contents...birational motives, i bruno kahn and r. sujatha preliminary version contents introduction...

BIRATIONAL MOTIVES, I

BRUNO KAHN AND R. SUJATHA

Preliminary version

Contents

Introduction 2

Part 1. Birational and stable birational categories 81. Review of some basic notions 82. Places and morphisms 173. Equivalences of categories 224. Stable birationality and R-equivalence 365. An interesting subcategory 426. Nonconnected varieties 447. Rational maps and proper birational maps 458. Examples and open questions 49

Part 2. Pure birational motives 499. Definition and first properties 4910. Locally abelian schemes 6011. Chow birational motives and locally abelian schemes 67

Part 3. Triangulated birational motives 7212. Birational finite correspondences 7213. Triangulated category of birational motives 8014. Birational motivic complexes 8315. Main theorems 9016. Unramified cohomology 94

Part 4. Appendices 98Appendix A. Complements on the localisation of categories 98Appendix B. Some lemmas on derived categories 119References 125

Date: February 27, 2004.1

2 BRUNO KAHN AND R. SUJATHA

Introduction

In this article, we try to conciliate two important ideas in algebraicgeometry: motives and birational geometry.

There are many reasons to do this; the one which motivated us orig-inally was to understand unramified cohomology from a motivic pointof view. Although this is not yet achieved in this paper (but see (2)at the end of this introduction), it led us to rather large developmentsand surprising structure theorems. For an application of an elementarypart of our theory to function fields over finite fields, see [32].

In order to give the reader a brief resumé of our results, we assumefamiliarity with Voevodsky’s construction of his triangulated categoriesof motives and recall the naturally commutative diagram of categories[66, 68], where F is a perfect field

Sm(F ) → SmCor(F ) −−−→ DM effgm(F )ff−−−→ DM eff− (F )

ff

x ffx ff

xSmproj(F )→SmCorproj(F ) −−−→ Choweff(F )

in which ff means fully faithful. Here Sm(F ) is the category of smoothconnected F -varieties, Smproj(F ) is its full subcategory consisting ofsmooth projective varieties, SmCor(F ) is the category of smooth va-rieties with morphisms finite correspondences, SmCorproj(F ) its full

subcategory consisting of smooth projective varieties, Choweff(F ) thecategory of effective Chow motives, DM effgm(F ) the triangulated cate-

gory of effective geometrical motives and DM eff− (F ) the triangulatedcategory of effective motivic complexes (for the Nisnevich topology).Then, for F of characteristic 0, we construct a diagram to which theformer maps naturally

S−1r Sm(F ) → S−1r SmCor(F )\ff−−−→ DMogm(F )

ff−−−→ DMo−(F )ox o

x ffx

S−1r Smproj(F )→S−1r SmCorproj(F )\ ∼−−−→ Chowo(F )x Alb

yT−1 place(F )op AbS(F )

Here ∼ means an equivalence of categories, \ denotes karoubian en-velope (or pseudo-abelian envelope, or idempotent completion), S−1rand T−1 denote localisation with respect to certain sets of morphisms

BIRATIONAL MOTIVES, I 3

Sr and T , place(F ) is the category of function fields over F with mor-phisms given by F -places, Chowo(F ) is the category of birational Chowmotives, DMogm(F ) the triangulated category of birational geometricalmotives, DMo−(F ) the triangulated category of rationally invariant mo-tivic complexes and AbS(F ) the category of locally abelian schemes.

As a very special case, this diagram shows that any function field hasa birational Chow motive which is natural with respect to F -places. Italso shows that the situation is strikingly simpler in the birational casethan in the “regular” case. In particular, the equivalence between alocalisation of the category of finite correspondences on smooth projec-tive varieties with that of birational Chow motives, although not oneof the most difficult results of this paper, is crucial in our proof thatthe functor Chowo(F )→ DMogm(F ) is fully faithful.

Let us now give a few more details on the various categories intro-duced above and some intuition of the situation. First, if one wants tomake sense of a “birational category”, one is confronted at the outsetwith two different ideas: use the notion of place of Zariski-Samuel [70]or use the geometric idea of a birational map. It turns out that bothideas work but give rather different answers.

The first idea gives the category place(F ), that we like to call thecoarse birational category. For the second idea, one has to be a littlecareful: the näıve attempt at taking as objects smooth varieties andas morphisms birational maps does not work because, as was pointedout to us by Hélène Esnault, one cannot compose birational maps ingeneral. On the other hand, one can certainly start from the categorySm(F ) and localise it with respect to the multiplicative set Sb of bira-tional morphisms. We like to call the resulting category S−1b Sm(F )the fine birational category. By hindsight, the problem mentionedjust above can be understood as a problem of calculus of fractionsin S−1b Sm(F ).

These definitions raise the following issues. First, in place(F ), theHom sets are very big. Second, in Sm(F ), the set Sb does not admitcalculus of left or right fractions in the sense of Gabriel-Zisman [18].And finally, there is no obvious comparison functor between the coarseand the fine birational categories at this stage.

In order to answer these issues at least to an extent, we introduce an“incidence category” SmP(F ), whose objects are smooth connected F -varieties and morphisms from X to Y are given by pairs (f, v), wheref is a morphism X → Y , v is a place F (Y ) 99K F (X) and f, v arecompatible in an obvious sense (see Definition 2.0.5 below). This cat-egory maps to both place(F )op and Sm(F ) by obvious forgetful func-tors. Denote by Smproj P(F ) the full subcategory of SmP(F ) consistingof smooth projective varieties. Note that the set Sb lifts naturally to


SmP(F ) and restricts to Smproj(F ) and Smproj P(F ) (same notation).Then:

T1 Theorem 1 (cf. Theorem 3.2.4). Assume F of characteristic 0. Thenlocalisation with respect to Sb yields a naturally commutative diagramof categories, in which vertical functors are equivalences of categorieswhile horizontal functors are full and essentially surjective:

S−1b Smproj P(F ) // //

o²²

S−1b Smproj(F )

o²²

S−1b SmP(F )// //

o²²

S−1b Sm(F )

place(F )op.

