Graphs Universality

(1)

Graphs Universality

Arnaud Vanhaecke [email protected]

Valdo Tatitscheff [email protected]

Abstract

Following P. Belkale’s and P. Brosnan’s paper [1], we study Kontsevich’s conjecture about polynomial countability of some schemes overZarising naturally from graphs. After a contextualization of this conjecture, we prove that it is false by means of geometric motives.

1 Feynman graphs, graph polynomials and Kontsevich’s conjecture

1.1 Historical overview and physical motivations for Feynman graphs

Feynman graphs are graphs which arise with some additional structure from the quantum-and- relativistic description of particles’ interactions.

Special relativity, achieved in 1905 by Einstein, is a theory of "fast moving objects", which takes into account the invariance of the speed of light through Minkowski’s metric. It predicts deviations from Newton’s theory due to this invariance, and explains the behavior of light in a natural way, while retrieving Maxwell’s dynamical equations for the electromagnetic (EM) field.

The basic objects in this theory are fields, that is functions of space-time, which have a tensorial behavior with respect to Minkowski’s metric, aspace-time metric.

On the other hand, quantum mechanics describe the behavior of objects at a very small scale, given by Planck’s constant ~. It has been built by several great scientists in the 20’s, such as Schrödinger, Heisenberg, Dirac, Einstein... In the framework of quantum mechanics, particles are described bywave functions, i.e probability waves which are functions of space-time as well, and take values inC. They express, by their square module, the density of the probability measure that describes the chance of finding the particle in a measurable part of space at a given time, with respect to Lebesgue’s measure on the space R³. The behavior in time of the wave function is described by Schrödinger’s equation, which is a diffusion equation, and thus can’t be relativistic. The implicit metric is the euclidean one.

The most astonishing feature of quantum theories is quantization. The possible states in such a theory are unit vectors of a separable Hilbert space, and physical observables are hermitian operators, such as the hamiltonian, i.e the energy operator. The set of possible measured values of an observable is the spectrum of the corresponding operator, and thus has countably many elements. One says that the observable quantities are quantized. For example, the energy of a quantum harmonic oscillator can only take values of the form~ω(¹₂+n), whereωis a parameter of the system, and nis an integer which labels the particular states which have been chosen to be the Hamiltonian’s eigenvectors.

Schrödinger’s equation has relativistic generalizations, namely Dirac’s equation, which describes fermions ("matter particles"), andKlein-Gordon’s equationwhich describes bosons ("non- matter particles"). Both of them can be retrieved by studying the representations of the Poincaré group (the symmetry group of relativistic flat space-time, R⁴ with Minkowski’s metric) in the process of building a quantum system (a Hilbert space) which has these underlying symmetries.

During the 40’s, Feynman, Schwinger and other scientist tried to give a quantum and relativistic description of interactions between matter and light. This theory was called Quantum Electrodynamics (QED), the first of the so-called Quantum Field Theories (QFTs) which are relativistic quantum theories. Between the 60’s and the early 80’s, it became clear that the weak and strong nuclear interactions could also be described correctly by QFTs.

In classical electrodynamics, it is a local differential equation which contains the information of the interaction and deviates the motion at each space-time point from the non-interacting particle’s motion. One has that in QED, the EM field is quantized too, and thus the motion is

"piecewise affine". The EM field only effects the particle’s motion at a discrete set of space-time points : when it absorbs or emits a photon, the quantum of the EM field.

(3)

Let’s look at the example of two electrons scattering : first, the classical description. Two electrons are projected against one another at high speed. At first very they are very far away from each other, and as they get closer their speeds decrease. There is a time denote t2 in the schematic below at which the speed of both electrons is zero, and after which their distance increases, as well as their speed. Let’s sketch a drawing to represent the way the scattering occurs.

Figure 1: The scattering experiment.

The particles move on a one-dimensional axis, but instead of comprehending it it on a line, time being a parameter, we represent time as the vertical direction, and make a two-dimensional drawing to describe this one dimensional event. Time flows from bottom to top.

Figure 2: We see the experiment as a two-dimensional diagram. Here the interaction is classically understood.

On the other hand, in the QED description, the same process would be described by a perturbative expansion. The simpler the interaction’s structure, the greater its probability. The simplest scattering consists of an exchange of one photon (γ), which instantaneously changes the momentum of the electrons, by absorption or emission. But there are many ways to do it ! The photon can be emitted by the electron coming from the left, and absorbed by the other one, or emitted by the electron coming from the right, and absorbed by the other one. Let’s describes these processes by simple sketches. For convenience again, time is seen as a dimension, fixed to be the vertical one.

The space-time points at which absorption or emission occur are possibly everywhere, in such a way that to get the total amplitude of the electron-electron scattering by one photon exchange, we have to integrate on all eventualities. Schematically, we represent all these situations by drawing a single diagram, in which the photon seems to travel instantaneously; in fact it contains the information of the two upper diagrams.

Let’s take an experimental point of view for the next few lines. Suppose you’re in a particle collider, knowing the momenta (the "speed") of incoming electrons. After a lot of experiments,

(4)

Figure 3: On the left, the left electrons emits the photon, whereas on the right, this same electron absorb the photon produced by the other one.

