GEOMETRY OVER THE FIELD WITH ONE ELEMENT

GEOMETRY OVER THE FIELD WITH ONE ELEMENT OLIVER LORSCHEID 1. MOTIVATION Two main sources have led to the development o several notions o F 1 -geometry in the recent ive years. We will concentrate on one o these, which originated as remark in a paper by Jacques Tits ([10]). For a wide class o schemes X (including aine space A n, projective space P n, the Grassmannian Gr(k, n), split reductive groups G), the unction N(q) = #X(F q ) is described by a polynomial in q with integer coeicients, whenever q is a prime power. Taking the value N(1) sometimes gives interesting outcomes, but has a 0 o order r in other cases. A more interesting number is the lowest non-vanishing coeicient o the development o N(q) around q 1, i.e. the number lim q 1 N(q) (q 1) r, which Tits took to be the number #X(F 1 ) o F 1 -points o X. The task at hand is to extend the deinition o the above mentioned schemes X to schemes that are deined over F 1 such that their set o F 1 -points is a set o cardinality #X(F 1 ). We describe some cases, and suggest an interpretation o the set o F 1 -points: #P n 1 (F 1 ) = n = #M n with M n := {1,..., n}. # Gr(k, n)(f 1 ) = ( n k) = #Mk,n with M k,n = {subsets o M n with k elements}. I G is a split reductive group o rank r, T G r m G is a maximal torus, N its nomalizer and W = N()/T (), then the Bruhat decomposition G(F q ) = w W BwB(F q) (where B is a Borel subgroup containing T ) implies that N(q) = w W (q 1)r qw d or certain d w 0. This means that #G(F 1 ) = #W. In particular, it is natural to ask whether the group law m : G G G o a split reductive group may be deined as a morphism over F 1. I so, one can deine group actions over F 1. The limit as q 1 o the action GL(n, F q ) Gr(k, n)(f q ) Gr(k, n)(f q ) induced by the action on P n 1 (F q ) should be the action S n M k,n M k,n induced by the action on M n = {1,..., n}. The other, more loty motivation or F 1 -geometry stems rom the search or a proo o the Riemann hypothesis. In the early 90s, Deninger gave criteria or a category o motives that would provide a geometric ramework or translating Weil s proo o the Riemann hypothesis or global ields o positive characteristic to number ields. In particular, the Riemann zeta unction ζ(s) should have a cohomological interpretation, where an H 0, an H 1 and an H 2 -term are involved. Manin proposed in [7] to interpret the H 0 -term as the zeta unction o the absolute point Spec F 1 and the H 2 -term as the zeta unction o the absolute Tate motive or the aine line over F 1. 1

2 OLIVER LORSCHEID 2. OVERVIEW OVER RECENT APPROACHES We give a rough description o the several approaches towards F 1 -geometry, some o them looking or weaker structures than rings, e.g. monoids, others looking or a category o schemes with certain additional structures. In the ollowing, a monoid always means a abelian mutliplicative semi-group with 1. A variety is a schem that deines, via base extension, a variety X k over any ield k. 2.1. Soulé, 2004 ([9]). This is the irst paper that suggests a candidate o a category o varieties over F 1. Soulé consideres schemes together with a complex algebra, a unctor on inite rings that are lat over and certain natural transormations and a universal property that connects the scheme, the unctor and the algebra. Soulé could prove that smooth toric varieties provide natural examples o F 1 -varieties. In [6] the list o examples was broadened to contain models o all toric varieties over F 1, as well as split reductive groups. However, it seems unlikely that Grassmannians that are not projective spaces can be deined in this ramework. 2.2. Connes-Consani, 2008 ([1]). The approach o Soulé was modiied by Connes and Consani in the ollowing way. They consider the category o scshemes together with a unctor on inite abelian groups, a complex variety, certain natural transormations and a universal property analogous to Soulé s idea. This category behaves only slightly dierent in some subtle details, but the class o established examples is the same (c. [6]). 2.3. Deitmar, 2005 ([3]). A completely dierent approach was taken by Deitmar who uses the theory o prime ideals o monoids to deine spectra o monoids. A F 1 -scheme is a topological space together with a shea o monoids that is locally isomorphic the spectrum o a ring. This theory has the advantage o having a very geometric lavour and one can mimic algebraic geometry to a large extent. However, Deitmar has shown himsel in a subsequent paper that the F 1 -schemes whose base extension to are varieties are nothing more than toric varieties. 2.4. Toën-Vaquié, 2008 ([11]). Deitmar s approach is complemented by the work o Toën and Vaquié, which proposes locally representable unctors on monoids as F 1 -schemes. Marty shows in [8] that the Noetherian F 1 -schemes in Deitmar s sense correspond to the Noetherian objects in Toën-Vaquié s sense. We raise the question: is the Noetherian condition necessary? 2.5. Borger, in progress. The category investigated by Borger are schemes X together with a amily o morphism {ψ p : X X} p prime, where the ψ p s are lits o the Frobenius morphisms Frob p : X F p X F p and all ψ p s commute with each other. There are urther approaches by Durov ([4], 2007) and Haran ([5], 2007), which we do not describe here. In the ollowing section we will examine more closely a new ramework or F 1 -geometry by Connes and Consani in spring 2009. 3. F 1 -SCHEMES À LA CONNES-CONSANI AND TORIFIED VARIETIES The new notion o an F 1 -scheme due to Connes and Consani ([2]) combines the earlier approaches o Soulé and o themselves with Deitmar s theory o spectra o monoids and Toën-Vaquié s unctorial viewpoint. First o all, Connes and Consani consider monoids with 0 and remark that the spaces that are locally isomorphic to spectra o monoids with 0, called M 0 -schemes, are the same as locally representable unctors o monoids with 0. (Note that they do not make any Noetherian hypothesis). There is a natural notion o morphism in this setting. The base extension is locally given by taking the semi-group ring, i.e. i A is a monoid with zero 0 A and X = Spec A is its spectrum, then X := X F1 := Spec ( [A]/(1 0 A 0 [A] ) ).

