry over F1. The key idea is
that one should look for varieties having “enough cyclotomic points”, an thought that re-
lates to Kazuo Habiro’s notion ([19]) of the cyclotomic completion of a polynomial ring,
which finds the interpretation as the ring of analytic functions on the set of all roots of
unity. Matilde Marcolli presents in [32] an alternative model to the BC-system for the
noncommutative geometry of the cyclotomic tower. For that purpose, she uses the mul-
tidimensional analogues to the Habiro ring defined by Manin, and constructs a class of
multidimensional BC-endomotives. Marcolli’s endomotives turn out to be closely related
to Λ-rings, in the sense of [2].
Aiming at unifying the different notions of varieties over F1 (after Soule´ and Connes-
Consani) as well as establishing new examples of F1-varieties, the authors of this text
introduced in [26] the notion of torified varieties. Of particular interest were Grassmann
varieties, which are shown to be torified varieties and to provide candidates of F1-varieties.
However, these candidates fail to satisfy the constraints of Soule´’s and Connes-Consani
F1-geometries. Independently of this work, Connes and Consani introduced in [8] a new
notion of scheme over F1, which simplified the previous approaches by Soule´ and them-
selves by merging Deitmar’s and Toe¨n-Vaquies viewpoints into it. We will show in this
paper that this notion is closely related to torified varieties. The second author showed in
[27] that Connes-Consani’s new notion of F1-geometry is indeed suitable to realize Tits’
original ideas on Chevalley groups over F1.
Another promising notion of F1-geometry was given recently by James Borger in [3],
who advocates the use of Λ-ring structures as the natural descent data from Z to F1.
The aim of this paper is to give an overview of the new land of geometries over the field
with one element. Firstly, we review briefly the different developments and building bricks
of F1-geometries. Secondly, we tie up and lay paths and bridges between the different F1-
geometries by describing functors between them. This will finally lead to a commutative
diagram of F1-geometries and functors between them that can be considered as a first map
of F1-land.
The paper is organized as follows. In part 1, we review the different notions of F1-
geometries. We describe a theory in detail where it seems important to gain an insight
into its technical nature, but we will refrain from a detailed treatment in favor of a rough
impression where technicalities lead too far for a brief account. In these cases, we provide
the reader with references to the literature where the missing details can be found.
In the order of the paper, the following approaches towards F1-geometries are reviewed:
Deitmar’s F1-schemes (which we call M-schemes in the following) are described in sec-
tion 1.1, Toe¨n and Vaquie´’s F1-schemes in section 1.2, Haran’s F-schemes in section 1.3

4 JAVIER L ´OPEZ PE ˜NA AND OLIVER LORSCHEID
and Durov’s generalized schemes in section 1.4. In section 1.5, we give a taste of Soule´’s
definition of a variety over F1 and describe his and Connes-Consani’s variations on this
notion. In section 1.6, we present Connes and Consani’s new proposal of F1-geometry
including the notion of an M0-scheme. In section 1.7, we review the authors’ definition
of a torified variety. Finally, we give in section 1.8 an insight into Borger’s Λ-schemes.
In part 2, we review and construct functors between the categories introduced in the
first part of the paper. The very central objects of F1-geometries are toric varieties. As
we will see, these can be realized in all considered F1-geometries. We begin in section
2.1 with recalling the definition of a toric variety and following Kato ([22]) to show that
toric varieties are F1-schemes after Deitmar, i.e. M-schemes. In section 2.2 we describe
Florian Marty’s work on comparing Deitmar’s and Toe¨n-Vaquie´’s notions of F1-schemes.
In section 2.3, we lay the path from M-schemes to M0-schemes and Connes-Consani’s
F1-schemes. In section 2.4, we recall from Haran’s paper [20] that M-schemes and
M0-schemes are F-schemes. In section 2.5, we see that the same result holds true for
Durov’s generalized schemes (with 0) in place of F-schemes. In section 2.6, we review
Peter Arndt’s work in progress about comparing Haran’s and Durov’s approaches towards
F1-geometry. In section 2.7, we refer to Borger’s paper [3] to establish M-schemes as
Λ-schemes.
In section 2.8, we recall from the authors’ previous paper [26] that toric varieties are
affinely torified. In section 2.9, we give an idea of why affinely torified varieties define
varieties over F1 (after Soule´ resp. Connes and Consani). In section 2.10, we extend the
definition of a torified variety to a generalized torified scheme in order to show that the idea
behind the notion of a torified variety and an F1-scheme after Connes and Consani are the
same. All these categories and functors between them will be summarized in the diagram
in Figure 1 of section 2.11.
We conclude the paper with a review of the realization of Tits’ ideas on Chevalley
groups over F1 by the second author in section 2.12.
Acknowledgments: The authors thank the organizers and participants of the Nashville
conference on F1-geometry for a very interesting event and for numerous inspiring dis-
cussions that finally lead to an overview as presented in the present text. In particular, the
authors thank Peter Arndt, Pierre Cartier, Javier Fresa´n, Florian Marty, Andrew Salch and
Christophe Soule´ for useful discussions and openly sharing the pieces of their unpublished
works with us. The authors thank Susama Agarwala, Snigdhayan Mahanta, Jorge Plazas
and Bora Yakinoglu for complementing the authors’ research on F1 with many other valu-
able aspects.
1. THE BUILDING BRICKS OF F1-GEOMETRIES
In this first part of the paper, we present an introduction to several different approaches
to F1-geometry. Some of the original definitions have been reformulated in order to unify
notation, simplify the exposition or make similarities with other notions apparent. In what
follows–unless explicitly mentioned otherwise–all rings will be assumed to be commu-
tative rings with 1. Monoids are commutative semi-groups with 1 and will be written
multiplicatively. Schemes are understood to be schemes over Z, and by variety, we will
mean a reduced scheme of finite type.
1.1. M-schemes after Kato and Deitmar. In [9], Deitmar proposes the definition of a
geometry over the field with one element by following ideas of Kato (cf. [22]) of mimick-
ing classical scheme theory but using the category M of commutative monoids (abelian
multiplicative semigroups with 1) in place of the usual category of (commutative and uni-
tal) rings. To a far extent, this idea leads to a theory that is formally analogous to algebraic
geometry.
Given a monoid A, an ideal of A is a subset a such that aA ⊆ a. An ideal a is prime if
whenever xy ∈ a, then either x ∈ a or y ∈ a (or equivalently, if the setA\a is a submonoid