Figure 4: A more concise representation.

keeping only those in which the momenta of the electrons in the final state are close to chosen values. It is not possible to tell which final electron comes from which initial one, so we have to take this into account. Eventually, this scattering by one photon exchange is described by two "inequivalent" graphs. From now on, we will always keep the time axis as the vertical one, but we will not draw it anymore. Moreover, we label the edges of the graph by the momentum of the corresponding particles. The momenta are subject to momentum conservation, that is, the sum of momenta entering a vertex is equal to the sum of momenta leaving a vertex. We thus need an orientation on the graph, and because electrons are particles and have distinct anti-particles, we have put arrows on the electrons lines in the graphs. That is independent of the momenta-orientation of the graph, which will rather be drawn as little arrows near the edges.

In our example we can, and do, use the orientation induced by the arrows on the edges. We only have to fix the orientation of the photon edge.

These are Feynman graphs. Generically, in QFTs, interactions between particles can be sketched by Feynman diagrams: there is an isomorphism between the different ways of interacting, and the topologically inequivalent Feynman graphs. As looking at one Feynman diagram, one can write the corresponding Feynman amplitude, a complex number depending on the external momenta and particle masses, which can be used to compute the probability of the given

(5)

Figure 5: Two ways of interacting by one photon exchange.

event : first, one restricts to the ways of interacting one is interested in, then one computes the corresponding amplitudes, sums them all, and takes the squared module: this is the probability one is looking for.

Eventually, let’s go back to the concept of perturbative expansion. In a QFT, each field interaction, that is, each vertex on a Feynman graph, is linked to a number (thecoupling constant α) which describes the probability of the interaction. If this probability is small, then the more vertex the graph has, the more unlikely this way of interacting is; we said earlier that, in order to compute the likelihood of getting some final state (= external upper edges) from some initial state (= external lower edges), one should draw all possible graphs with these initial and final states, compute the amplitudes, sum them all, and take the square module of the sum. The "possible" graphs are those allowed by the theory, which fixes the possible forms for the interactions.

For example, in electron-photon QED, each vertex in a Feynman graph connects two electron lines and one photon line. A photon-photon-electron interaction is for example forbidden. Fixing the possible ways of interacting is exactly fixing the type of QFT we’re looking at. Moreover, when the probability of each interaction is small, i.e α is small, we can make a perturbative expansion, and a very good description of some process is given by the probability associated to the simplest Feynman graphs, those with a low number of vertices: indeed the contribution the higher order graphs becomes negligible, because the amplitude associated to a Feynman graph is proportional toα^{#V ertices}. Such an expansion is called a perturbative expansion, and the theory a perturbative QFT (pQFT). QED is a miracle, becauseαQEDis very small, of order ₁₃₇¹ , so one has a very good understanding of a process just by looking at the very first Feynman graphs.

However, the more graphs computed, the better the prediction.

In our example, the simplest graphs are of order two. We have already drawn the two classes of inequivalent order-two Feynman graphs describing this process.The next-to-simpler graphs are of order four; here are few of them.

This is how Feynman graphs and Feynman amplitudes arise in particle physics ; let’s now see how Feynman amplitudes involve the computation of graph polynomials.

(6)

Figure 6: A few graphs of order 4 for the electron-electron scattering.

1.2 From Feynman graph to graph polynomials through the calculus of Feynman amplitudes

In what follows, we will only consider the most simple Feynman graphs (scalar graphs), in order to have to most simple expressions of Feynman amplitudes. We work in euclidean flat D-dimensional space-time R^D. The previous discussion about Feynman graphs leads to the following generalization.

Definition 1.2.1. A Feynman graph Gis a triple (V, Eînt, Eêxt), where V is the finite set of vertices, Eînt is the set of internal edges, and Eêxt the set of external edges. Eînt is a set of 2-elements subsets ofV, andEêxt a set of 1-element subsets of V.

In order to express the corresponding amplitude, we assign to each oriented internal edge~i a mass mi ∈ R+ and a D-momentum k_~i ∈ R^D, and to each external edge ~e a D-momentum q~e∈R^D. Momenta arise together with an orientation of the graph. From now on, oriented edges will be denoted by letters with an arrow on top of it. External momenta are subject to overall momentum conservation:

X

~e∈E^ext

q~e= 0

where the~e’s are taken to be incoming. Letp_~_edenote the momentum in general, that is,p_~_e=k_~_e

(7)

ife∈E^int andp_~_e=q_~_e ife∈E^ext. The momenta are subject to local momentum conservation, that is,

∀v∈V, X

~e incoming at v

p_~_e= 0 To the graphGone assigns the Feynman amplitudeIG defined by :

I_G(q_e, e∈E^ext, m_i, i∈E^int) = (Z

R^D

)^#E^int Y

i∈E^int

d^Dk_i k²_i +m²_i

Y

v∈V

δ( X

~

e incoming at v

p_~_e) Theδ-functions lead to trivial integrations, but some of the internal momenta are not constrained by the external ones ; after all trivial integrations, if one is left withl d^Dki, one says that the graph Gis an l-loops graph, and thatIG is an l-loops integral. This definition ofloops do not coincide with the one on graphs, in fact, the "loop number" is thegraph genusg= #E^int−#V+1:

the number of "independent" cycles in the graph. Let’s show this with the notion ofA-flows on graphs.