In particular, we get a full and essentially surjective functorplace(F )op → S−1b Sm(F ), justifying the terminology of “coarse” and“fine” birational categories. In the course of the proof, we get that,at least, the first axiom of calculus of right fractions (“Ore condition”)holds for proper morphisms in Sb, which is by no means obvious a pri-ori (see Proposition 3.3.9). Hence in S−1b Sm(F ) any morphism maybe written as a single fraction fs−1, with f a regular map and s ∈ Sb.On the other hand, the second axiom definitely does not hold.

If X,Y are two connected smooth projective varieties over F , we getin this way a surjective map from the set of F -places from F (Y ) toF (X) to the set of morphisms from X to Y in S−1b Sm

proj(F ). Thisdefines a natural equivalence relation on the first set, which is stableunder composition of places. Unfortunately we don’t know how todescribe this equivalence relation in concrete terms, and this appearsto be a very challenging question.

If we now define T to be the multiplicative set of (trivial) places inplace(F ) given by purely transcendental extensions of function fields,and Sr as the multiplicative set of stable birational morphisms in Sm(F ),i.e. dominant morphisms such that the corresponding extension offunction fields is purely transcendental, then we may localise furtherthe diagram of Theorem 1, getting part of the second diagram in thisintroduction (see Theorem 3.2.4). We call the corresponding categoriesthe coarse and fine stable birational categories.

Having the above categories at hand is well and good but, as forusual algebraic geometry, it is very difficult to compute with them. Sowe apply the classical idea to enlarge morphisms by adding algebraiccorrespondences and making the categories additive.


In this programme, having Voevodsky’s categories in mind, the firstlogical step (even if for us it came rather late in the development) is toextend part of Theorem 1 to finite correspondences, i.e. to study thefunctor

S−1b SmCorproj(F )→ S−1b SmCor(F )

where Sb denotes the image of Sb under the natural functor Sm(F )→SmCor(F ), i.e. the multiplicative set of graphs of birational mor-phisms. In Proposition 12.1.1, we prove that this functor is an equiv-alence of categories if F has characteristic zero. (The issue is faithful-ness.) Although the idea of the proof is simple (interpret finite corre-spondences as maps to symmetric powers as in [60]), details are notreally straightforward and the proof takes several pages. It uses thecategory SmP(F ) at a crucial step.

Although, in view of the above, the most natural definition of bira-tional Chow motives would be to localise effective Chow motives withrespect to Sr, this is not the way we proceed and we go through amore convoluted way, even though in the end our definition coincideswith the one just outlined (see Corollary 9.2.5). We define the categoryChowo(F ) as the pseudo-abelian envelope of the quotient of Choweff(F )by the ideal I characterised as follows: for X, Y two smooth projectivevarieties, I(h(X), h(Y )) is the subgroup of CHdimY (X × Y ) consistingof those correspondences whose restriction to U×Y is 0 for some denseopen subset U of X. The fact that I is an ideal, i.e. is stable underleft and right composition by any correspondence, is not obvious andamounts to a generalisation of the argument in [17, Ex. 16.1.11]. Infact, I is even a monoidal ideal, see Lemma 9.3.4, which means that thetensor structure of Choweff(F ) passes to Chowo(F ). In characteristic0, I can be described more concretely as the set of those morphismsfactoring through an object of the form M⊗L, where L is the Lefschetzmotive; in characteristic p, this remains true at least if one tensors allmorphisms with Q (Theorem 9.3.9). Hence, to obtain birational Chowmotives, we do something orthogonal to what is done habitually: in-stead of inverting the Lefschetz motive, we kill it!

This rather surprising picture becomes a little more natural if weconsider the parallel one for Voeovdsky’s triangulated motives. To getDMogm(F ), we simply invert open immersions (or, equivalently, the im-

age of Sb in DMeffgm(F )) and add projectors. The resulting category is a

tensor triangulated category. It is easy to see, only assuming F perfect,that DMogm(F ) = DM

effgm(F )/DM

effgm(F )(1) (Proposition 13.1.3 b)); this

is made intuitive by thinking of the Gysin exact triangles. Then one


easily sees that the functor Choweff(F ) → DM effgm(F ) of Voevodskyinduces a functor Chowo(F )→ DMogm(F ).

The main result of this paper is the computation ofHom(M̄(X), M̄(Y )[i]) for two smooth projective varieties X,Yand i ∈ Z, where M̄(X) and M̄(Y ) denote their motives in DMogm(F ).It may be stated as follows:

T2 Theorem 2 (cf. Th. 15.1.3). a) The functor Chowo(F )→ DMogm(F )is fully faithful.b) For X, Y, i as above, we have

Hom(M̄(X), M̄(Y )[i]) =

{CH0(YF (X)) if i = 0

0 if i 6= 0.

The proof of this theorem is rather intricate. We follow the methodVoevodsky used to compute Hom groups in DM effgm(F ): we introduceanother category DMo−(F ) of “rationally invariant motivic complexes”and construct a functor DMogm(F ) → DMo−(F ) that we show to befully faithful. In fact the definition of DMo−(F ) is simple: it is thefull subcategory of DM eff− (F ) consisting of those objects C such thatHiNis(X,C)

∼−−→ HiNis(U,C) for any dense open immersion of smoothschemes U → X. So DMo−(F ) is defined naturally as a subcategoryof DM eff− (F ), while DM

ogm(F ) is defined as a quotient of DM

effgm(F ).

This is not surprising, as DM eff− (F ) is a subcategory of the category offunctors from DM effgm(F ) to abelian groups. However, we have to showthat the embedding

eq0 (1) i : DMo−(F )→ DM eff− (F )has a left adjoint ν≤0. This is done fairly easily in Lemma 14.2.1. Thefact that Voevodsky’s full embedding DM effgm(F )→ DM eff− (F ) descendsto a full embedding DMogm(F ) → DMo−(F ) is then formal (Theorem14.2.2).

At this point, it remains to compute the group Hom(M̄(X), M̄(Y )[i])within DMo−(F ). This turns out to be very delicate and we can onlyrefer the reader to Section 15 for the proof.

