Projective geometry in characteristic one and the epicyclic category

# Projective geometry in characteristic one and the epicyclic category

Alain Connes Collège de France, 3 rue d’Ulm, Paris F-75005 France
I.H.E.S. and Ohio State University.
Caterina Consani Department of Mathematics, The Johns Hopkins University
Baltimore, MD 21218 USA.
###### Abstract

We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield of “max-plus integers” . Finite dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of . The associated projective spaces are finite and provide a mathematically consistent interpretation of J. Tits’ original idea of a geometry over the absolute point. The self-duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.

Characteristic one, Cyclic category, Semifields, Projective geometry, Hyperstructures.
###### :
19D55, 12K10, 51E26, 20N20.
thanks: The second author is partially supported by the NSF grant DMS 1069218 and would like to thank the Collège de France for some financial support.

## 1 Introduction

In this paper we establish a bridge between the combinatorial structure underlying cyclic homology and the -operations on one side and the framework of geometry in characteristic one on the other. The combinatorial system supporting cyclic homology and the -operations is best encoded by the cyclic category [CoExt] and its natural extension to the epicyclic category [G1, bu1] which play an important role in algebraic topology and algebraic -theory (cf.[good0]). In [cycarch], we showed the relevance of cyclic homology of schemes and the -operations for the cohomological interpretation of the archimedean local factors of L-functions of arithmetic varieties, opening therefore the road to applications of cyclic homology in arithmetic.
Mathematics in characteristic one has two algebraic incarnations: one is provided by the theory of semirings and semifields supporting tropical geometry and idempotent analysis, while the other one is centered on the more flexible notions of hyperrings and hyperfields on which certain number-theoretic constructions repose. In our recent work [CC1, CC2, CC3, CC4, CC5] we explained the relevance of these two algebraic theories to promote the development of an absolute geometry.
In this paper we provide the geometric meaning of the cyclic and the epicyclic categories in terms of a projective geometry in characteristic one and we supply the relation of the above categories with the absolute point. In §LABEL:sectF1 we show that the epicyclic category is isomorphic to a category of projective spaces over the simplest infinite semistructure of characteristic one, namely the semifield of “max-plus integers” (here denoted multiplicatively). The objects of are projective spaces where the semimodules  over are obtained by restriction of scalars from the one-dimensional free semimodule using the endomorphisms of . These endomorphisms form the multiplicative semi-group : for each integer the corresponding endomorphism is the Frobenius : . Let denote by the semimodule over obtained from by restriction of scalars using , then for the projective spaces provide the complete collection of objects of . The morphisms in are projective classes of semilinear maps of semimodules over which fulfill the condition . One also derives the definition of a full (but not faithful) functor to the category of finite sets which associates to a semimodule  over the quotient space (cf. Remark LABEL:rem (a)). If one restricts the construction of the morphisms in to maps which are linear rather than semilinear, one obtains a subcategory canonically isomorphic to the cyclic category : the inclusion functor corresponds to the inclusion of the categories .
It is traditional to view the category of finite sets as the limit for of the category of finite dimensional vector spaces over a finite field and the symmetric group as the limit case of the general linear group . There is however one feature of the category of finite dimensional vector spaces over a field which is not preserved by this analogy, namely the self-duality provided by transposition of linear maps. Indeed, the cardinality of the set of maps between two finite sets is a highly asymmetric function of the sets, whereas for vector spaces over the cardinality of is the symmetric function , for ().
The geometric interpretation provided in this paper of the epi/cyclic categories and of the functor refines and clarifies the above correspondence. In §LABEL:dual we prove that the well known self-duality of the cyclic category is described by transposition of linear maps. On the other hand, the failure of the extension of the property of self-duality to the epicyclic category is explained by the fact that the transpose of a semilinear map fails to be semilinear when the associated morphism of fields is not surjective. In our construction the semilinearity of the maps is encoded by the functor to the multiplicative monoïd of natural numbers (viewed as a small category with a single object) which associates to a morphism in the integer such that . also provides, using the functor , a geometric interpretation of the cyclic descent number of arbitrary permutations as the measure of their semilinearity: cf. Proposition LABEL:propdescent.
One can finally formulate a mathematically consistent interpretation of J. Tits’ original idea [Tits] of a geometry over the absolute point which is provided in our construction by the data given by the category () and the functor . Notice that the cardinality of the set underlying the projective space is and that this integer coincides with the limit, for , of the cardinality of the projective space . The fullness of the functor shows in particular that any permutation arises from a geometric morphism of projective spaces over .
Even though the above development of a (projective) geometry in characteristic one is formulated in terms of algebraic semistructures, in §LABEL:hypersec we show how one can obtain its counterpart in the framework of hyperstructures by applying a natural functor , where is the smallest finite hyperfield of signs (cf.[CC4]) that minimally contains the smallest finite idempotent semifield . In [CC5] we have shown that by implementing the theory of hyperrings and hyperfields one can parallel successfully J. M. Fontaine’s -adic arithmetic theory of “perfection” and subsequent Witt extension by combining a process of dequantization (to characteristic one) and a consecutive Witt construction (to characteristic zero). In view of the fact that this dequantization process needs the framework of hyperstructures to be meaningful, it seems evident that the arithmetical standpoint in characteristic one requires a very flexible algebraic theory which encompasses semistructures. On the other hand, several successful developments of the theory of semirings in linear algebra and analysis show that the context of semistructures is already adequate for many applications. The only reasonable conclusion one can draw is that for the general development of mathematics in characteristic one ought to keep both constructions available and select the most appropriate one in relation to the specific context in which each problem is formulated.