A-flows, g-loops graphs and genus

Definition 1.2.2. LetAbe a ring with unit. WriteE for the unionE^int∪E^ext, andEfor the set of oriented edges inG. AnA-flow on the graphGis an application

ω: E→A satisfying the two following conditions :

i) ∀−uv→∈E, ω(−uv→) =−ω(−vu→) ii) ∀v∈V we have

X

~

eincoming atv

ω(~e) = 0

where in the second condition, v is always taken to be the well for the incident edges.

Suppose that the external edges are enumerated as e1,· · ·, en. We write ΩA(G, q1,· · · , qn) the set of A-flows on G with boundary conditions (q1,· · ·, qn).It is the set of A-flow ω on G satisfying:

∀ei∈E^ext, ω(e~i) =qi

where all e~i are incoming edges. We wish to understand the structure of ΩA(G, q1,· · · , qn) to know how many degrees of freedom are left after allδ integrations.

Let ˜Gbe the induced graph obtained by consideringGwithout its external edges, but with all of its vertices and its internal edges.

Remark 1.2.1. If ˜Gis a tree, then ΩA(G, q₁,· · · , q_n) is a singleton{ω0}.

Let~γ = (e~1, .., ~em) be an oriented cycle in ˜G(~γis a subgraph of Gwhose vertices are all of order 2, together with the orientation induced by the orientation of one edge). We can associate anA-flowω_~_γ ∈ΩA(G,0,· · · ,0) to~γbe letting :

ω~γ(−uv→) = 1if −uv→∈~γ ω~γ(−uv→) =−1 if ~vu∈~γ

ω_~_γ(−uv→) = 0otherwise

The following proposition proves that ag-genus graph ˜Gis a g-loop graph, and reciprocally.

(8)

Theorem 1.2.1. Let T be a spanning tree ofG, and˜ ωT the unique A-flow on T satisfying the boundary conditions. The flows associated to oriented cycles generateΩA(G, q1,· · ·, qn)−ωT as an A-module. Moreover, ifG is a Feynman graph of genusg,ΩA(G, q1,· · ·, qn)'ωT +A^g. Proof. Consider the graphT∪ {e}, wheree∈E( ˜G)−E(T). It contains a unique cycleγewhich gets oriented as soon as we chose an orientation for e. Let ~γ~e be the corresponding oriented cycle. Letω∈ΩA(G, q1,· · · , qn). One definesω⁰ ∈ΩA(G,0,· · ·,0) by

ω⁰= X

uv∈E( ˜G)−E(T)

ω(−uv→)ωγ−uv→

It is clear that ω andω⁰ coincide on all edges which are not inT. But the difference ω−ω⁰ in zero on these edges, so its restriction toT is anA-flow onT satisfying the boundary conditions.

Thus,ω=ω_T +ω⁰, and that concludes the proof.

Eventually, if the genus ofGisg, the non-redundant integration measure inI_G is a product of g times d^Dk_i-like measures. Note thus that I_G may diverge, but we can, by means of a regularization procedure, derivate from it a well-defined and finite quantity which contains all physical information.

From Feynman amplitude to graphs polynomials Let’s go back toIG: IG(qe, e∈E^ext, mi, i∈E^int) = (Z

R^D

)^#E^intod_i∈Eint

d^Dki

k_i²+m²_i Y

v∈V

δ( X

~e incoming at v

p~e) To get a more suitable expression, one may introduce a new real variable for each internal edge.

This is known asSchwinger’s trick: ∀i∈E^int, α_i∈R^∗+. Using the fact that:

1

X =Z ∞ 0

e^−αXdα

forX >0, we get:

IG(qe, mi) =Z

(R^∗+)^#E^int

Z

R^D∗#E^int

Y

i∈E^int

d^Dkidαie⁻ P

i∈Eint(k_i²+m²_i) Y

v∈V

δ(X

v

p~e) Then, one uses a general form of the Gauss identity:

Z

R^D

e^Q(x)dx= π^D²

p(detA)e^b^T^A⁻¹^b where

Q(x) =−x^TAx+ 2b^Tx

is a quadratic form on R^D with A ∈ M_D×D(R) symmetric and positive definite, and b ∈R^D. We eventually get the following expression forIG, where Cis a constant term:

IG(qe, mi) =C Z

(R^∗₊)^#E^int

Y

i∈E^int

dαi

ΨG^D²

e⁻

ΦG(qe)

ΨG −P

Eintm²_iα_i

(9)

The ΨG ∈Z[αe, e∈ E^int] and ΦG(qe)∈Z[αe, qe, e∈E^int] are the Symanzik polynomials. ΨG, which is also called Kirchhoff’s polynomial can be written:

ΨG=X

T

Y

e /∈T

αe

where the sum runs over all spanning trees of ˜G. The second Symanzik polynomial ΦG can be expressed using spanning 2-trees and is closely related to ΨG.

Another way to come to these polynomials is known as Feynman’s parameter technique. We will not describe it here, and only give the result of the manipulation, which is:

Z ^g Y

r=1

d^Dkr n

Y

j=1

1

(p²_j+m²_j)^ν^j = ˜C Z

x_j≥0,j∈J1,nK

(

n

Y

j=1

x^ν_j^j⁻¹dxj)δ(1−

n

X

i=1

xj) Ψ^ν−2(g+1)G

(ΦG+ ΨG(Pn

j=1xjm²_j))^ν−2g where ˜C is a constant term, theνj are non-zero integers andν=Pn

j=1νj.

Written like this, it is clearer that computation of Feynman’s amplitudes involves period integrals, as defined by Kontsevich and Zagier [8].