We end this paper by relating the previous constructions to moreclassical objects. We define a tensor additive category AbS(F ) of locallyabelian schemes, whose objects are those F -group schemes that areextensions of a lattice (i.e. locally isomorphic for the étale topologyto a free finitely generated abelian group) by an abelian variety. Wethen show that the classical construction of the Albanese variety of a


smooth projective variety extends to a tensor functor

Alb : Chowo(F )→ AbS(F )which becomes full and essentially surjective after tensoring morphismsby Q (Proposition 11.2.1). So, one could say that AbS(F ) is the rep-resentable part of Chowo(F ). We also show that, after tensoring withQ, Alb has a right adjoint-right inverse, which identifies AbS(F ) ⊗Qwith the thick subcategory of Chowo(F )⊗Q generated by motives ofvarieties of dimension ≤ 1.

This work is only the beginning of our investigations on birationalmotives. Let us just mention for the moment future lines of research:

(1) We hope to use this formalism to study the unramified coho-mology of BG, where G is a linear algebraic group.

(2) The localisation of the Morel-Voevodsky A1-homotopy categoryof schemes H(F ) [46] with respect to Sr should be studied, aswell as that of Voevodsky’s effective stable A1-homotopy cate-gory of schemes. For the latter, this is very likely any “chunk”of the slice filtration of [67]; this analogy was pointed out to usby Chuck Weibel.

(3) The relationship between these ideas and the proofs that theBloch-Kato conjecture implies the Beilinson-Lichtenbaum con-jecture [61, 20, 21, 31] should definitely be investigated: a startis given in [31, Def. 2.14 and Lemma 2.15]. The best contextfor this seems to be in the previous item.

(4) Equally the relationship with Déglise’s category of generic mo-tives [12, 13]: at this point it is not completely clear what thisrelationship is.

(5) Finally, the exact relationship between the category Chowo(F )and Beilinson’s “correspondences at the generic point” [4] shouldbe investigated as well.

Acknowledgements. We would like to thank Shreeram Abyankhar,Yves André, Luca Barbieri-Viale, Antoine Chambert-Loir, Jean-LouisColliot-Thélène, Frédéric Déglise, Hélène Esnault, Najmuddin Fakhrud-din, Eric Friedlander, Ofer Gabber, Jens Hornbostel, Annette Huber-Klawitter, Bernhard Keller, Nikita Karpenko, Marc Levine, GeorgesMaltsiniotis, Louis Mahé, Vikram Mehta, Fabien Morel, Amnon Nee-man, Joël Riou, Raphaël Rouquier, Takeshi Saito, Marco Schlichting,Michel Vaquié, Claire Voisin, Vladimir Voevodsky and Chuck Weibelfor various discussions and exchanges while writing this paper. Wegratefully acknowledge the support of CEFIPRA project 2501-1 (as


well as CEFIPRA Project 1601-2 for the first author) during the prepa-ration of this work.

Notation. F is the base field. All varieties are F -varieties and allmorphisms are F -morphisms. If X is irreducible, ηX denotes its genericpoint.

Part 1. Birational and stable birational categories

1. Review of some basic notionspro

1.1. Pro-objects. We refer to [SGA4-I, Exp. 1, §8.10] or to [2, Ap-pendix] for this notion and its basic properties. We shall only recallthat

• (Opposite of “filtering”, [SGA4-I, Exp. 1, Def. 2.7]) A categoryI is cofiltering if it is nonempty and(1) for any two objects i, j, there exists a diagram

i

k

@@¢¢¢¢¢¢¢¢

ÁÁ<

>>>>

>

i′

@@¢¢¢¢¢¢¢¢

ÁÁ===

====

= i.

j′

@@¢¢¢¢¢¢¢¢


(3) Any two maps iv

⇒uj may be equalised by some map w :

i′ → i (i.e. , uw = vw).(A category I is filtering if the opposite category Iop is cofilter-ing.)• (Opposite of “cofinal”, [2, p. 149] or [SGA4-I, Prop. 8.1.3]) A

functor ϕ : I → J between two cofiltering categories is final if(1) For any i ∈ I, there is a j ∈ J and a map ϕ(i)→ j.(2) Any two such maps ϕ(i)

v

⇒uj may be equalised by a map

w : i′ → i, i.e. uϕ(w) = vϕ(w).(A functor ϕ : I → J between two filtering categories is cofinalif the opposite fonctor ϕop : Iop → Jop is final.)• A pro-object of a category C is a contravariant functor X :Iop → C from an essentially small filtering category I to C, orequivalently a covariant functor from an essentially small cofil-tering category. It turns out that in this paper, it is covariantfunctors from cofiltering categories that will appear naturally,and this is why we formulate standard notions in this context.Pro-objects form a category pro –C with the following formulafor morphisms from (X, I) to (Y, J):

eq1.3 (1.1) Hom((X, I), (Y, J)) = lim←−j

lim−→i

C(X(i), Y (j)).

(This formula does not depend on whether X and Y arecontravariant or covariant, but of course they should have thesame variance.)

The “constant object” functor c : C → pro –C is fully faithful andwe may (and will) thus identify C with a full subcategory of pro –C. Ifa functor C → D is fully faithful, then so is pro –C → pro –D: this isobvious.

If ϕ : I → J is a functor between two essentially small cofilteringcategories and X : J → C is a pro-object, then Y = X ◦ ϕ : I → C isanother pro-object and there is an obvious morphism of pro-objects

eq1.5 (1.2) ϕ∗ : X → Ygiven by the identity map from X(ϕ(i)) to Y (i) for all i ∈ I. If ϕ isfinal, then ϕ∗ is an isomorphism (dual of [SGA4-I, Exp. 1, 8.2.3]).

cf1.2. Calculus of fractions [18]. Let C be a category and S a setof morphisms of C. Recall from [18, I.1] the category C[S−1] and thecanonical functor PS : C → C[S−1]: PS is universal among functorsfrom C that render all arrows of S invertible.


We shall denote by 〈S〉 the set of morphisms s of C such that PS(s)is invertible: this is the saturation of S. We says that S is saturated ifit is equal to its saturation.

Recall (opposite of loc. cit. , I.2) that S admits a calculus of rightfractions when it verifies the following conditions:

(1) S is multiplicative: The identities are in S; S is stable undercomposition.