GEOMETRY OVER THE FIELD WITH ONE ELEMENT 3 An F 1 -scheme is a triple ( X, X, ), where X is an M 0 -scheme, X is a scheme and : X X is a morphism such that (k) : X (k) X(k) is a bijection or all ields k. Note that an M 0 -scheme X deines the F 1 -scheme ( X, X, id X ). We give irst examples o F 1 -schemes o this kind. The aine line A 1 F 1 is the spectrum o the monoid {T i } i N {0} and, indeed, we have A 1 F 1 F1 A 1. The multiplicative group G m,f1 is the spectrum o the monoid {T i } i {0}, which base extends to G m as desired. Both examples can be extended to deine A n F 1 and G n m,f 1 by considering multiple variables T 1,..., T n. More generally, all F 1 -schemes in the sense o Deitmar deliver examples o M 0 and thus F 1 -schemes in this new sense. In particular, toric varieties can be realized. To obtain a richer class o examples, we recall the deinition o a toriied variety as given in a joint work with Javier López Peña ([6]). A toriied variety is a variety X together with morphism : T X such that T i I m, where I is a inite index set and d Gdi i are non-negative integers and such that or every ield k, the morphism induces a bijection T (k) X(k). We call : T X a toriication o X. Note that T is isomorphic to the base extension X o the M 0 -scheme X = i I Gdi m,f 1. Thus every toriied variety : T X deines an F 1 -scheme ( X, X, ). In [6], a variety o examples are given. Most important or our purpose are toric varieties, Grassmannians and split reductive groups. I X is a toric variety o dimension n with an = {cones τ R n }, i.e. X = colim τ Spec [A τ ], where A τ = τ n is the intersection the dual cone τ R n with the dual lattice n R n. Then the natural morphism τ Spec [A τ ] X is a toriication o X. The Schubert cell decomposition o Gr(k, n) is a morphism w M k,n A dw Gr(k, n) that induces a bijection o k-points or all ields k. Since the aine spaces in this decomposition can be urther decomposed into tori, we obtain a toriication : T Gr(k, n). Note that the lowest-dimensional tori are 0-dimensional and the number o 0-dimensional tori is exactly #M k,n. Let G be a split reductive group o rank r with maximal torus T G r m, normalizer N and Weyl group W = N()/T (). Let B be a Borel subgroup containing T. The Bruhat decomposition w W BwB G, where BwB Gr m A dw or some d w 0, yields a toriication e G : T G analogously to the case o the Grassmannian. This deines a model G = ( G, G, e G ) over F 1. Note that in this case the lowest-dimensional tori are r-dimensional and that the number o r-dimensional tori is exactly #W. Clearly, there is a close connection between toriied varieties and the F 1 -schemes in the sense o Connes and Consani with the idea that Tits had in mind. However, the natural choice o morphism in this category is a morphism : X Ỹ o M 0 -schemes together with a morphism : X Y o schemes such that X X commutes. Unortunately, the only reductive groups G whose group law m : G G G extends to a morphism µ : G G G in this sense such that (G, µ) becomes a group object in the category o F 1 -schemes are algebraic groups o the orm G G r m (inite group). In the ollowing section we will show how to modiy the notion o morphism to realize Tits idea. Ỹ Y 4. STRONG MORPHISMS Let X = ( X, X, ) and Y = (Ỹ, Y, e Y ) be F 1 -schemes. Then we deine the rank o a point x in the underlying topological space X as rk x := rk O X,x, where O X,x is the stalk