AN OVERVIEW OF GEOMETRIES OVER THE FIELD WITH ONE ELEMENT 5
of A). The set specA of all the prime ideals of the monoid A can be endowed with the
Zariski topology, for which a set V ⊆ specA is closed if there is an ideal a ⊆ A such that
V = V (a) = {q ∈ specA| a ⊆ q}. A commutative monoid always contains a minimal
prime ideal, the empty set, and a unique maximal prime ideal, mA := A \A×, where A×
is the group of units of A. For any submonoid S ⊆ A the localization of A at S is defined
by S−1A = {as | a ∈ A, s ∈ S}/ ∼, where a/s ∼ b/t if there exists some u ∈ S such
that uta = usb. The localization QuotA := A−1A is called the total fraction monoid of
A. The monoidA is integral (or cancellative) if the natural map A→ QuotA is injective.
For each prime ideal p ⊆ A we construct the localization at p as Ap := (A \ p)−1A.
Let X = specA be endowed with the Zariski topology. Given an open set U ⊆ X ,
define
OX(U) :=


s : U →
∐
p∈U
Ap| s(p) ∈ Ap and s is loc. a quotient of elements in A



where s is locally a quotient of elements in A if s(q) = a/f for some a, f ∈ A with
f /∈ q for all q in some neighborhood of p. We call OX the structure sheaf of X . The
stalks are OX,p := lim−→p∈U OX(U)
∼= Ap. Taking global sections yields OX(X) =
Γ(specA,OX) ∼= A.
A monoidal space is a pair (X,OX) where X is a topological space and OX is a sheaf
of monoids. A morphism of monoidal spaces is a pair (f, f#) where f : X → Y is a
continuous map and f# : OY → f∗OX is a morphism of sheaves. The morphism (f, f#)
is local if for each x ∈ X , we have (f#x )−1(O×X,x) = O×Y,f(x). For every monoid A and
X = specA, the pair (X,OX) is a monoidal space, called an affine M-scheme, and every
morphism of monoids ϕ : A → B induces a local morphism between the corresponding
monoidal spaces. A monoidal space (X,OX) is called a M-scheme (over F1) if for all
x ∈ X there is an open set U containing x such that (U,OX |U ) is an affine M-scheme.
Let Z[A] denote the semi-group ring of A. The base extension functor − ⊗F1 Z that
sends specA to SpecZ[A] has a right-adjoint given by the forgetful functor from rings to
monoids ([9, Theorem 1.1]). Both functors extend to functors between M-schemes and
schemes over Z ([9, section 2.3]). We will often write XZ for X ⊗F1 Z.
1.2. Schemes over F1 in the sense of Toe¨n and Vaquie´. Toe¨n and Vaquie´ introduce in
their paper [41] F1-schemes as functors on monoids that can be covered by representable
functors. To avoid confusion with the other notions of F1-schemes in the present text, we
will call the F1-schemes after Toe¨n and Vaqiue´ “TV-schemes”. We will explain the notion
of a TV-scheme in this section.
As in the previous section, we let M be the category of monoids. Then the category
of affine TV-schemes is by definition the opposite category Mop. If A is a monoid in M,
then we write SpecTV A for the same object in the opposite category. The category of
presheaves Psh(Mop) on Mop consists of functors Mop → Sets as objects and natural
transformations between them as morphisms. The affine TV-scheme X = SpecTV A
can be regarded as a presheaf X : Mop → Sets by sending Y = SpecTV B to X(Y ) =
Hom(Y,X) = Hom(A,B). This defines an embedding of categoriesMop →֒ Psh(Mop).
For a monoid A, let A-Mod be the category of sets with A-action together with equi-
variant maps. A homomorphism f : A→ B of monoids is flat if the induced functor
−⊗A B : A-Mod −→ B-Mod
commutes with finite limits.
Let X = SpecTV A and Y = SpecTV B. A morphism ϕ : Y → X in Mop is
called Zariski open if the dual morphism f : A → B is an flat epimorphism of finite
presentation. A Zariski cover of X = SpecTV A is a collection {Xi → X}i∈I of Zariski

6 JAVIER L ´OPEZ PE ˜NA AND OLIVER LORSCHEID
open morphisms in Mop such that there is a finite subset J ⊂ I with the property that the
functor ∏
j∈J
−⊗A Aj : A-Mod −→
∏
j∈J
Aj-Mod
is conservative. This defines the Zariski topology on SpecA. The category Sh(Mop) of
sheaves is the full subcategory of Psh(Mop) whose objects satisfy the sheaf axiom for the
Zariski topology. Toe¨n and Vaquie´ show that SpecA is indeed a sheaf for every monoid
A.
Let F : Mop → Sets be a subsheaf of an affine TV-schemeX = SpecA. Then F ⊂ X
is Zariski open if there exists a family {Xi → X}i∈I of Zariski open morphisms in Mop
such that F is the direct image of ∐
i∈I
Xi → X
as a sheaf. Let F be a subsheaf of a sheaf G : Mop → Sets. Then F ⊂ G is Zariski open
if for all affine TV-schemes X = SpecA, the natural transformation F ×G X → X is a
monomorphism in Sh(Mop) with Zariski open image.
Let F : M → Sets be a functor. An open cover of F is a collection {Xi →֒ F}i∈I of
Zariski open subfunctors such that
∐
i∈I
Xi → F
is an epimorphism in Sh(Mop).
A TV-scheme is a sheaf F : Mop → Sets that has an open cover by affine TV-schemes.
Given an affine TV-scheme X = SpecTV A, we define its base extension to Z as the
scheme XZ = SpecZ[A]. This extends to a base extension functor
−⊗F1 Z : {TV-schemes} −→ {schemes}.
Remark 1.1. In their paper [41], Toe¨n and Vaquie´ define schemes w.r.t. any cosmos, that
is, a symmetric closed-monoidal category that admits small limits and small colimits. It is
possible to speak of monoids in such a category. In the case of the category Sets, we obtain
M as monoids and the schemes w.r.t. Sets are TV-schemes as described above. The cate-
gory of Z-modules yields commutative rings with 1 as monoids, and the schemes w.r.t. the
category of Z-modules are schemes in the usual sense. There are several other interesting
categories obtained by this construction that can be found in [41], also cf. Remark 1.2.
1.3. Haran’s non-additive geometry. In [20], Haran proposes an “absolute geometry” in
which the field with one element and rings over F1 are realized as certain categories.
Haran’s definition of the field with one element is the category whose objects are pointed
finite sets and together with morphisms that are maps ϕ : X → Y such that f|X\ϕ−1(0Y )
is injective (where 0Y is the base point of Y ).
Neglecting the base point yields an equivalence of this category with the category F of
finite sets together with partial bijections, i.e. the morphism set from X to Y is
FY,X = HomF(X,Y ) := {ϕ : U ∼−→ V | U ⊆ X,V ⊆ Y }.
The category F is endowed with two functors ⊕ (disjoint union) and ⊗ (cartesian prod-
uct). These functors induce two structures (F,⊕, [0] = ∅) and (F,⊗, [1] = {∗}) of sym-
metric monoidal categories on F. An F-ring is a categoryA with objects Ob(A) = Ob(F)
being finite sets, endowed with a faithful functor F → A that is the identity on objects, and
two functors ⊕,⊗ : A × A → A extending the ones in F and each making A into a sym-
metric monoidal category. For example, every ring together with the matrix algebras with
coefficients in the ring is an F-ring. This embeds the category of rings into the category of
F-rings.