(2) Ore condition: Each diagram

eq1 (1.3)

Y ′

s

yX

u−−−→ Ywhere s ∈ S can be completed in a commutative diagram

X ′u′−−−→ Y ′

s′y s

yX

u−−−→ Ywith s′ ∈ S.

(3) Exchange of equalisers : If f, g : X ⇒Y are two morphisms andif s : Y → Y ′ is a morphism of S such that sf = sg, there existsa morphism t : X ′ → X of S such that ft = gt:

X ′ t−−→ Xf

⇒gY

s−−→ Y ′.

Under calculus of right fractions, the category C[S−1] has a very nicedescription S−1C, with the following set of morphisms between twoobjects X, Y (opposite of loc. cit. I.2.3):

eq1.1 (1.4) S−1C(X, Y ) = lim−→X′∈S/X

C(X ′, Y )

where S/X is the category (cofiltering by calculus of fractions) whoseobjects are morphisms s : X ′ → X with s ∈ S and morphisms aremorphisms in S above X.

In this article, we shall encounter situations where Condition (2) isverified only for certain pairs (u, s) and where Condition (3) is notverified. This leads us to give more careful definitions.

d1.2 1.2.1. Definition. a) A pair (u, s) as in Condition (2) above admits a(weak) calculus of right fractions within S if there exists a pair (u′, s′)as in the said condition. It admits a cartesian calculus of right fractionswithin S if the pull-back of (1.3) exists and provides such a pair (u′, s′).


b) The set S admits a weak calculus of right fractions if only Conditions(1) and (2) above are verified. It admits a cartesian weak calculus ofright fractions if each pair (u, s) with s ∈ S admits a cartesian calculusof right fractions.

We shall take the practice to abuse notation and write S−1C ratherthan C[S−1] even when calculus of fractions is not verified.1.3. Equivalence relations.

d1.5 1.3.1. Definition. Let C be a category. An equivalence relation onC consists, for all X, Y ∈ C, of an equivalence relation ∼X,Y =∼ onC(X,Y ) such that f ∼ g ⇒ fh ∼ gh and kf ∼ kg whenever it makessense.

Given an equivalence relation ∼ on C, we may form the factor cate-gory C/ ∼, with the same objects as C and such that C/ ∼ (X, Y ) =C(X,Y )/ ∼. This category and the projection functor C → C/ ∼ areuniversal for functors from C which equalise equivalent morphisms.1.3.2. Example. Let A be an Ab-category (sets of morphisms areabelian groups and composition is bilinear). An ideal I in A is givenby a subgroup I(X, Y ) ⊆ A(X, Y ) for all X,Y ∈ A such that IA ⊆ Iand AI ⊆ I. Then the ideal I defines an equivalence relation on A,compatible with the additive structure.

Let ∼ be an equivalence relation on the category C. We have thecollection S∼ = {f ∈ C | f is invertible in C/ ∼}. The functor C →C/ ∼ factors as a functor S−1∼ C → C/ ∼. Conversely, let S ⊂ C be aset of morphisms. We have the equivalence relation ∼S on C such thatf ∼S g if f = g in S−1C, and the localisation functor C → S−1C factorsas C/ ∼S→ S−1C. Neither of these two factorisations is an equivalenceof categories in general.

1.3.3. Exercise. Let A be a commutative ring and I ⊆ A an ideal.a) Assume that the set of minimal primes of A that do not contain

I is finite (e.g. that A is noetherian). Show that the following twoconditions are equivalent:

(i) There exists a multiplicative subset S of A such that A/I 'S−1A (compatibly with the maps A→ A/I and A→ S−1A).

(ii) I is generated by an idempotent.

(Hint: show first that, without any hypothesis, (i) is equivalent to

(iii) For any a ∈ I, there exists b ∈ I such that ab = a.)b) Give a counterexample to (i)⇒ (ii) in the general case (hint: take

A = kN, where k is a field).


place1.4. Valuations and places [70, Ch. VI], [7, Ch. 6]. Recall [7,Ch. 6, §2, Def. 3] that a place from a field K to a field L is a mapλ : K ∪ {∞} → L ∪ {∞} such that λ(1) = 1 and λ preserves sum andproduct whenever they are defined. We shall usually denote places bydotted arrows:

λ : K 99K L.Then Oλ = λ−1(L) is a valuation ring of K and λ|Oλ factors as

Oλ →→ κ(λ) ↪→ Lwhere κ(λ) is the residue field ofOλ. Conversely, the data of a valuationring O of K with residue field κ and of a field homomorphism κ → Luniquely defines a place from K to L (loc. cit. , Prop. 2). It is easilychecked that the composition of two places is a place.

If K and L are extensions of a field F , we say that λ is an F -placeif λ|F = Id and then write F (λ) rather than κ(λ).

In this situation, let X be an integral F -scheme of finite type withfunction field K. A point x ∈ X is a centre of a valuation ring O ⊂ K ifO dominates the local ring OX,x. If O has a centre on X, we sometimessay that O is finite on X. As a special case of the valuative criterionof separatedness (resp. of the valuative criterion of properness), x isunique (resp. exists) for all O if and only if X is separated (resp.proper) [23, Ch. 2, Th. 4.3 and 4.7].

On the other hand, if λ : K 99K L is an F -place, then a pointx ∈ X(L) is a centre of λ if there is a map ϕ : SpecOλ → X lettingthe diagram

SpecOλϕ

%%LLLLL

LLLLLL

SpecKoo

²²SpecL

λ∗OO

x // X

commute. Note that the image of ϕ is then a centre of the valuationring Oλ and that ϕ uniquely determines x.

In this paper, when X is separated we shall denote by c(O) ∈ X thecentre of a valuation ring O and by c(λ) ∈ X(L) the centre of a place λ,and carefully distinguish the two notions (one being a scheme-theoreticpoint and the other a rational point).

ratmaps1.5. Rational maps. LetX, Y be two F -schemes of finite type. Recallthat a rational map from X to Y is a pair (U, f) where U is a denseopen subset of X and f : U → Y is a morphism. Two rational maps(U, f) and (U ′, f ′) are equivalent if there exists a dense open subset U ′′


contained in U and U ′ such that f|U ′′ = f ′|U ′′ . We denote by Rat(X, Y )the set of equivalence classes of rational maps, so that

Rat(X, Y ) = lim−→MapF (U, Y )where the limit is taken over the open dense subsets of X.