1.3 Konsevitch’s conjecture

LetG= (V, E) be a graph and ΨG be the polynomial:

ΨG=X

T

Y

e /∈T

α_e∈Z[x₁,· · ·, x_#E]

LetV(ΨG) denote the zero locus inA^#E_Z of ΨG overZ, which is a hypersurface inA^#E_Z , and let YG be its complement in A^#E_Z . LetQ denote the set of prime powers. We define the function

|YG|:Q →Zto be the function which associate to aq∈ Qthe number ofFq-points of the variety YG, that is, the number of distinct solutions (x1,· · ·, x#E)∈F^#Eq of the equation

PG(x1,· · ·, x#E)6= 0

For example, considerG=C₄ the 4-cycle (http://arxiv.org/pdf/math/9806055.pdf). Then P_G =x₁+x₂+x₃+x₄

x1, x2, andx3can be arbitrary, as soon as−x46=x1+x2+x3, thus:

|YG|(q) =q³(q−1)

In a lecture given at the Rutgers University Gelfand Seminar on December 8, 1997, M. Kontsevich stated the conjecture that|Y_G| ∈Z[q], i.e is a "universal polynomial" inqin the sense that it does not depend on the characteristic of the fieldF^q for example. Let’s define some useful notions to express the conjecture in a more formal way.

Conjecture 1.3.1 (Kontsevich’s conjecture). |Y_G| is polynomially countable for every graph G. Since|V(P_G)|+|Y_G|=q^#E, it is equivalent to say that for every graphG,|V(P_G)|is polynomially countable.

This conjecture is motivated by the fact that it holds for a very generals set of graphs : for example, Stembridge verified this conjecture for all graphs with less than 12 edges [14]; moreover, it holds for graphs which have a simple structure. For instance, ifG=C_n is the cycle withn vertices,V(P_G) is isomorphic toAⁿ⁻¹_Z , and thus|Y_G|=qⁿ−qⁿ⁻¹. For trees, we will prove later that the conjecture holds too.

(10)

1.4 Main theorem, and quick outline of the proof

In the following we will make an intensif use of the language of schemes and categories. We will refer to [7], [9] or even [6] for a detailed exposition of this theory. We will writeSchemefor the category of schemes and for a scheme S we will write Scheme/S for the relative objects with respect to S, that is the category of S-schemes. We will recall some notions of point-functors and representable functors in 2.1.

The combinatorial theorem

Definition 1.4.1. LetCMot⁺ be the group generated by all functions|X|, whereX is a scheme of finite type overZ. Because|X×Y|=|X| ×|Y|,CMot⁺is a ring. Moreover, it is aZ[q]-module since|A¹_Z| =q. We call CMot⁺ the ring of effective combinatorial motives. In this context let S be the multiplicative system in Z[q] generated by the functions q 7→ qⁿ −q for n >1. We define the ring of combinatorial motivesCMot as the localization ofCMot⁺ with respect to S : CMot=S⁻¹CMot. Since the functions which spanSnever vanish onQ, elements of the local ring CMot are everywhere-defined functions from Q to Q. Moreover let R=S⁻¹Z[q]. Let CGraphs denote theR-module generated by the functions|YG|for all graphsG.

Our goal is to prove the following theorem : Theorem 1.4.1. CGraphs=CMot

If we manage to prove it, the theorem implies that Kontsevich’s conjecture is false. If it were true, for any schemeX of finite type overZ, the function|X|would be inR, and thus, be rational. But if we take X to be the subscheme of A¹_Z defined by the equationE(x) =px for pa given prime, we have|X|(q) = qif p|q, and |X|(q) = 1 otherwise (0 is always a solution of the equation). In other words,|X| cannot be rational inq in that case. We will see later that we can show that Kontsevich’s conjecture is false even before proving this theorem, by giving a non-explicit counter-example.

Geometric form of the theorem We will in fact prove a more general theorem, that is a geometric version of theorem 1.4.1. We introduce the geometric counterpart of the former definition :

Definition 1.4.2. We defineGeoMot⁺ to be the Grothendieck ring of symbols [X] associated to objects of finite type in the category Schemeof schemes. That is the free abelian group on this symbols subject to the following relations :

(a) [X] = [Y] ifX ∼=Y.

(b) ifV ⊆X is a closed subscheme and U =X−V, then [X] = [U] + [V].

The ring structure is induced by the relation [X]·[Y] = [X×Y]; it thus admits [SpecZ] as a unit. This ring is called thethe ring of geometric effective motives.

Moreover we set L := [AZ] the Tate motive where AZ := SpecZ[X] is the affine line, and we consider the saturated multiplicative system S = hLⁿ −L | n > 2i^sat. We now define GeoMot := S⁻¹GeoMot the ring of geometric motives. Note that GeoMot⁺ is a Z[L]-module henceGeoMot_S is a R-module withR:=S⁻¹Z[L].

We have a natural group morphism ev: GeoMot→CMotcalled the evaluation that associates to a motive [X] its function|X|, extended byZ-linearity. This morphism will be detailed in the conclusion to the proof.

(11)

We will prove the following theorem, and by means of the evaluation morphism the preceding theorem will follow as a direct consequence.