AN OVERVIEW OF GEOMETRIES OVER THE FIELD WITH ONE ELEMENT 7
An ideal of an F-ringA is a collection of subsets {IY,X ⊆ AY,X}X,Y ∈|F| that is closed
under the operations ◦, ⊕, ⊗, i.e
A ◦ I ◦A ⊆ I, A⊗ I ⊆ I and I ⊕ I ⊆ I.
An ideal a ⊆ A is called homogeneous if it is generated by a[1],[1]. A subset A ⊆ A[1],[1] is
called an H-ideal if for all a1, . . . , an ∈ A, b ∈ A[1],[n], b′ ∈ A[n],[1] we have b◦(a1⊕· · ·⊕
an) ◦ b′ ∈ A. If a is a homogeneous ideal, then a[1],[1] is an H-ideal, and conversely every
H-ideal A generates a homogeneous ideal a such that a[1],[1] = A. An H-ideal p ⊆ A[1],[1]
is prime if its complementA[1],[1]\p is multiplicatively closed. The set SpecA of all prime
H-ideals can be endowed with the Zariski topology in the usual way. Localization of an
F-ringA with respect to a multiplicative subset S ⊆ A[1],[1] also works exactly as classical
localization theory for commutative rings, with the localization functor having all the nice
properties we might expect. We will denote by Ap =
(
A[1],[1] \ p
)−1A the localization of
A with respect to the complement of a prime H-ideal p.
As in Deitmar’s geometry, we can use this localization theory to build structure sheaves.
Let A be an F-ring and U ⊆ SpecA an open set. For X,Y ∈ |F| let OA(U)Y,X denote
the set of functions
s : U −→
∐
p∈U
(Ap)Y,X
such that s(p) ∈ (Ap)Y,X and s is locally a fraction, i.e. for all p ∈ U , there is a neigh-
borhood V of p in U , an a ∈ AY,X and an f ∈ A[1],[1] \
∐
q∈V q such that s(q) = a/f
for all q ∈ V . This construction yields a sheaf OA of F-rings, which is called the structure
sheaf of A. For each H-prime p the stalk OA,p := lim−→p∈U OA(U) is isomorphic to Ap.
Moreover, the F-ring Γ(SpecA,OA) of global sections is isomorphic to A.
An F-ringed space is a pair (X,OX) where X is a topological space and OX is a
sheaf of F-rings. An F-ringed space (X,OX) is F-locally ringed if for each p ∈ X the
stalk OX,p is local, i.e. contains a unique maximal H-ideal mX,p. A (Zariski) F-scheme
is an F-locally ringed space (X,OX) such that there is an open covering X =
⋃
i∈I Ui
for which the canonical maps Pi : (Ui,OX |Ui) → SpecOX(Ui) are isomorphisms of
F-locally ringed spaces. The category of F-schemes is the category of inverse systems (or
pro-objects) in the category of Zariski F-schemes.
For F-ringsA,B,C and morphismsA→ B andA→ C, we can construct the (relative)
tensor productB⊚AC. Using this construction, it follows that the category of (Zariski) F-
schemes contains fibered sums. In particular, we can take A = F and C = F(Z), in order
to obtain an extension of scalars from F to F(Z); however, the category of F(Z)-algebras
is not equivalent to the category of rings (this is pretty much due to the existence of F-rings
which are not matrix rings, cf. [1] for details), so this functor does not provide an extension
of scalars from F-schemes to usual schemes.
The embedding of the category of rings inside the category of F-rings show that every
scheme can be regarded as an F-scheme. A similar construction allows to produce an
F-ring out of a monoid, providing a relation with M-schemes, cf. section 2.4 for further
details. It is worth noting that all the examples mentioned here are what Haran calls rings
of matrices; there are examples (cf. [20, §2.3, Examples 3 and 5] of more exotic F-rings
that are not rings of matrices. Haran succeeds (cf. [20, §6.3]) in defining the completion
SpecZ of the spectrum of Z, which is one step in Deninger’s program to prove the Riemann
hypothesis.
1.4. Durov’s generalized schemes. In [16], Durov introduces a generalization of classi-
cal algebraic geometry, into that Arakelov geometry fits naturally. As a byproduct of his
theory of generalized rings, he obtains a model for the field with one element and a notion
of a geometry over F1. Here, we outline Durov’s construction briefly. For full details, cf.
[16], or consider [17] for a summary.

8 JAVIER L ´OPEZ PE ˜NA AND OLIVER LORSCHEID
Any ring R can be realized as the endofunctor in the category of sets that maps a set
X to RX , the free R-module generated by X . The ring multiplication and unit translate
into properties of this functor, making it into a monad that commutes with filtered direct
limits (cf. for instance [39] or [28, Chapter VI]). Motivated by this fact, Durov defines a
generalized (commutative) ring as an monad in the category of sets that is commutative
(cf. [16, §5.1] for the precise notion of commutativity) and commutes with filtered direct
limits. If the set A([0]) = A(∅) is not empty, A is said to admit a constant, or we say
that A is a generalized ring with zero (cf. [16, §5.1]). There is a natural notion of a module
over a monad, which allows to construct the category of modules over any generalized ring
A. Every generalized ring A has an underlying monoid |A| := A([1]), so we can define
a prime ideal of A as any A-submodule p of |A| such that the complement |A| \ p is a
multiplicative system. The set SpecA of all the prime ideals in A can be endowed with the
Zariski topology in the usual way.
The notions of localization, presheaves and sheaves of generalized rings are defined
analogously to the usual theory of schemes resp. to the theory of the former sections. We
can talk about generalized ringed spaces (X,OX) consisting of a topological spaceX with
a sheaf of generalized rings OX . A generalized ringed space is local if for every p ∈ X
the generalized ring OX,p has a unique maximal ideal. Every generalized ring defines
a locally generalized ringed space (SpecA,OA), which is called the generalized affine
scheme associated to A. A generalized scheme is then a locally generalized ringed space
which is locally isomorphic to a generalized affine scheme. In this setting, the category
of F1-schemes consists precisely of those generalized schemes (X,OX) for that the set
Γ(X,OX(0)) is not empty, also called generalized schemes with zero (cf. [16, §6.5.6]).
Examples of generalized schemes in the sense of Durov include usual schemes via the
aforementioned realization of a ring as a generalized ring, as well as schemes over the
spectrum of monoids or semi-rings, in particular in Durov’s theory it is possible to speak
about schemes over the natural numbers, the completion Z∞ or the tropical semi-ring T.
As Haran does in the context of his non-additive geometry, Durov defines the completion
SpecZ of the spectrum of Z as a generalized scheme with zero. However, Durov’s con-
struction forces the fibered product SpecZ ×F1 SpecZ to be isomorphic again to SpecZ,
making it unsuitable to pursue Deninger’s program.
1.5. Varieties over F1 in the sense of Soule´ and its variations. The first suggestion of a
category that realizes a geometry over F1 was given by Soule´ in [36]. This notion found
several variations ([37], [7]) that finally lead to the notion of F1-schemes as given by
Connes and Consani in [8], cf. the following section. We do not give the formal definitions
from [36], [37] and [7] in its completeness, but try to give an overview of how ideas
developed.
1.5.1. S-varieties. We begin with the notion of varieties over F1 as given in [36]. To avoid
confusion, we will call these varieties over F1 “S-varieties”. For a precise definition, cf.
[36] or [26].
An affine S-variety X consists of
• a functor X : {finite flat rings} → {finite sets},
• a complex (non necessarily commutative) algebra AX ,
• an affine scheme XZ = SpecA of finite type,
• a natural transformation evX : X ⇒ Hom(AX ,−⊗Z C),
• an inclusion of functors ι : X ⇒ Hom(A,−) and
• an injection ιC : A⊗Z C →֒ AX

AN OVERVIEW OF GEOMETRIES OVER THE FIELD WITH ONE ELEMENT 9
such that the diagram
(1.1) X(R) ι(R) //
evX (R)