SupposeX integral for simplicity. Then there is a largest open subsetU of X on which a given rational map f : X Ã Y is defined [23, Ch.I, Ex. 4.2]. The (reduced) closed complement X − U is called thefundamental set of f (notation: Fund(f)). We say that f is dominantif f(U) is dense in Y .

Similarly, let f : X → Y be a birational morphism. The complementof the largest open subset of X on which f is an isomorphism is calledthe exceptional locus of f and is denoted by Exc(f).exc

Note that the sets Rat(X,Y ) only define a precategory (or diagram,or diagram scheme, or quiver) Rat(F ), because rational maps cannotbe composed in general. To clarify this, let f : X Ã Y and g : Y Ã Zbe two rational maps, where X, Y, Z are varieties. We say that f andg are composable if f(ηX) /∈ Fund(g), where ηX is the generic point ofX. Then there exists an open subset U ⊆ X such that f is defined onU and f(U) ∩ Fund(g) = ∅, and g ◦ f makes sense as a rational map.This happens in two important cases:

• f is dominant;• g is a morphism.

This composition law is associative wherever it makes sense. In par-ticular, we do have the category Ratdom(F ) with objects F -varietiesand morphisms dominant rational maps. Similarly, Var(F ) acts onRat(F ) on the left.

1.6. The graph trick. We shall often use this well-known and basicdevice. For future reference, we formalise it here in a rather fancy way.Let U, Y be two F -varieties. Let C = C(U, Y ) denote the category oftriples (X, j, f), where j : U → X is an open immersion (X a variety)and f : X → Y is a morphism. On the other hand, let D = D(U, Y ) bethe category of triples (X, j, g), where j : U → X is an open immersionand g : U → Y is a morphism. There is an obvious functor

T : C → D(X, j, f) 7→ (X, j, fj).

p1.1 1.6.1. Proposition. The functor T has a right adjoint S. If Y isproper, then the counit TS(E) → E is proper (on the X components)for any E ∈ D.


Proof. Given E = (X, j, g) ∈ D, consider the graph Γg ⊂ U × Y . Bythe first projection, Γg

∼−−→ U . Let Γ̄g be the closure of Γg in X × Y ,viewed as a reduced scheme, and j′ : U → Γ̄g the corresponding openimmersion. We define

S(X, j, g) = (Γ̄g, j′, π)

where π is induced by the second projection. Clearly, if Y is properthen the first projection Γ̄g → X is proper. The fact that this defines aright adjoint (i.e. that j′ is minimal among solutions to the extensionproblem) is easily checked and is left to the reader. 2

1.7. Structure theorems on varieties. Here we collect some well-known results, for future reference.

nagata 1.7.1. Theorem (Nagata [49]). Any variety X can be embedded into aproper variety X̄. We shall sometimes call X̄ a compactification of X.

Chow 1.7.2. Theorem (Relative Chow lemma, [EGA 2, Th. 5.6.1]). Forany morphism f : Y → X, there exists a projective birationalmorphism p : Ỹ → Y such that f ◦ p is quasi-projective.

Hironaka 1.7.3. Theorem (Hironaka [25]). If charF = 0,a) For any variety X there exists a projective birational morphism

f : X̃ → X with X̃ smooth. (Such a morphism is sometimes calleda modification.) Moreover, f may be chosen such that it is an iso-morphism away from the inverse image of the singular locus of X. Inparticular, any smooth variety X may be embedded as an open subsetof a smooth proper variety (projective if X is quasi-projective).b) For any proper birational morphism p : Y → X between smoothvarieties, there exists a proper birational morphism p̃ : Ỹ → X whichfactors through p and is a composition of blow-ups with smooth centres.

In several places we shall have to use resolution of singularities, henceimplicitely assume characteristic 0. The expression “Under resolutionof singularities” will mean that we assume that the conclusions of The-orem 1.7.3 are valid. We shall specify this by putting an asterisk to thestatement of the corresponding result.

dJ 1.7.4. Theorem (de Jong [14, Th. 4.1]). For any variety X there ex-

ists an alteration X̃ → X, with X̃ smooth (recall that an alteration isa proper, generically finite morphism).


zoo11.8. A zoo of categories. In this part, we shall mainly work withinthe category of F -varieties Var(F ): objects are F -varieties (i.e. in-tegral separated F -schemes of finite type) and morphisms are all F -morphisms. We will occasionally view Var(F ) as a sub-precategory ofRat(F ). However we will also have to work with a dizzying array ofsubcategories of Var(F ). To define these subcategories, we shall re-strict the type of varieties, the type of morphisms, or both. In orderto normalise notation we choose to denote restriction of the type ofvarieties by a superscript and the type of morphisms by a subscript.

Types of varieties. We shall use the following three main types: proper,quasi-projective and smooth, as well as their intersections. We shalluse the following exponents to denote them:

• prop for proper• qp for quasi-projective• sm for smooth• proj for projective (= proper and quasi-projective)• smqp for smooth quasi-projective• smprop for smooth proper• smproj for smooth projective.

Types of morphisms. We shall use two main types: dominant mor-phisms and “good” morphisms, the latter for lack of a better terminol-ogy:

1.8.1. Definition. A morphism f : X → Y is good if it sends thesmooth locus of X into the smooth locus of Y , both being supposed tobe nonempty (this is automatic if F is perfect).

The reason for introducing this class of morphisms will appear inProposition 3.3.3. If Y is smooth then any morphism to Y is good; thecomposition of two good morphisms is good. We shall use the followingindices to denote them:

• dom for dominant• sm for good• smdom for good and dominant.