Theorem 1.4.2. Graphs=GeoMot

Reduction of the problem In order to prove the theorem above, we introduce some additional notions which will be useful. First of all, ifGis a graph, one defines

QG =X

T

Y

e∈T

xe

It is constructed just as the polynomial ΨG, summing over all spanning treesT ofG, except that the product is on all edges which are in the spanning tree. LetV(Q_G) be the variety of zeros of QG over Z; one defines an algebraic variety XG =A^#E_Z −V(QG) with associated scheme XG. Stanley showed [13] that Kontsevich’s conjecture is equivalent to :

Conjecture 1.4.1. For all graphsG,|XG| ∈Z[q].

We will see that the [XG] and the [YG], whereYG is the scheme associated to the algebraic variety YG, generate the same R-submodule of GeoMot. Still motives of the form [XG] have a more tractable expression than the [YG], especially whenGis simple, and has anapex, because of the matrix-tree theorem as we will see.

Definition 1.4.3. A vertex v ofGis said to be an apex if∀u∈V, u6=v, vu∈E. Taking an arbitrary simple graphG= ((v₁,· · ·, v_n), E), we can construct a graphG^∗with apex by adding a vertexv₀ and connecting it by edges to all other vertices.

Using the matrix-tree theorem, Stanley showed that for any fieldk,X_G^∗(k) is isomorphic to the set ofn×nnon-degenerate, symmetrick-matricesM such thatM_ij = 0 ifi6=j and if there is no edge fromv_i tov_j. We will construct a corresponding schemeZ⁰_G that will have this set of k-points. Stanley observed for those schemes thatZ⁰_G'XG^∗, as we will prove.

We are thus led to a third and weaker form of the conjecture:

Conjecture 1.4.2. For every simple graph G,|Z⁰_G|is polynomially countable.

We will see that that form of the conjecture is false too. Let’s define an operation of complementation on graphs. ForGa graph, we denoteG^∗ the graph one gets by adding an apex.

Definition 1.4.4. LetGbe a graph, and H a subgraph of G.

1. We define the graphG−H as the graph obtained fromGby removing all edges which are in H, but leaving all vertices. We have (G−H)^∗=G^∗−H.

2. IfGis simple withnvertices,Gis contained in the complete graphK_n. Thecomplement G⁰ ofGis defined to be the graph K_n−G.

3. IfGis a graph, we denoteDGthe graph obtained by adding a disjoint vertex toG. Remark1.4.1. The operations of adding an apex, complementation, and adding a disjoint vertex, are related by the equality : (G⁰)^∗= (DG)⁰

The last step before proving our theorem is to shift our attention fromGto its complement.

Letkbe a field andZG=Z⁰_G0be the scheme whomk-points (as we will construct it) corresponds to alln×n k-matrices satisfying Mij = 0 if there is an edge betweenvi andvj.

(12)

Outline of the proof

Definition 1.4.5. LetGraphs_∗ be the R-submodule ofGeoMot generated by the motives [ZG] forGsimple graphs. We trivially have the inclusionGraphs_∗⊂Graphs because

[ZG] = [Z⁰_G0] = [X(G⁰)^∗] We are in fact going to prove the following theorem : Theorem 1.4.3. GeoMot⊂Graphs_∗

We will introduce a new R-submodule Matroids, generated in some way by objects called matroids. The first part of the proof will be the study of a special kind of schemes, theincidence schemes (I.S), which will serve to prove that Matroids ⊂ Graphs_∗. Then, Mnëv’s universality theorem (M.U) will give that matroids spanGeoMot, that isGeoMot=Matroids. The complete form of the demonstration is gathered in this theorem.

Theorem 1.4.4. We have the following inclusions Matroids GeoMot

Graphs_∗ Graphs

I.S.

M.U.

2 Schemes and Geometric Motives

In this section we develop most of the tools and definitions needed for the proof of the main theorem. We can not give full detailed proofs but we will give solid references so that the interested reader can easily find proofs of the facts we announce. Moreover the reader is supposed to be familiar with basic category theory, which may be found in the classical [10], and basic facts on schemes which can be found in the classical [6] and [7] or the more recent [9].

2.1 Representable functors

We give here only a concise description of representable functors and the Yoneda lemma. For more information and detailed exposition of these objects you may refer to [6, Chap. 0, 1] or to the detailed [5].

LeC be a category.

Definition 2.1.1. Let X be an object in C. Then we define the functor of points hX :=

Hom(., X).

Lemma 2.1.1(Yoneda lemma). Let F:C^o→Set be a functor, there is a canonical bijection Nat(F, h_X)∼=F(X)

Proof. We construct two inverse maps.

→ Let N: F → hX be a natural transform. By taking the value in X we get an arrow NX:h(X)→F(X). We now take the image of the identityX −−→^id^X X, this defines a map

α: Nat(F, h_X) → F(X) N 7→ N_X(id_X).

(13)

← Let ξ∈F(X), F gives a map Hom(Y, X)→Hom(F(X), F(Y)). So to an arrow Y −→^f X we can associate F(f)(ξ)∈F(Y). The same yields on the arrows, we thus get a map

β: F(X) → Nat(F, hX) ξ 7→ F(.)(ξ).

The fact that these map are inverses of one another is calculation left to the reader.

We get a straightforward corollary :

Corollary 2.1.1(Weak Yoneda lemma). Nat(hY, hX)∼= Hom(X, Y)

This lemma will allow us to encode canonically natural transformations with sets.