Hom(A,R)
−⊗ZC

Hom(AX , R⊗Z C)
ι∗C // Hom(A⊗Z C, R⊗Z C)
commutes for all finite flat rings R and such that a certain universal property is satisfied.
This universal property characterizes XZ together with ι and ιC as the unique extension of
the triple (X,AX , evX) to an affine S-variety. We defineXZ as the base extension of X to
Z.
A morphism X → Y between affine S-varieties consists of
• a natural transformationX → Y ,
• a C-linear map AY → AX and
• a morphismXZ → YZ of schemes
such that they induce a morphism between the diagrams (1.1) corresponding to X and Y .
The idea behind the definition of global S-varieties is to consider functors on affine
S-varieties. More precisely, an S-variety X consists of
• a functor X : {affine S-varieties} → {finite sets},
• a complex algebra AX ,
• a scheme XZ of finite type with global sections A,
• a natural transformation evX : X ⇒ Hom(AX ,A(−)),
• an inclusion of functors ι : X ⇒ Hom((−)Z, XZ) and
• an injection ιC : A⊗Z C →֒ AX
such that the diagram
X(V ) ι(V ) //
evX(V )

Hom(VZ, XZ)
global
sections

Hom(AX ,AV )
ι∗C // Hom(A⊗Z C,AV )
commutes for all finite flat rings R and such that a certain universal property is satisfied.
This universal property plays the same roˆle as the universal property in the definition of an
affine S-variety. We define XZ as the base extension of X to Z. Morphisms of S-varieties
are defined analogously to the affine case.
A remarkable property of these categories is that the dual of the category of finite
flat rings embeds into the category of affine S-varieties and that the category of affine
S-varieties embeds into the category of S-varieties. The essential image of the latter em-
bedding are those S-varieties whose base extension to Z is an affine scheme. Furthermore,
a system of affine S-varieties ordered by inclusions that is closed under intersections can
be glued to an S-variety.
In his paper, Soule´ constructs for every smooth toric variety an S-variety whose base
extension to Z is isomorphic to the toric variety. This result was extended in [26, Theorem
3.11], also cf. section 2.9.
The paper contains a definition of a zeta function for those S-varieties X that admit a
polynomial counting function, i.e. that the function q 7→ #XZ(Fq) on prime powers q is
given by a polynomial in q with integer coefficients. This provides, up to a factor 2π, a
first realization of Deninger’s motivic zeta functions of h0 and h2 (as in the introduction)
as zeta functions of the “absolute point SpecF1” and the “affine line over F1”.
1.5.2. S∗-varieties. In [37], Soule´ describes the first modification of this category. The
idea is to exchange the functor on finite flat rings by a functor on finite abelian groups.
Correspondingly, the functors Hom(AX ,− ⊗Z C) and Hom(A,−) in the definition of

10 JAVIER L ´OPEZ PE ˜NA AND OLIVER LORSCHEID
an affine S-variety are exchanged by Hom(AX ,C[−]) and Hom(A,Z[−]), respectively,
where Z[D] and C[D] are the group algebras of a finite abelian group D. We call the ob-
jects that satisfy the definition of an affine S-variety with these changes affine S∗-varieties.
Morphisms between affine S∗-varieties and the base extension to Z is defined analogously
to the case of affine S-varieties. Also the definition of S∗-varieties and morphisms between
S∗-varieties are defined in complete analogy to the case of S-varieties.
Soule´ proves similar results for S∗-varieties as for S-varieties: Affine S∗-varieties are
S∗-varieties. A system of affine S∗-varieties ordered by inclusions that is closed under
intersections can be glued to an S∗-variety. Every smooth toric variety has a model as an
S∗-variety (see also section 2.9). The definition of zeta functions transfers to this context.
1.5.3. CC-varieties. In [7], Connes and Consani modify the notion of an S∗-variety in
the following way. They endow the functor X from finite abelian groups to finite sets
with a grading, i.e. for every finite abelian group D, the set X(D) has a decomposition
X(D) = ∐n∈N Xn(D) into a disjoint union of sets. Secondly, they exchange the complex
algebra by a complex variety. We call the objects that satisfy the definition of an affine S∗-
variety with these changes affine CC-varieties. Morphisms between affine CC-varieties
and the base extension to Z are defined analogously to the case of affine S-varieties and
S∗-varieties.
The category of affine CC-varieties is embedded into a larger category that plays the
roˆle of the category of locally ringed spaces in the theory of schemes. CC-varieties are
defined as those objects in the larger category that admit a cover by affine CC-varieties.
The definition of zeta functions transfers to this context. Connes and Consani show
certain examples of CC-schemes, where the zeta function can be read off from the graded
functor of the CC-scheme.
The main application of this paper is the construction of models of split reductive groups
as CC-varieties over “F12”. This is a first construction of objects in F1-geometry that
contributes to Tits’ ideas from [40]. These results were extended in [26]; in particular,
they also hold in the context of S-varieties and S∗-varieties, cf. section 2.9. However, these
categories are not suitable to define a group law for any split reductive group over F1, cf.
[7, p. 25] and [26, Section 6.1]. For how this can be done in the context of CC-scheme, see
section 2.12 of the present text.
1.6. Schemes over F1 in the sense of Connes and Consani. In their paper [8], Connes
and Consani merge Soule´’s idea and its variations with Kato’s resp. Deitmar’s monoidal
spaces and Toe¨n and Vaquie´’s sheaves on monoids. Roughly speaking, a scheme over F1
in the sense of Connes and Consani, which we call “CC-scheme” to avoid confusion, is a
triple consisting of a locally representable functor on monoids, a scheme and an evaluation
map. Unlike Kato/Deitmar and Toe¨n-Vaquie´, all monoids are considered together with a
base point as in Haran’s theory ([20])–the locally representable functor is a functor from
M0 (see section 1.3) to Sets.
We make these notions precise. First of all, we can reproduce all steps in the construc-
tion of M-schemes as in section 1.1 to define M0-schemes. Namely, an ideal of a monoid
A with 0 is a subset I ⊂ A containing 0 such that IA ⊂ I . A prime ideal is an ideal p ⊂ A
such that A − p is a submonoid of A. As in section 1.1, we can define localizations and
the Zariski topology on SpecM0 A = {prime ideals of A}, the spectrum of A.
A monoidal space with 0 is a topological space together with a sheaf in M0. Together
with its Zariski topology and its localizations, SpecA is a monoidal space with 0. A
geometric M0-scheme is a monoidal space with 0 that has an open covering by spectra of
monoids with 0.
Similar to section 1.2, M0-schemes are defined as locally representable functors on
M0. In detail, an M0-functor is a functor from M0 to Sets. Every monoid A with 0
defines an M0-functorX = SpecM0 A by sending a monoidB with 0 to the set X(B) =