We shall often use the following abbreviation which hopefully willnot create any confusion: Sm(F ) = Varsm(F ). Hence the followingother abbreviations:

• Smprop(F ) = Varsmprop(F )• Smqp(F ) = Varsmqp(F )• Smproj(F ) = Varsmproj(F )


On the other hand, note that all morphisms in Sm(F ) are good, sothat Smsm(F ) = Sm(F ) and Smsmdom(F ) = Smdom(F ), and similarlyfor the subcategories of Sm(F ). This makes altogether 24 categories!

zoo21.9. A zoo of multiplicative systems. To muddy the water a littlemore, let us introduce collections of morphisms of Var(F ) that we aregoing to invert. The two main ones are

• Birational morphisms Sb: s ∈ Sb if s is dominant and inducesan isomorphism of function fields• Stably birational morphisms Sr: s ∈ Sr if s is dominant and

induces a purely transcendental extension of function fields

In addition, we shall occasionally encounter the following oversets ofSb and Sr (living in Rat(F )):

• Birational maps S̃b: dominant rational maps which induce anisomorphism of function fields

• Stably birational maps S̃r: dominant rational maps which in-duce a purely transcendental extension of function fields

as well as the following subsets of Sb:

• So: open immersions• Spb : proper birational morphisms• Swb : the multiplicative subset of Spb generated by blow-ups with

smooth centres (the exponent w is meant to recall “weak fac-torisation”)

and of Sr:

• Spr : proper stably birational morphisms• Sh: the multiplicative subset of Spr generated by morphisms of

the form pr2 : X ×P1 → X.• Swr : the multiplicative subset of Spr generated by Swb and Sh.

The following lemma is elementary and therefore we give it here.

sb=so 1.9.1. Lemma. In Var(F ),a) The sets Sb and So have the same saturation (hence they give iso-morphic localisations). The same is true in Vardom(F ), Varsm(F ),Varsmdom(F ), Sm(F ) and Smdom(F ).b) The sets Spr and S

pb ∪Sh have the same saturation. The same is true

in Varqp(F ) *and in Sm(F ) under resolution of singularities.c) *(Weak weak factorisation.) Under resolution of singularities, thesets Spb and S

wb have the same saturation in Sm(F ).


Proof. a) It suffices to observe that, by definition, a birational mor-phism s : X → Y sits in a commutative diagram

X

s

²²

U

j>>~~~~~~~~

j′ ÃÃ@@@

@@@@

@

Y

where j, j′ ∈ So. This proves the lemma for Var(F ), Vardom(F ), Sm(F )and Smdom(F ) since birational morphisms are dominant. In the remain-ing cases, we just observe that any open immersion is good.

b) For a morphism s : Y → X in Spr , it suffices to consider a com-mutative diagram

Ỹt

¡¡¢¢¢¢

¢¢¢¢

u

$$IIIII

IIIIII

Y

s ÂÂ???

????

? X × (P1)n

πyyttt

tttttt

t

X

obtained by the graph trick, with t, u ∈ Spb . When we are in Sm, wefurther desingularise Ỹ thanks to Theorem 1.7.3 a).

c) (cf. [22, p. 47, (iii)]). Let p : Y → X be in Spb . By Theorem1.7.3 b), there exists q : Ỹ → Y in Spb such that pq ∈ Swb . Applyingthe theorem a second time, there exists r : Z → Ỹ in Spb such thatqr ∈ Swb . So q is invertible in (Swb )−1 Sm(F ) and therefore p is invertiblein (Swb )

−1 Sm(F ). 2

2. Places and morphisms

d1.1 2.0.2. Definition. We denote by place(F ) the category with objectsfinitely generated extensions of F and morphisms F -places. We denoteby field(F ) the subcategory of place(F ) with the same objects, but inwhich morphisms are F -homomorphisms of fields. We shall sometimescall the latter trivial places.

r1.1 2.0.3. Remark. If λ : K 99K L is a morphism in place(F ), then itsresidue field F (λ) is finitely generated over F , as a subfield of thefinitely generated field L. On the other hand, given a finitely gener-ated extension K/F , there exist valuation rings of K/F with infinitely


generated residue fields as soon as trdeg(K/F ) > 1, cf. [70, Ch. VI,§15, Ex. 4].

l1 2.0.4. Lemma. Let f, g : X → Y be two morphisms, with X integraland Y separated. Then f = g if and only if f(ηX) = g(ηX) =: y andf, g induce the same map F (y)→ F (X) on the residue fields.Proof. Let ϕ : X → Y ×F Y be given by (f, g). Then the diagonal∆Y is closed, so Ker(f, g) = ϕ

−1(∆Y ) is closed in X and contains ηX(compare [EGA, Ch. I, Prop. 5.1.5 and Cor. 5.1.6]). 2

d1 2.0.5. Definition. Let X, Y be two integral F -schemes of finite type,with Y separated, f : X Ã Y a rational map and v : F (Y ) 99K F (X)a place. We say that f and v are compatible if

• v is finite on Y (i.e. has a centre in Y ).• The corresponding diagram

ηXv∗−−−→ SpecOvy

yU

f−−−→ Ycommutes, where U is an open subset ofX on which f is defined.

l2 2.0.6. Proposition. Let X, Y, v be as in Definition 2.0.5. Suppose thatv is finite on Y , and let y ∈ Y be its centre. Then a rational mapf : X Ã Y is compatible with v if and only if

• y = f(ηX) and• the diagram of fields

F (v)v

##GGG

GGGG

GG

F (y)

OO

f∗ // F (X)

commutes.

In particular, there is at most one such f .

Proof. Suppose v and f compatible. Then y = f(ηX) because v∗(ηX)

is the closed point of SpecOv. The commutativity of the diagram thenfollows from the one in Definition 2.0.5. Conversely, if f verifies the twoconditions, then it is obviously compatible with v. The last assertionfollows from Lemma 2.0.4. 2


c1.1 2.0.7. Corollary. a) Let Y be an integral F -scheme of finite type, andlet O be a valuation ring of F (Y )/F with residue field K and centrey ∈ Y . Assume that F (y) ∼−−→ K. Then, for any rational map f :X Ã Y with X integral, such that f(ηX) = y, there exists a uniqueplace v : F (Y ) 99K F (X) with valuation ring O which is compatiblewith f .b) If f is an immersion, the condition F (y)

∼−−→ K is also necessaryfor the existence of v.c) In particular, let f : X Ã Y be a dominant rational map. Then f iscompatible with the trivial place F (Y ) ↪→ F (X), and this place is theonly one with which f is compatible.