## 2 The epicyclic category

In this section we show that the notion of archimedean set and related category (that we introduced in [topos]), provides a natural framework for the definition of the variants (cf.[bu1, good0]) of the cyclic category of [CoExt] and of the epicyclic category (originally due to Goodwillie). All these categories can be obtained by restricting to archimedean sets whose underlying set is the set of integers with the usual total order. In this section we study the categories , and obtained by dropping the above restriction.

### 2.1 The category Arc of Archimedean sets

We recall from [topos] the following notion

###### Definition 2.1.

An archimedean set is a pair of a non-empty, totally ordered set and an order automorphism such that , and fulfilling the following archimedean property:

 ∀x,y∈X, ∃n∈N:y≤θn(x).

For each positive integer we introduce the following category

###### Definition 2.2.

The objects of the category are archimedean sets ; the morphisms in are equivalence classes of maps

 f:X→X′,  f(x)≥f(y)  ∀x≥y;f(θ(x))=θ′(f(x)),∀x∈X (1)

where the equivalence relation identifies two such maps and if there exists an integer such that , .

For we shall drop the index : coincides with the category of archimedean sets.

###### Proposition 2.3.

The full subcategory of whose objects are the archimedean sets , where is endowed with the usual order, is canonically isomorphic to the -cyclic category considered in [bu1, good0].

###### Proof.

One checks that the category of the archimedean sets such as is an extension (likewise ) of the small simplicial category by means of a new generator of the cyclic group , for each , that fulfills the relations (cf.  [good0], p. 235) \linenomath

 τ(n+1)an=id, τn∘σ0 =σn∘τ2n+1 τn∘σj =σj−1∘τn+1,, ∀j∈{1,…,n} τn∘δ0 =δn, τn∘δj =δj−1∘τn−1, ∀j∈{1,…,n}.
\endlinenomath

###### Definition 2.4.

We denote with the archimedean set whose automorphism is defined by the translation , for a fixed .

Such object gives rise to the object of : the shifted indexing will be more convenient for our applications.

### 2.2 The correspondences ¯¯¯¯Ψk

Let be an archimedean set and let be an integer. Then the pair is also an archimedean set that we denote as

 Ψk(X,θ):=(X,θk). (2)

For and two archimedean sets and a morphism in connecting them, one has ( fixed). Thus defines a morphism . However the two maps and which define the same morphism in the category are in general no longer equivalent as morphisms . More precisely, one derives a correspondence rather than a functor that satisfies the following properties

###### Proposition 2.5.

Let , then for a fixed positive integer the set

 ¯¯¯¯Ψk(h):={Ψk(f)∣f∈h}

is finite with exactly elements.

Let be composable morphisms in , then one has

 ¯¯¯¯Ψk(g∘h):=¯¯¯¯Ψk(g)∘¯¯¯¯Ψk(h)={u∘v∣u∈¯¯¯¯Ψk(g), v∈¯¯¯¯Ψk(h)}.

For any positive integers .

###### Proof.

Let be a morphism in , then the composite is equivalent to in the set , while the class of in is represented by , for . Thus is the finite set of classes of , for . These elements are pairwise inequivalent since the maps are pairwise distinct for .
Let , and . Let and be maps in the corresponding equivalence classes (fulfilling (1)). Then , and one also has by construction . By replacing by and by one substitutes with with and only the class of modulo matters for the corresponding morphism from .
For any morphism in one easily check that .∎

### 2.3 Two functors Arca⟶Arcb when b|a

The correspondences are best described in terms of two functors and , which we now describe in slightly more general terms.

Let : when , the functor is the natural “forgetful” functor. It is the identity on objects and associates to an equivalence class (Definition 2.2) of morphisms the unique class it defines in .

The definition of given in §2.2 determines, for every positive integer , a functor

 Ψk:Arckt⟶Arct (3)

this because the two maps and , which define the same morphism in the category , are equivalent as morphisms of the set . One thus obtains the following commutative diagram where the lower horizontal arrow is the correspondence \linenomath

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