AN OVERVIEW OF GEOMETRIES OVER THE FIELD WITH ONE ELEMENT 11
Hom(SpecM0 B, SpecM0 A) = Hom(A,B). An affine M0-scheme is a M0-functor that
is isomorphic to SpecM0 A for some monoid A with 0.
A subfunctor Y ⊂ X of an M0-functor X is called an open subfunctor if for all
monoids A with 0 and all morphisms ϕ : Z = SpecM0 A → X , there is an ideal I of
A such that for all monoidsB with 0 and for all ρ ∈ Z(B) = Hom(A,B),
ϕ(ρ) ∈ Y (B) ⊂ X(B) ⇐⇒ ρ(I)B = B.
An open cover of X is a collection {Xi → X}i∈I of open subfunctors such that the map∐
i∈I Xi(H) → X(H) is surjective for every monoid H that is a union of a group with 0.
An M0-scheme is an M0-functor that has an open cover by affine M0-schemes.
If X is an M0-scheme, then the stalk at x is OX,x = lim→ X(U), where U runs through
all open neighborhoods of x. A morphism of M0-schemes ϕ : X → Y is a natural
transformation of functors that is local, i.e. for every x ∈ X and y = ϕ(x), the induced
morphism ϕ♯x : OY,y → OX,x between the stalks satisfies that (ϕ♯x)−1(O×X,x) = O×Y,y .
Connes and Consani claim that the M0-functor Hom(SpecM0 −, X) is a M0-scheme
for every geometric M0-schemeX and that, conversely, every M0-scheme is of this form
([8, Prop. 3.16]). For this reason, we shall make no distinction between geometric M0-
schemes and M0-schemes from now on.
The association
A→
(
Z[A]/1 · 0A − 0Z[A]
)
where 0A is the zero of A and 0Z[A] is the zero of Z[A] extends to the base extension
functor
−⊗M0 Z : {M0 − schemes} −→ {schemes}.
We denote the base extension of an M0-scheme X to Z by XZ = X ⊗M0 Z.
The ideas from sections 1.5.1–1.5.3 find now a simplified form. A CC-scheme is a
triple (X˜,X, evX) where X˜ is an M0-scheme, X is a scheme, viewed as a functor on the
category of rings, and evX : X˜Z ⇒ X is a natural transformation that induces a bijection
evX(k) : X˜Z(k) → X(k) for every field k. The natural notion of a morphism between CC-
schemes (Y˜ , Y, evY ) and (X˜,X, evX) is a pair (ϕ˜, ϕ) where ϕ˜ : Y˜ → X˜ is a morphism
of M0-schemes and ϕ : Y → X is a morphism of schemes such that the diagram
X˜Z
eϕZ //
evX

Y˜Z
evY

X
ϕ // Y
commutes. The base extension of X = (X˜,X, evX) to Z is XZ = X .
Remark 1.2. The definition of an M0-scheme is in the line of thoughts of Toe¨n and
Vaquie´’s theory as described in section 1.2. Indeed, monoid objects in the category of
pointed sets are monoids with 0. It seems to be likely, but it is not obvious, that the notion
a scheme w.r.t. the category of pointed sets are M0-schemes. Andrew Salch is working on
making this precise. Furthermore, Salch constructs a cosmos such that monoid objects on
this category correspond to triples (X˜,X, eX) where X˜ is an M0-functor, X is a functor
on rings and eX : X˜Z ⇒ X is a natural equivalence.
It is however questionable whether CC-schemes have an interpretation as schemes w.r.t.
some category in the sense of Toe¨n and Vaquie´ since it is not clear what an affine CC-
scheme should be. See Remark 2.5 for a more detailed explanation.
1.7. Torified varieties. Torified varieties and schemes were introduced by the authors of
this text in [26] in order to establish examples of varieties over F1 in the sense of Soule´ (cf.
section 1.5.1) and Connes-Consani (cf. section 1.5.3).

12 JAVIER L ´OPEZ PE ˜NA AND OLIVER LORSCHEID
A torified scheme is a scheme X endowed with a torification eX : T → X , i.e. T =∐
i∈I Gdim is a disjoint union of split tori and eX is a morphism of schemes such that the
restrictions eX |Gdim are immersions and eX(k) : T (k) → X(k) is a bijection for every field
k. Examples of torified schemes include (cf. [26, §1.3] or [27]):
• Toric varieties, with torification given by the orbit decomposition.
• Grassmann and Schubert varieties, where the torification is induced by the Schu-
bert cell decomposition.
• Split reductive groups, with torification coming from the Bruhat decomposition.
Let X and Y be torified schemes with torifications eX : T =
∐
i∈I Gdim → X and
eY : S =
∐
j∈J G
fj
m → Y , respectively. A torified morphism (X,T ) → (Y, S) consists of
a pair of morphisms ϕ : X → Y and eϕ : T → S such that the diagram
X
ϕ // Y
T
eX
OO
eϕ // S
eY
OO
commutes and such that for every i ∈ I there is a j ∈ J such that the restriction eϕ|Gdim :
Gdim → G
fj
m is a morphism of algebraic groups.
A torified scheme (X,T ) is affinely torified if there is an affine open cover {Uj} of X
that respects the torification, i.e. such that for each j there is a subset Tj =
∐
i∈Ij G
dj
m such
that the restriction eX |Tj is a torification of Uj . A torified morphism is affinely torified if
there is an affine open cover {Uj} of X respecting the torification and such that the image
of each Uj is an affine subscheme of Y . An (affinely) torified variety is an (affinely) torified
scheme that is reduced and of finite type over Z.
Toric varieties and split reductive groups are examples of affinely torified varieties, cf.
[26, §1.3], whilst the torifications associated to Grassmann and Schubert varieties are in
general not affine.
1.8. Λ-schemes after Borger. An approach that is of a vein different to all the other F1-
geometries is Borger’s notion of a Λ-scheme (see [3]). A Λ-scheme is a scheme with an
additional decoration, which is interpreted as descent datum to F1. In order to give a quick
impression, we restrict ourselves to introduce only flat Λ-schemes in this text.
A flat Λ-scheme is a flat scheme X together with a family Φ = {ϕp : X → X}p prime
of pairwise commuting endomorphism of X such that the diagram
X ⊗Z Fp

Frobp // X ⊗Z Fp

X
ϕp // X
commutes for every primes number p. Here, Frobp : X⊗ZFp → X⊗ZFp is the Frobenius
morphism of the reduction of X modulo p.
The base extension of this flat Λ-scheme is the scheme X . In particular every reduced
scheme X that has a family Φ as described above is flat.
Examples of Λ-schemes are toric varieties. For more examples, consider [3, sections 2]
or section 2.7 of this text.
2. PATHS AND BRIDGES
In this second part of the paper, we review and construct various functors between the
different F1-geometries. In some central cases, we will describe functors in detail, in other
cases, we will give a reference. We will also describe some work in progress by Arndt and
Marty on such functors. All these functors are displayed in the large diagram in Figure 1.