Proof. This follows immediately from Proposition 2.0.6. 2

p1 2.0.8. Proposition. Let f : X → Y , g : Y → Z be two morphismsof varieties. Let v : F (Y ) 99K F (X) and w : F (Z) 99K F (Y ) be twoplaces. Suppose that f and v are compatible and that g and w arecompatible. Then g ◦ f and v ◦ w are compatible.Proof. We first show that v ◦ w is finite on Z. By definition, thediagram

ηYw∗−−−→ SpecOwy

ySpecOv −−−→ SpecOv◦w

is cocartesian. Since the two compositions

ηYw∗−→ SpecOw → Z

andηY → SpecOv → Y g−→ Z

coincide (by the compatibility of g and w), there is a unique induced(dominant) map SpecOv◦w → Z. In the diagram

ηXv∗−−−→ SpecOv −−−→ SpecOv◦wy

yy

Xf−−−→ Y g−−−→ Z

the left square commutes by compatibility of f and v, and the rightsquare commutes by construction. Therefore the big rectangle com-mutes, which means that g ◦ f and v ◦ w are compatible. 2

d2 2.0.9. Definition. We denote by VarP(F ) the following category:


• Objects are integral separated F -schemes of finite type, i.e. F -varieties.• Let X, Y ∈ VarP(F ). A morphism ϕ ∈ VarP(X,Y ) is a pair

(v, f) with f : X → Y a morphism, v : F (Y ) 99K F (X) a placeand v, f compatible.• The composition of morphisms is given by Proposition 2.0.8.

If C is one of the categories introduced in Subsection 1.8, we shalldenote by CP(F ) the subcategory of VarP(F ) whose objects are theobjects of C and whose morphisms are the pairs (v, f) where f ∈ C.

We now want to do an elementary study of the two forgetful functorsappearing in the diagram below:

l3 (2.1)

VarP(F )Φ1−−−→ Var(F )

Φ2

yplace(F )op.

Clearly, Φ1 and Φ2 are essentially surjective. Concerning Φ1, we havethe following partial result on its fullness:

l6 2.0.10. Lemma. Let f : X Ã Y be a rational map, with X integral andY separated. Assume that y = f(ηX) is a regular point (i.e. A = OY,yis regular). Then there is a place v : F (Y ) 99K F (X) compatible withf .

Proof. By Corollary 2.0.7 a), it is sufficient to produce a valuationring O containing A and with the same residue field as A.

The following construction is certainly classical. Let m be the max-imal ideal of A and let (x1, . . . , xd) be a regular sequence generat-ing m, with d = dimA = codimY y. For 0 ≤ i < j ≤ d + 1, letAi,j = A[x

−11 , . . . , x

−1i ]/(xj, . . . , xd) (for i = 0 we invert no xk, and for

j = d+1 we mod out no xk). Then, for any (i, j), Ai,j is a regular localring of dimension j− i−1. In particular, Fi = Ai,i+1 is the residue fieldof Ai,j for any j ≥ i + 1. We have A0,d+1 = A and there are obviousmaps

Ai,j → Ai+1,j (injective)Ai,j → Ai,j−1 (surjective).

Consider the discrete valuation vi associated to the discrete valuationring Ai,i+2: it defines a place, still denoted by vi, from Fi+1 to Fi. Thecomposition of these places is a place v from Fd = F (Y ) to F0 = F (y),whose valuation ring dominates A and whose residue field is clearlyF (y). 2


rk2.1 2.0.11. Remark. In Lemma 2.0.10, the assumption that y is a regularpoint is necessary. Indeed, take for f a closed immersion. By [7,Ch. 6, §1, Th. 2], there exists a valuation ring O of F (Y ) whichdominates OY,y and whose residue field κ is an algebraic extension ofF (y) = F (X). However we cannot choose O such that κ = F (y) ingeneral. For example, if Y is a singular curve with normalisation Ỹ ,

then O will also dominate any point ỹ ∈ Ỹ above y, and the extensionF (ỹ)/F (y) is in general nontrivial.

Here are other counterexamples, for which we are indebted to Mahé.Suppose k = R. Let K be a function field which is not formally real.Then for any valuation v of K, the residue field of v is not formallyreal. Therefore, if X is a model of K and x ∈ X has a formally realresidue field, there is no valuation with centre x and the same residuefield. (In particular, any smooth point has a non formally real residuefield.) The easiest case where this gives a normal counterexample is(0, 0, 0) on the affine cone x2 + y2 + z2 = 0.

Now concerning Φ2, we have:

l2bis 2.0.12. Lemma. Let X,Y be two varieties and λ : F (Y ) 99K F (X) aplace. Assume that λ is finite on Y . Then there exists a unique rationalmap f : X Ã Y compatible with λ.Proof. Let y be the centre of Oλ on Y and V = SpecR an affineneighbourhood of y, so that R ⊂ Oλ, and let S be the image of R inF (λ). Choose a finitely generated F -subalgebra T of F (X) containingS, with quotient field F (X). Then X ′ = SpecT is an affine model ofF (X)/F . The composition X ′ → SpecS → V → Y is then compatiblewith v. Its restriction to a common open subset U of X and X ′ definesthe desired map f . The uniqueness of f follows from Proposition 2.0.6.2

2.0.13. Remark. Let Z be a third variety and µ : F (Z) 99K F (Y ) beanother place, finite on Z; let g : Y Ã Z be the rational map com-patible with µ. If f and g are composable, then g ◦ f is compatiblewith λ ◦ µ: this follows easily from Proposition 2.0.8. However it maywell happen that f and g are not composable. For example, assumeY smooth. Given µ, hence g (that we suppose not to be a morphism),choose y ∈ Fund(g) and find a λ with centre y, for example by themethod in the proof of Lemma 2.0.10. Then the rational map f corre-sponding to λ has image contained in Fund(g).