Definition 2.1.2. Let F:C^o→Setbe a functor, we say that the pair (X, ξ) withX ∈Ob(C) andξ∈Ob(C)represents the functorF ifξgives an isomorphismh_X∼=F in the sens of Yoneda lemma. If such a pair exists we say that the functor F is representable. In the case whereX representsF we will writeX(T) :=F(T) for all objectsT inC.

Example2.1.1. LetP:Set^o→Setbe the functor that associates to a setXthe set of its subsets P(X). This functor is representable and is represented by ({0,1},{1}). It is not really difficult to check it properly, you may think of the proof that the number of subsets of a finite setX is 2^#X.

We give another important example although we don’t give the complete proof.

Example 2.1.2. Let S be a base scheme. Then the functor O: Scheme^o_/S → Set, where Scheme/S is the category of schemes over S (or the relative category to the object S), associating to an S-scheme X its global sections OX(X). We can see that if S = SpecR this functor is represented by the affine line AS := SpecR[X] with the global section X giving the isomorphism in the sens of Yoneda lemma. More generally for a non-affine scheme this functor is representable, to do this properly we would have to show that we can glue together these affine schemes.

2.2 Motives and Zariski fibrations

For more information on motives see [3]. In this section we will consider S and all the other considered schemes to be Noetherian schemes. We extend the definition 1.4.2 over any noetherian base schemeS.

Definition 2.2.1. We define GeoMot⁺_S to be the Grothendieck ring of symbols [X −→^f S] associated to objects of finite type in the category Scheme_/S of schemes over a scheme S (This implies thatX is also noetherian), that is the free abelian group on this symbols subject to the following relations :

(a) [X−→^f S] = [Y −→^g S] ifX ∼=Y inScheme_/S.

(b) ifV ⊆X is a closed subscheme and U =X−V, then

[X −→^f S] = [U −−→^f|^U S] + [V −−→^f|^V S].

(14)

where f|U denotes the restriction of f to U. The ring structure is induced by the relation [X −→^f S]·[Y −→^f S] = [X×_SY −→^f S]; it thus admits [S−→^id S] as a unit written1. We will simply write [X]S for [X −→^f S] if the structure map f is clear or doesn’t matter, and as in definition ÜÜ the absence of subscript means that we take our motives overs SpecZ. This ring is called theDenef-Loeser ring of effective motives.

Moreover we set L:= [AS] the Tate motive whereAS is the affine line over S, and we consider the saturated multiplicative system S=hLⁿ−L|n>2i^sat. We now defineGeoMot_S :=

S⁻¹GeoMot the Denef-Loeser ring of motives. Note that GeoMot⁺_S is a Z[L]-module hence GeoMotS is aR-module withR:=S⁻¹Z[L].

We get a straightforward lemma that will be useful.

Lemma 2.2.1. [X]S = 0if X is the empty scheme. moreover[Xred]S = [X]S.

Proof. The first point comes from the definition. The second point is becauseX_red is a closed sub-scheme whose complement is empty.

Proposition 2.2.1. An arrow S−→^u T induces base change (ring) morphisms given by : u^∗: GeoMotT → GeoMotS

[X]T 7→ [X×T S]S, and ifS is of finite type overT :

u_∗: GeoMotS → GeoMotT

[X−→^f S] 7→ [X −−→^u◦f T]. This proposition is clear.

Remark 2.2.1. If the morphism uin proposition 2.2.1 is implicit we will just write MT :=u^∗M forM ∈GeoMotS.

Definition 2.2.2. A mapX−→^f Y inScheme_/S is called aZariski fibrationwith fiberF if there is an open covering{Yi,→Y} such that

X×Y Yi∼=F×SYi.

A covering{Yi,→Y}with this property will be said totrivialize the fibration.

Remark 2.2.2. Note that this definition works in any category with a Grothendieck topology, this is why the covering is written {Yi ,→Y} instead of Y =SY_i. It is routine to check that it generalizes the notion of a locally trivial fibration in topology. For more informations on Grothendieck topologies see [5].

Proposition 2.2.2. Let X −→^f Y be a Zariski fibration overS with fiber F of schemes of finite type overS.

(i) ForT −→^u S the pullbackX×ST −^f×−−−^S→^u Y ×ST is a Zariski fibration with fiber F. (ii) For a Zariski fibration Y −→^g Z overS with fiber G, X −−→^g◦f Z is a Zariski fibration with

fiberF×SG.

(iii) Letv:Y →S be the structure map ofY then[X]Y =v^∗[F]S withv^∗ defined in proposition 2.2.1.

(15)

Proof. Let{Yi,→Y}be a covering that trivialize the fibrationX −→^f Y.

(i) We only have to check that (X×ST)×_(Y_×_S_T₎(Yi×ST) = (X×Y Yi)×ST which comes from the universal property of the fibered product.

(ii) Let{Zi,→Z} be a covering that trivialize the fibrationY −→^g Z and let ˜Z_i,j:=Y_i×ZZ_j. with the canonical maps {Z˜_i,j ,→ Z} is a covering and by a small calculation, and using the fact thatY_i∼=Y ×_Y Y_i we can show that it trivializesg◦f with fiberF×_SG. (iii) Let U ⊆ Y be an open subset such that it trivializes the fibration, and let V be its

complement. We may then write X = (X×Y U)t(X×Y V) asY-schemes and thus get [X]Y = [F×SU]Y+[X×YV]Y and in the same way [F×SY]Y = [F×SU]Y+[F×SV]Y. By (i) the restriction off toX×Y V is a Zariski fibration and we get the result by noetherian induction.