AN OVERVIEW OF GEOMETRIES OVER THE FIELD WITH ONE ELEMENT 13
2.1. Toric varieties as M-schemes. In this section, we recall the definition of a toric
variety, describe the reformulation of toric varieties in terms of a fan of monoids as done
by Oda ([35]) and relate them to M-schemes as done by Kato ([22]). For more details on
toric varieties, consider [18] and [29].
A (strictly convex and rational) cone (in Rn) is an additive semi-group τ ⊂ Rn of the
form τ = ∑i∈I tiR≥0 where {ti}i∈I ⊂ Qn ⊂ Rn is a linearly independent set. A face σ
of τ is a cone of the form σ = ∑i∈J tiR≥0 for some subset J ⊂ I . A fan is a non-empty
collection ∆ of cones in Rn such that
(1) each face of a cone τ ∈ ∆ is in ∆ and
(2) for all cones τ, σ ∈ ∆, the intersection τ ∩ σ is a face of both τ and σ.
In particular, the face relation makes ∆ into a partially ordered set.
If τ is a cone in Rn, we define Aτ as the intersection τ∨ ∩ Zn of the dual cone τ∨
with the lattice Zn ⊂ Rn. Since the generators ti of τ have rational coordinates, Aτ is a
finitely generated (additively written) monoid. An inclusion σ ⊂ τ of cones induces an
open embedding of schemes SpecZ[Aσ] →֒ SpecZ[Aτ ]. A toric variety (of dimension n)
is a scheme X together with a fan ∆ (in Rn) such that
X ≃ lim−→
τ∈∆
SpecZ[Aτ ].
A morphism ψ : ∆ → ∆′ of fans is map ψ˜ between partially ordered sets together with
a direct system of semi-group morphisms ψτ : τ → ψ˜(τ) (with respect to the inclusion
of cones). The dual morphisms induce monoid homomorphisms ψ∨τ : Aeψ(τ) → Aτ .
Taking the direct limit over the system of scheme morphisms SpecZ[ψ∨τ ] : SpecZ[Aτ ] →
SpecZ[Aeψ(τ)] yields a morphism ϕ : X → X ′ between toric varieties. Such a morphism
is called a toric morphism.
Note that a toric variety is determined by its fan and a toric morphism between toric
varieties is determined by the morphism between the fans. This leads to a completely
combinatorial description of the category of toric varieties in terms of monoids as follows
(cf. [22, section 9]).
Recall the definition of the quotient group QuotA and of an integral monoid from
section 1.1. The monoid A is saturated if it is integral and if for all a ∈ A, b ∈ QuotA
and n > 0 such that bn = a, we have that b ∈ A. A fan in Zn is a collection ∆ of (additive)
submonoidsA ⊂ Zn such that
(1) all A ∈ ∆ are finitely generated and saturated, A× = 1 and Zn/QuotA is
torsion-free,
(2) for all A ∈ ∆ and p ∈ SpecMA, we have A \ p ∈ ∆, and
(3) for all A,B ∈ ∆, we have A ∩ B = A \ p = B \ q for some p ∈ SpecMA and
q ∈ SpecMB.
In particular, ∆ is a diagram of monoids via the inclusions A \ p →֒ A.
This yields the following alternative description: a toric variety is a scheme X together
with a fan ∆ in Zn such that
X ≃ lim−→
A∈∆
SpecZ[A].
Note that SpecZ[A] is the base extension of the M-scheme SpecMA to Z, and that for
X˜ = lim−→
A∈∆
SpecMA,
X˜Z ≃ X . We introduce some further definitions to state Theorem 2.1 where we follow
ideas from [11, section 4].
An M-scheme is connected if its topological space is connected. An M-scheme is
integral / of finite type / of exponent 1 if all its stalks are integral / of finite type / (multi-
plicatively) torsion-free.

14 JAVIER L ´OPEZ PE ˜NA AND OLIVER LORSCHEID
Theorem 2.1 ([26, Thm. 4.1]). The association as described above extends to a functor
K : {toric varieties} −→ {M-schemes}
that induces an equivalence of categories
K :
{
toric varieties
} ∼−→
{
connected integral M-schemes
of finite type and of exponent 1
}
with −⊗F1 Z being its inverse.
2.2. Comparison of M-schemes with TV-schemes. A project of Florian Marty is the
investigation of the connection between M-schemes and TV-schemes. In the second arXiv
version of his paper [33], one finds partial results. From a private communication, we get
the following picture.
It is possible to associate to each affine TV-schemeX = SpecTV A a topological space
Y whose locale is the locale of X (defined by the Zariski open subsheaves as introduced
in section 1.2) and such that Y is homeomorphic to the topological space SpecMA as
considered in section 1.1. More precisely, this construction yields a functor
M1 : {affine TV-schemes} −→ {affine M-schemes}
that is an equivalence of categories with inverse M2 : SpecMA 7→ Hom(A,−). More-
over, Marty also proves in [33] that the notions of open immersions for M-schemes and
TV-schemes coincide.
It seems to be likely, but it is not obvious, that the functors M1 and M2 extend to an
equivalence of categories
{TV-schemes}
M1 //______ {M-schemes}
M2
oo_ _ _ _ _ _ .
2.3. M-schemes as M0- and CC-schemes. The category of M-schemes embeds into
the category of CC-schemes. We proceed in two steps.
Firstly, we construct a functor from M-schemes to M0-schemes. Consider the fully
faithful functor M +0−→ M0 that associates to a monoid A the monoid A+0 = A ∪ {0}
with 0 whose multiplication is extended by 0 · a = 0 for every a ∈ A+0. This induces a
faithful functor
{monoidal spaces} +0−→ {monoidal spaces with 0},
that we denote by the same symbol, by composing the structure sheaf OX of a monoidal
space X with M +0−→ M0. We reason in the following that this functor restricts to a
functor from the category of M-schemes to the category of M0-schemes.
The prime ideals of a monoid A (cf. section 1.1) coincide with the prime ideals of A+0
(cf. section 1.6) since we asked a prime ideal of a monoid with 0 to contain 0. Recall that
localizations, the Zariski topology, monoidal spaces (without and with 0) for M-schemes
and M0-schemes are defined in complete analogy. Thus the above functor restricts to a
faithful functor
{M-schemes} +0−→ {M0-schemes}.
Secondly, we define the functor
F : {M0-schemes} −→ {CC-schemes}
as sending an M0-scheme X to the CC-scheme (X,XZ, idXZ). A morphism ϕ : Y → X
of M0-schemes defines the morphism (ϕ,ϕZ) of CC-schemes. The condition that
YZ
ϕZ //
idYZ

XZ
idXZ

YZ
ϕZ // XZ

AN OVERVIEW OF GEOMETRIES OVER THE FIELD WITH ONE ELEMENT 15
commutes is trivially satisfied and shows thatF is faithful. The composition of the functors
+0 and F yields a functor
{M-schemes} −→ {CC-schemes}.
2.4. M- and M0-schemes as F-schemes. Let M be a monoid with 0. We can define an
F-ring F〈M〉 by setting F〈M〉Y,X = HomF〈M〉(X,Y ), that is, the set of Y ×X matrices
with values in M with at most one nonzero entry in every row and column. This is an F-
ring with composition defined by matrix multiplication. This is possible because all sums
that occur in the product of two matrices with at most one non-zero entry in every row and
column range over at most one term. This construction yields a faithful functor
F〈−〉 : M0 −→ {F-rings}
(cf. [20, §2.3, Example 2]), which has a right adjoint functor, namely, the functor −[1],[1]
that takes any F-ring A to the monoid A[1],[1] = HomA([1], [1]). Composition with the
functor +0 from section 2.3 yields a faithful functor from M to the category of F-rings,
which admits a right adjoint given by composing −[1],[1] with the forgetful functor.
Since ideals, localization and gluing in the category of F-rings are defined in terms of
the underlying monoid A[1],[1], which is in complete analogy to the construction of M0-
schemes, we obtain a faithful functor
F〈−〉 : {M0-schemes} −→ {F-schemes}.
Composing with +0 yields a faithful functor from the category of M-schemes to the cate-
gory of F-schemes.
2.5. M0-schemes as generalized schemes with zero. In this section we mention some
relations between the category of M-schemes and the one of Durov’s generalized schemes
with zero. What follows is based on a work in progress by Peter Arndt (cf. [1]).
Associated to any monoid with zeroM , we can construct the monad TM : Sets → Sets
that takes any set X into TM (X) := (M ×X)/∼, where we identify all the elements of
the form (0, x) and assume that if X = ∅ the quotient consists of one element (the empty
class). This monad is algebraic since cartesian products commute with filtered colimits, so
we have a functor T− from M0 to the category of generalized rings with 0. The functor
T− has as a right adjoint, namely, the functor | − | that associates to a generalized ring A
its underlying monoid |A| := A([1]).
The functor T− commutes with localizations because localizations in generalized rings
are defined in terms of localizations of the underlying monoids. Thus it naturally extends
to an embedding of categories
T− : {M0-schemes} −→ {Generalized schemes with 0}.
2.6. Relation between Durov’s generalized rings with zero and Haran’s F-rings. In
this section, we explain the relation between the categories of generalized rings with zero
defined by Durov and the one of F-rings defined by Haran, following some remarks by
Durov (cf. [16, §5.3.25]) and a work in progress by Peter Arndt (cf. [1]).
Given a generalized ring with zero T , we can construct the F-ring TD defined by the
sets of morphisms
TDY,X = HomTA(X,Y ) := {T (f)| f ∈ HomSets(X,Y )}
obtained by applying T to set maps between X and Y . This construction yields a functor
D : {Generalized rings with 0} −→ {F-rings},
admitting a left adjointA, where for every F-ringR the monadAR is defined byAR([n]) :=
R[1],[n].
It is worth noting that the functor D that sends generalized rings to F-rings is not
monoidal. This is due to the fact that the product ⊚ in the category of F-rings is not