4 OLIVER LORSCHEID (o monoids) at x and O X,x denotes its group o invertible elements. We deine the rank o X as rk X := min x X{rk x} and we let X rk := Spec O X,x, rk x=rk X which is a sub-m 0 -scheme o X. A strong morphism ϕ : X Y is a pair ϕ = (, ), where : Xrk is a morphism o M 0 -schemes and : X Y is a morphism o schemes such that X rk e Y X Y commutes. This notion comes already close to achieving our goal. In the category o F 1 -schemes together with strong morphisms, the object (Spec{0, 1}, Spec, id Spec ) is the terminal object, which we should deine as Spec F 1. We deine X (F 1 ) := Hom strong (Spec F 1, X ), which equals the set o points o X rk as every strong morphism Spec F 1 X is determined by the image o the unique point {0} o Spec{0, 1} in X rk. We see at once that #X (F 1 ) = #M k,n i X is a model o the Grassmannian Gr(k, n) as F 1 -scheme and that #G(F 1 ) = #W i G = ( G, G, e G ) is a model o a split reductive group G with Weyl group W. Furthermore, i the Weyl group can be lited to N() as group, i.e. i the short exact sequence o groups 1 T () N() W 1 splits, then rom the commutativity o G rk rk G G G (e G,e G ) m m G rk e G G we obtain a morphism m : Grk G rk G rk o M 0 -schemes such that µ = ( m, m) : G G G is a strong morphism that makes G into a group object. However, SL(n) provides an example where the Weyl group cannot be lited. This leads us, in the ollowing section, to introduce a second kind o morphisms. 5. WEAK MORPHISMS The morphism Spec O X,x M 0 to the terminal object M0 = Spec{0, 1} in the category o M 0 -schemes induces a morphism X rk = Spec O X,x X := M0. x X rk x X rk Given : Xrk, there is a unique morphism X Y such that X rk X Y commutes. Let X rk denote the image o : Xrk X. A weak morphism ϕ : X Y is a pair ϕ = (, ), where : Xrk is a morphism o M 0 -schemes and : X Y

GEOMETRY OVER THE FIELD WITH ONE ELEMENT 5 is a morphism o schemes such that X rk X rk ( X ) ( Y ) commutes. The key observation is that a weak morphism ϕ = (, ) : X Y has a base extension : X Y to, but also induces a morphism : X (F 1 ) Y(F 1 ). With this in hand, we yield the ollowing results. Y rk 6. ALGEBRAIC GROUPS OVER F 1 The idea o Tits paper is now realized in the ollowing orm. Theorem 6.1. Let G be a split reductive group with group law m : G G G and Weyl group W. Let G = ( G, G, e G ) be the model o G as described beore as F 1 -scheme. Then there is morphism m : G G G o M 0 -schemes such that µ = ( m, m) is a weak morphism that makes G into a group object. In particular, G(F 1 ) inherits the structure o a group that is isomorphic to W. We have already seen that X (F 1 ) = M k,n when X is a model o Gr(n, k) as F 1 -scheme. Furthermore, we have the ollowing. Theorem 6.2. Let G be a model o G = GL(n) as F 1 -scheme and let X be a model o X = Gr(k, n) as F 1 -scheme. Then the group action : GL(n) Gr(k, n) Gr(k, n), induced by the action on P n 1, can be extended to a strong morphism ϕ : G X X such that the group action ϕ(f 1 ) : S n M k,n M k,n, o G(F 1 ) = S n on X (F 1 ) = M k,n is induced by the action on M n = {1,..., n}. REFERENCES [1] A. Connes, C. Consani. On the notion o geometry over F 1. arxiv: 0809.2926 [math.ag], 2008. [2] A. Connes, C. Consani. Schemes over F 1 and zeta unctions. arxiv:0903.2024v2 [math.ag], 2009. [3] A. Deitmar. Schemes over F 1. Number ields and unction ields two parallel worlds, Progr. Math., vol. 239, 2005. [4] N. Durov. A New Approach to Arakelov Geometry. arxiv: 0704.2030v1 [math.ag], 2007. [5] S. M. J. Haran. Non-additive geometry. Compositio Math. Vol.143 (2007) 618 688. [6] J. López Peña, O. Lorscheid. Toriied varieties and their geometries over F 1. arxiv:0903.2173 [math.ag], 2009. [7] Y. Manin. Lectures on zeta unctions and motives (according to Deninger and Kurokawa). Astérisque No. 228 (1995), 4, 121 163. [8] F. Marty. Relative ariski open objects. arxiv:0712.3676 [math.ag], 2007. [9] C. Soulé. Les variétés sur le corps à un élément. Mosc. Math. J. 4 (2004), 217 244. [10] J. Tits. Sur les analogues algébriques des groupes semi-simples complexes. Colloque d algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956 (1957), pp. 261 289. [11] B. Toën and M. Vaquié. Au-dessous de Spec. Journal o K-Theory (2008) 1 64. MAX-PLANCK INSTITUT FÜR MATHEMATIK, VIVATSGASSE, 7. D-53111, BONN, GERMANY E-mail address: oliver@mpim-bonn.mpg.de