Remark 2.2.3. In this proposition we used a classical technic called noetherian induction, this consist of iterating the described process to get a strictly decreasing sequence of closed subsets on which the conclusion is true. As the topological space is noetherian this process stops and gives the desired result on the initial scheme. This naturally only works because we supposed our schemes to be noetherian, as they are of finite type over a noetherian base. For more precise exposition on noetherian induction see.

Definition 2.2.3. Let X −→^f Y be a map in Scheme/S and v: Y → S be the structure map ofY. Then f is said to be amotivic fibration over S of fiber A∈GeoMotS if [X]Y =v^∗A. In particular proposition 2.2.2 assures that a Zariski fibration is a motivic fibration.

Proposition 2.2.3. Let X −→^f Y motivic fibration.

(i) ForT −→û S the pullbackX×ST −^f×−−−^S→û Y ×ST is a motivic fibration with fiberF. (ii) For V ⊆ X a closed sub-scheme of X with complement V if V −−→^f|^V Y and U −−→^f|Û Y

are motivic fibrations with fibers A and B respectively then [F] = A+B. Under these assumptions the fact that f is a motivic fibration is a consequence.

(iii) For a Zariski fibrationY −→^g Z overS with fiber G,X −−→^g◦f Z is a motivic fibration with fiber[F]·[G].

Proof. The proofs of (i) and (iii) are the same as in proposition 2.2.2. Also (ii) is immediate.

Proposition 2.2.4. Let X −→^f Y be a map in Scheme/S. Suppose there exists a motive M ∈ GeoMot_S such that for every fieldK and for every pointη∈Y(K)the map f:X_η→SpecK is a motivic fibration with fiberM_K thenf is a motivic fibration with fiberM.

Here we use the notationsM_K forM_Spec_K andX_η forX×_Spec_KSpeck(η) wherek(η) is the residue field inη. The proof relies on a lemma :

Lemma 2.2.2. Le M ∈GeoMotS where S is a reduced irreducible base scheme. Let K(S) be the function field ofS, ifMK(S) = 0then there exists a nonempty open subset U ⊆S such that MU = 0.

(16)

Proof. The lemma comes from the facts that if a mapf:X →Y between two schemes of finite type overS such that fK is an isomorphism then there is a nonempty open subsetU ⊆S such thatfU is an isomorphism and that ifA⊂XK is a closed subset with complementB then there is an nonempty open subsetU ⊆SandA⁰, B⁰ ⊂XU two disjoint sub-schemes respectively closed and open withA⁰_K=AandB_K⁰ =B and such thatXU =A⁰∪B⁰. These facts may be found in [6].To prove the proposition we may assume that Y is reduced by lemma 2.2.1 and that it is irreducible by considering only an irreducible component. Apply the hypothesis forK=K(Y) the function field ofY, we get [XK]K−MK = 0. The lemma assures that there exists a nonempty open subset U1 ∈ Y such that [XU]U −MU. Let V =Y −U, the fV: XU → V satisfies the hypothesis of proposition 2.2.4. We then get the result by noetherian induction (see remark 2.2.3).

Remark 2.2.4. As we supposed our base to be noetherian this proposition will make life easier in the realm of motives as it will allow us to deduce relations between general motives by establishing these relations on fields.

2.3 Vector bundles, linear group schemes and their motives

In this section letSbe a base scheme,E andF be quasi-coherentOS-modules. We will suppose them locally free respectively of finite ranke and f. Moreover for every T ∈ Scheme/S with structural morphismeu:T →S we will write the pull-back ET :=f^∗(E) := f⁻¹E ⊗f⁻¹OS OT

which is a quasi-coherentOS-module, locally free of rankein our case.

Linear group schemes For a detailled proof of the propositions that aren’t proved see [6, Chap. 1 par. 9]. Moreover this reference gives also lots of details around these schemes.

Definition 2.3.1. Letφ:E →F be a morphisme. Ther-degeneracy locus D_r(φ) is the closed subset of zeros of the morphism

r+1

^φ:

r+1

^E →

r+1

^F.

This givesDr(φ) the structure of a closed sub-scheme ofS. Moreover we putZr(φ) =Dr(φ)− D_r−1(φ). This gives a stratification ofS in closed sub-schemes.

Proposition 2.3.1. The functor F_E defined by

T F_E(T) = Hom_OT(ET,OT)

is represented by a couple(V(E), π_E)whereV(E)∈Scheme_/S andπ_E:EV(E)→OV(E). The proof is done in [6, Chap. 1 par. 9, p. 372]

Definition 2.3.2. V(E) defined in proposition 2.3.1 is called thevector bundleonS defined by E. We define asub-bundleas the vector bundle associated to a locally free submodule ofE such that the quotient is still locally free. We will say that such a module (resp. submodule) defines a vector bundle (resp. a sub-bundle).

Proposition 2.3.2. The functor defined by

T Hom_OT(ET,FT) is representable.

(17)

Proof. As in linear algebra we defineHom(E,F) :=V( ˇF⊗_O_SE). We then immediately get that the couple (Hom(E,F), π_E,F) with the morphismπ_E,F:EHom(E,F)→FHom(E,F) represents the given functor.