16 JAVIER L ´OPEZ PE ˜NA AND OLIVER LORSCHEID
compatible with the tensor product of generalized rings, cf. [20, Remark 7.19] and [1] for
further details.
It seems to be likely that the pair of functors above lift to the corresponding categories
of schemes, providing an adjunction
{Generalized schemes with 0}
D // {F-schemes}
A
oo .
One of the interesting applications of the functor A is that we can define a base change
functor from F-rings to usual rings by composing A with Durov’s base change functor.
This extension of scalars is left adjoint to the inclusion of rings into F-rings. Complete
details on this extension of scalars functor and its properties will be given in [1].
2.7. M-schemes as Λ-schemes. In [3], Borger describes different examples of (flat) Λ-
schemes. We will recall his constructions briefly to explain, why the category of M-
schemes embeds into the category of Λ-schemes.
Given a monoidA, we can endow the (flat) scheme X = SpecZ[A] with the following
Λ-scheme structure Φ = {ϕp : X → X}p prime: for each prime p, the endomorphism
ϕp : X → X is induced by the algebra map Z[A] → Z[A] given by a→ ap for every a ∈
A. Every morphism of monoids A → B induces therefore a morphism of the associated
Λ-schemes SpecZ[B] → SpecZ[A].
Borger remarks in [3, section 2.3] that all small colimits and limits of Λ-schemes exist
and that they commute with the base extension to Z. Every M-scheme X has an open
cover by affine M-schemes Xi. By definition of the base extension to Z this yields an
open cover of XZ by Xi,Z. Since the Xi,Z have a structure of a Λ-scheme and XZ is the
limit over the Xi,Z and their intersections, X inherits the structure of a Λ-scheme. This
yields the functor
B : {M-schemes} −→ {Λ-schemes}.
Note, however, that B is not essentially surjective as certain quotients of Λ-schemes are
Λ-schemes that are not induced by an M-scheme. See [3, sections 2.5 and 2.6] for details.
2.8. Toric varieties and affinely torified varieties. As it was mentioned in section 1.7,
the orbit decomposition
∐
τ∈∆ Tτ = SpecZ[A×τ ] −→ X provides an affine torification of
a toric varietyX with fan ∆ (for details, cf. [26, §1.3.3]). Moreover, every toric morphism
between toric varieties induces a morphism between the corresponding affinely torified
varieties.
In other words, we obtain an embedding of categories
ι : {Toric varieties} −→ {Affinely torified varieties}.
2.9. Relation between affinely torified varieties, S-varieties and its variations. The
relation between affinely torified varieties and S-varieties or CC-varieties is established in
[26, §2.2, §3.3]. Theorems 3.11 and 2.10 in loc. cit. provide embeddings S and L of the
category of affinely torified varieties into the category of S-varieties resp. CC-varieties.
In order to show that the above functors define varieties over F1 in the sense of Soule´
or Connes-Consani, it is necessary to show that the universal property holds, which boils
down to prove that a certain morphism ϕC : X → V of affine complex varieties is actually
defined over the integers, where X is an affinely torified variety and V is an arbitrary
variety. The main idea of the proof is to consider for each irreducible component of X the
unique open torus Ti in the torification ofX that is contained in the irreducible component
(cf. [26, Corollary 1.4]). Since a split torus satisfies the universal property of an S-variety
resp. a CC-variety (cf. [36] and [7]), the morphism ϕC|Ti is defined over Z and extends to
a rational function ψ on X defined over the integers with the locus of poles contained in
the complement of Ti. But the extension to C of the rational function is ϕC, which has no
poles. Thus ψ cannot have a pole and is a morphism of schemes.

AN OVERVIEW OF GEOMETRIES OVER THE FIELD WITH ONE ELEMENT 17
Note that this proof that works for both S-varieties and CC-varieties can also be adopted
to S∗-varieties. This yields an embedding S∗ of the category of affinely torified varieties
into the category of S∗-varieties.
In [26], we find the construction of a partial functor FCC→S mapping CC-varieties to
“S-objects”, which, however, do not have to satisfy the universal property of an S-variety.
In a similar fashion, replacing the complex algebra AX by the complexified scheme XZ ⊗
C, or conversely replacing the complex scheme XC by the algebra of global sections,
provide ways to compare the categories of CC-varieties and S∗-varieties. But there are
technical differences between the two categories that prevent this correspondence to define
a functor. Namely, the notions of gluing affine pieces are different and it is not clear
how to define the grading for an CC-variety. Regarding the other direction, going from
the complex scheme XC to the algebra of global functions is a loss of information if the
CC-variety is not affine. All these issues suggest that the reader should consider these
categories as similar in spirit, but technically different.
2.10. Generalized torified schemes and CC-schemes. Torified schemes are closely con-
nected to Connes and Consani’s notion of F1-schemes. In this section, we generalize the
notion of a torified scheme, which allows the sheaves to have multiplicative torsion and
which provides an easier setting to compare the category of torified schemes with the cat-
egory of CC-schemes.
A generalized torified scheme is a triple X = (X˜,X, eX) where X˜ is a geometric M0-
scheme, X is a scheme, and eX : X˜Z → X is a morphism of schemes such that for every
field k the map eX(k) : X˜Z(k) → X(k) is a bijection. A morphism of torified schemes is
a pair of morphisms ϕ˜ : X˜ → Y˜ (morphism of M0-schemes) and ϕ : X → Y (morphism
of schemes) such that the diagram
X˜Z
eϕ //
eX

Y˜Z
eY

X
ϕ // Y.
commutes. We denote the natural inclusion of the category of affinely torified varieties into
the category of generalized torified schemes by ι. We also have an obvious inclusion
F ′ : {M0-schemes} −→ {generalized torified schemes}
constructed in the same fashion as the functor F in section 2.3.
Let X be a scheme together with a torification eX : T =
∐
i Gdim −→ X . Then
T = X˜Z for theM0-scheme X˜ =
∐
i G
di
m,M0 . Using this description, the relation between(generalized) torified schemes and CC-schemes becomes apparent.
Theorem 2.2. The functor
I : {generalized torified schemes} −→ {CC-schemes}
(X˜,X, eX) 7−→ (X˜(−), X(−), eX)
is an equivalence of categories.
Proof. First of all note that the objects of both categories satisfy the condition that X˜Z(k) =
X(k) for every fields k. The theorem follows from the facts that every M0-scheme X˜(−)
is represented by a geometric M0-scheme ([8, Prop. 3.16]) and that a natural transforma-
tion of representable functors is induced by a morphism between the representing objects
(Yoneda’s lemma). 
This theorem establishes toric varieties, Grassmann and Schubert varieties and split
reductive groups as CC-schemes. We have an immediate equivalence I ◦ F ′ ≃ F .

18 JAVIER L ´OPEZ PE ˜NA AND OLIVER LORSCHEID
Remark 2.3. If X = (X˜,X, eX) is a generalized torified scheme, then X is torified if and
only if there is an isomorphism X˜ ∼=
∐
x∈ eX Grkxm,M0 where rkx is the rank of the group
O×eX,x and if the restriction eX |Grkxm is an immersion for every x ∈ X˜ . As a consequence we
obtain the following result.
Corollary 2.4. Let X = (X˜,X, eX) be a CC-scheme. If X˜ is integral and torsion-free
and if eX |eYZ is an immersion for every connected component Y˜ of X˜ , then X = XZ is
torified.
Remark 2.5. The torification of Gr(2, 4) given by a Schubert decomposition is not affine
(cf. [26, Section 1.3.4]), what makes it unlikely that the corresponding CC-scheme has
an open cover by open sub-CC-schemes of the form (X˜,X, eX) such that both X˜ and
X are affine. This is a reason for the flexibility of CC-schemes, and it seems that the
category of CC-schemes is different to categories that are obtained by gluing affine pieces.
In particular, cf. Remark 1.2.
2.11. The map of F1-land. The functors that we described in the previous section are
illustrated in the commutative diagram in Figure 1 of categories and functors. Note that
we place the category of schemes on the outside in order to have a better overview of the
different categories. In Figure 1, all the extensions of scalars from F1 to Z are denoted
by − ⊗F1 Z, the functor ι is the canonical inclusion of (affinely) torified schemes into
generalized torified schemes. All the other functors have been defined in the previous
sections. As we are dealing with categories and functors, the diagram is commutative only
up to isomorphism. Some of the functors admit adjoints also described in the text, that in
many cases have been left out of the diagram for the sake of clarity. The dashed arrows
corresponding to the functors D and A′ represent a work in progress.
A few remarks about the commutativity (up to isomorphism) of the diagram. Any path
that starts at toric varieties and ends up at schemes will always produce the toric variety
itself, so all these possible paths are equivalent. Commutativity of subdiagrams involving
S-varieties, CC-varieties was proven in [26], and extends verbatim to S∗-varieties. The
commutativity of the triangle involving CC-schemes, generalized schemes with 0 and H-
schemes (also involving the functor A adjoint of D is explained in Arndt’s work [1]. The
equivalence (−⊗F1 Z) ◦ B ≃ (−⊗F1 Z) ◦M1 follows immediately from the definitions.
2.12. Algebraic groups over F1. The aim of Connes and Consani’s paper [7] as described
in section 1.5.3 was to realize Tits’ idea from [40] of giving the Weyl group of a split
reductive group scheme (cf. [12, Expose´ XIX, Def. 2.7]) an interpretation as a “Chevalley
group over F1”. Connes and Consani obtained partial results, namely, that every split
reductive group schemeG has a model “over F12” and that the normalizerN of a maximal
split torus T of G can be defined as a group object over F12 . However, there is no model
of G as an algebraic group over F1 or F12 . For a further discussion of possibilities and
limitation in this context, see [26, section 6.1].
In the category of CC-schemes, the same reasons prevent split reductive group schemes
(except for tori) to have a model as an algebraic group over F1. However, the second
author of this paper showed that this category is flexible enough to invent new notions of
morphisms that yield the desired results. We review the definitions and the main result
from [27].
Let X˜ be an M0-scheme. Recall the definition of the rank rkx of x ∈ X˜ from Remark
2.3. We define the sub-M0-scheme X˜ rk →֒ X˜ as the disjoint union ∐SpecM0 O×eX,x over
all points x ∈ X˜ of minimal rank. A strong morphism between F1-schemes (Y˜ , Y, eY )
and (X˜,X, eX) is a pair (ϕ˜, ϕ), where ϕ˜ : Y˜ rk → X˜ rk is a morphism of M0-schemes and

A
N
OV
ERV
IEW
O
F
G
EO
M
ETR
IES
OV
ER
TH
E
FIELD
W
ITH
O
N
E
ELEM
EN
T
19
Affinely torified varieties
S-varieties
S∗-varieties
TV-schemes
Toric varietiesM0-schemes
Λ-schemes
CC-schemes
Generalized torified schemes
CC-varieties
M-schemes
F-schemes
∼ I
Generalized schemes with 0
+0
B
A′
T−
S
L
ι S∗
−⊗F1 Z
−⊗F1 Z
−⊗F1 Z
−⊗F1 Z
F
−⊗F1 Z
−⊗F1 Z
−⊗F1 Z
−⊗F1 Z
F〈 〉
Schemes
M1
M2
Schemes
SchemesSchemes
T
K
∼DFIG
U
R
E
1
.
Th
e
m
ap
ofF
1
-land

20 JAVIER L ´OPEZ PE ˜NA AND OLIVER LORSCHEID
ϕ : Y → X is a morphism of schemes such that the diagram
Y˜ rkZ
efZ //
eY

X˜ rkZ
eX

Y
f // X
commutes.
The morphism specO×X,x → ∗M0 to the terminal object ∗M0 = SpecM0{0, 1} in the
category of M0-schemes induces a morphism
t eX : X˜ rk =
∐
x∈ eX rk
specO×X,x −→ ∗ eX :=
∐
x∈ eX rk
∗M0 .
Given a morphism ϕ˜ : Y˜ rk → X˜ rk of M0-schemes, there is thus a unique morphism
teϕ : ∗eY → ∗ eX such that teϕ ◦ teY = t eX ◦ ϕ˜. Let X rk denote the image of eX : X˜ rkZ → X .
A weak morphism between F1-schemes (Y˜ , Y, eY ) and (X˜,X, eX) is a pair (ϕ˜, ϕ), where
ϕ˜ : Y˜ rk → X˜ rk is a morphism of M0-schemes and ϕ : Y → X is a morphism of schemes
such that the diagram
Y˜ rkZ
eϕZ //
t eY ,Z %%
LL
LL
LL
X˜ rkZ tfX,Z
&&LL
LL
LL
(∗eY )Z
t eϕ,Z // (∗ eX)Z
Y rk
ϕ //
88qqqqqq
X rk
88qqqqqq
commutes.
If X = (X˜,X, eX) is an CC-scheme, then we define the set of F1-points X (F1) as
the set of strong morphism from (∗M0 , SpecZ, idSpecZ) to (X˜,X, eX). An algebraic
group over F1 is a group object in the category whose objects are F1-schemes and whose
morphisms are weak morphisms. The base extension functor − ⊗F1 Z from this category
to the category of schemes is given by sending X = (X˜,X, eX) to XZ = X . On the other
hand, the group law of an algebraic group G over F1 induces a group structure on the set
G(F1). We realize Tits’ idea in the following form.
Theorem 2.6 ([27, Thm. 7.9]). For every split reductive group scheme G with Weyl group
W , there exists an algebraic group G over F1 such that GZ ≃ G as group schemes and
G(F1) ≃W as groups.
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MATHEMATICS RESEARCH CENTRE, QUEEN MARY UNIVERSITY OF LONDON, MILE END ROAD, LON-
DON E1 4NS, UNITED KINGDOM
E-mail address: jlopez@maths.qmul.ac.uk (J. Lo´pez Pen˜a)
MAX-PLANCK INSTITUT FU¨R MATHEMATIK, VIVATSGASSE, 7. D-53111, BONN, GERMANY
E-mail address: oliver@mpim-bonn.mpg.de (O. Lorscheid)