Remark2.3.1. The fact that (Hom(E,F), π_E_,_F) represents the functor means by Yoneda lemma that for any scheme T ∈ Scheme_/S and morphism ψ:ET →FT there is a unique mapT −→^f Hom(E,F) such that the following diagram commutes:

ET FT

EHom(E,F) FHom(E,F) ψ

id_E⊗_OSf˜ id_F⊗_OSf˜ π_E_,F

Definition 2.3.3. The schemeHom(E,F) defined in proposition 2.3.2 is called the homomor- phism scheme. This scheme admits by the morphismπ_E,F the stratification defined in definition 2.3.1. We put Hom₆r(E,F) := Dr(π_E,F) and Homr(E,F) := Zr(π_E,F) the determinantal schemes. ifE =OS^e andF =OS^f we will simply write these schemesHom(e, f) with the right subscript.

Example2.3.1. Let us explicit these schemes in the affine case,S= SpecA. LetA[y] be the polynomial ring withy={yij}^j=1···f_i=1···e a set ofe×f variables, we then getHom(OS^e,O_S^f) = SpecA[y].

Let m^k_l ∈ Z[y] be the collection of all k×k minors and Ik = h{m^k_l}li be the ideal generated by them. We then have Hom₆_r(OS^e,O_S^f) = Spec(A[y]/I_r+1) and moreover the subscheme Homr(OS^e,OS^f) =S

iSpec((A[y]/Ir+1)(m^r_i)) where the subscript (m^r_i) denotes localisation in the multiplicative set generated bym^r_i.

The next proposition will identifie these schemes as representing a natural functor, for the following we introduce therank of a mapψ:E →F as the integerrk(ψ) :=r if the cokernel of ψis a locally free of rankf−r.

Proposition 2.3.3. The scheme Homr(E,F)represents the functor T {ψ:ET →FT |rk(ψ) =r}

Proof. LetT ∈Scheme/S be a scheme andψ:ET →FT be a map, by the universal property of remark 2.3.1 we get a unique mapf:T →Hom(E,F) such that the diagram of the remark commutes. We only have to show thatf factors throughHom_r(E,F) if and only ifrk(ψ) =r. Now f factors through Hom_r(E,F) if and only if for an open covering {U_i ,→ T} the restrictionf|_U_i factors through Hom_r(E,F). Choosing this covering to be affine, by example 2.3.1 the factorization holds if and only if on any affine open subsetU ⊆T the ((r+1)×(r+1))- minors of ψ|U are zero while its (r×r)-minors are invertible. This is a commutative algebra problem which is solved in [4, Prop. 20.8,p. 495]

We also get a natural generalization of our beloved general linear group.

Definition 2.3.4. We defineGL(E) :=Home(E,E). We will simply writeGLeifE =OS^e

We get more of those :

T {Q:ET ⊗_O_SET →OT |Qsymmetric bilinear form}

is representable.

(18)

Proof. The functorT {B:ET ⊗_O_S ET →OT} is represented byHom(E,Eˇ) =V( ˇE ⊗_O_S Eˇ).

Moreover we have a transposition map ˇE ⊗_O_SEˇ→Eˇ⊗_O_SEˇthat induces a mapHom(E,Eˇ)→ Hom(E,Eˇ). We may check that the fixed points of this map define a closed sub-scheme that represents the given functor, the isomorphism being given byπ_E_,E˜.

Definition 2.3.5. The scheme that represents the functor in proposition 2.3.4 is called the symmetric scheme denoted by Sym(E) and Sym(e) if E =OS^e. It is, as in the proof, a sub- scheme ofHom(E,Eˇ) and thus it admits a stratification by the closed sub-schemesSym_r(E) :=

Homr(E,Eˇ)∩Sym(E).

We now define the Grassmannian, the proof of the existence of the corresponding scheme being too technical and out of the scope of this paper, we refer again to [6].

Definition 2.3.6. We define Grassr(E) to be the set of locallyOS-modules ofE of rankrsuch that the quotient is still locally free, it is the sub-bundles ofV(E).

T Grassr(ET) is representable.

This is proven at [6, Chap. 1, 9.4, p. 383].

Definition 2.3.7. The Grassmannian scheme is the scheme Grassr(E) that represents the functor in proposition 2.3.5. IfE =OS^e we will write it Grassr(e)

This comes along with a submodule of EGrass_r(E) that defines the isomorphism of schemes by Yoneda lemma. This submodule writtenS_r(E) is called theuniversal submodule. The vector bundle in the sens of definition 2.3.2 associated to this submodule is called the universal sub- bundleand is denoted bySub_r(E). AsV(EGrass_r(E)) =Grass_r(E) this is a (closed) sub-scheme ofGrassr(E).

In the same way we can take the quotient of EGrassr(E) by this submodule, this is called thecanonical quotient written Qr(E). The associated vector bundleQuot_r(E) called the Quot scheme.

Proposition 2.3.6. There are two natural maps

p:Homr(E,F)→Grassr(F)

q:Homr(E,F)→Grasse−r(E)

Proof. We have two natural transformation between the corresponding functor by associating, for T ∈ Scheme_/S, to a map ψ:ET → FT its image and its kernel sheaf. By Yoneda this respectively the mapspandq.

Definition 2.3.8. As the proof suggests itpand qare respectively called theImage map and thekernel map.

We now establish motivic formulas for these schemes, that are similar to the formulas we get when counting points

Proposition 2.3.7. We have in GeoMotS the formulas

Graphs Universality