We conclude this section with a useful lemma which shows that placesrigidify the situation very much.


l4 2.0.14. Lemma. a) Let Z,Z ′ be two models of a function field K, withZ ′ separated, and v a valuation of K with centres z, z′ respectively onZ and Z ′. Assume that there is a birational morphism g : Z → Z ′.Then g(z) = z′.b) Consider a diagram

Z

g

²²

X

f>>}}}}}}}}

f ′ ÃÃAAA

AAAA

A

Z ′

with g a birational morphism. Let K = F (X), L = F (Z) = F (Z ′) andsuppose given a place v : L 99K K compatible both with f and f ′. Thenf ′ = g ◦ f .Proof. a) Let f : SpecOv → Z be the dominant map determined byz. Then f ′ = g ◦ f is a dominant map SpecOv → Z ′. By the valuativecriterion of separatedness, it must correspond to z′. b) This followsfrom a) and Proposition 2.0.6. 2

3. Equivalences of categorieseqcat

In this section we prove our main localisation theorems. They are oftwo kinds: given a category C as in Subsection 1.8 and the multiplica-tive system Sb of birational morphisms as in Subsection 1.9, we provethat, except in the case of dominant morphisms:

• The functorS−1b Φ2 : CP(F )→ place(F )op

induced by the functor Φ2 of Diagram (2.1) is an equivalence ofcategories.• Given another such category D containing C, the induced func-

tor S−1b C → S−1b D is often (but not always) an equivalence ofcategories.

Some of these results depend on resolution of singularities: we shallmake this clear in their precise statements. Assuming resolution ofsingularities for simplicity, we find that there are finally only two typesof categories of the form S−1b C: one with C = Sm(F ) and the otherwith C = Var(F ). The latter category is much finer in that its Homsets are very small; however it does detect some nontrivial invariantsof arbitrary varieties. See Remark 3.3.5.


As a warm-up we start with the easier case of dominant maps: in thiscase all the categories we get are equivalent to field(F ). Here, resolutionof singularities only comes in to prove this fact in the case of smoothproper varieties. (Clearly we already need resolution of singularities toknow that every function field has a smooth proper model.)

3.1. Dominant maps. LetX,Y be two F -schemes of finite type, withX irreducible. Recall from Subsection 1.5 the set Rat(X,Y ) of rationalmaps from X to Y . There is a well-defined map

Rat(X, Y )→ Y (F (X))eq2.1 (3.1)(U, f) 7→ f|ηX

where ηX is the generic point of X.

l2.1 3.1.1. Lemma. If X is integral, the map (3.1) is surjective. If more-over Y is separated, it is bijective.

Proof. The first statement is clear, and the second one follows fromLemma 2.0.4. 2

On the other hand, let field(F ) be the category defined in Definition2.0.2 (finitely generated fields and inclusions of fields). Recall [23, Ch.I, Th. 4.4] that there is an anti-equivalence of categories

Ratdom(F )∼−−→ field(F )ophart (3.2)

X 7→ F (X).Actually this follows easily from Lemma 3.1.1. We want to revisit

this theorem from the point of view of the previous section. Recall fromSubsection 1.5 the precategory Rat(F ) and its subcategory Ratdom(F ),and from Subsection 1.8 the category Vardom(F ). We have the cor-responding categories Ratdom P(F ) and Vardom P(F ) (cf. Definition2.0.9). We have an obvious faithful functor

ρ : Vardom(F )→ Ratdom(F )which is the identity on objects. It is clear that ρ sends a birationalmorphism to an isomorphism. Hence ρ factors into a functor

eq3.0 (3.3) ρ̄ : S−1b Vardom(F )→ Ratdom(F ).and the same is true of the functor Vardom P(F )→ Ratdom P(F ).

p3.2 3.1.2. Theorem. a) The forgetful functors Vardom P(F )→ Vardom(F )and Ratdom P(F )→ Ratdom(F ) are isomorphisms of categories.


b) The multiplicative systems So, Sb and Spb all admit a cartesian cal-

culus of right fractions within Vardom(F ) and Vardom P(F ).c) In the commutative diagram

S−1b Vardom P(F )//

ρ̄

²²

S−1b Vardom(F )

ρ̄

²²Ratdom P(F ) //

²²

Ratdom(F )

field(F )op

all functors are equivalences of categories.d) The above results remain true if we replace Vardom(F ) by Var

qpdom(F ),

Varpropdom (F ), Varprojdom(F ), Varsmdom(F ), Var

propsmdom(F ), Var

qpsmdom(F ),

Varprojsmdom(F ) or, if F is perfect, by Smdom(F ) or Smqpdom(F ). *Un-

der resolution of singularities they remain true for Smpropdom (F ) and

Smprojdom(F ).

Proof. a) This follows immediately from Corollary 2.0.7 c).b) In view of a) it suffices to deal with Vardom(F ). In fact, any

pair (u, s) as in Definition 1.2.1 admits cartesian calculus of fractions.Similarly, condition (3) of §1.2 is trivially verified: with its notation,we have in fact sf = sg ⇒ f = g since f and g are dominant.

c) In view of a) and (3.2), it remains to see that the compositefunctor S−1b Vardom P(F ) → field(F )op is an equivalence of categories.It is clearly essentially surjective. The graph trick shows that it is full.It remains to show faithfulness. Let ϕ, ψ ∈ Hom(X,Y ) inducing thesame map F (Y ) ↪→ F (X). By calculus of right fractions, we may write

ϕ = fs−1, ψ = gs−1

where s : X ′ → X is in Sb. Since f and g are dominant, we have f = gby Lemma 2.0.4, hence ϕ = ψ.

d) The proofs are the same, mutatis mutandis. In the smooth case weneed perfectness for essential surjectivity, but we may replace the graphtrick by restriction to an open subset by replacing Sb by So (Lemma1.9.1 a)). In the smooth proper case we need resolution of singularitiesfor essential surjectivity, and also to desingularise the result of thegraph trick. 2

3.1.3. Remarks. 1) By calculus of right fractions, we have inS−1o Vardom(F )

Hom(X,Y ) = lim−→Hom(U, Y )


where U → X runs through the members of So (1.4). Thus we recoverthe formula for morphisms in Ratdom(F ).2) The functor (Spb )

−1 Vardom P(F )→ field(F )op is not full (hence is notan equivalence of categories). For example, let X be a proper varietyand Y an affine open subset of X, and let K be their common functionfield. Then the identity map K → K is not in the image of the abovefunctor. Indeed, if it were, then by calculus of fractions it would berepresented by a map of the form fs−1 where s : X ′ → X is properbirational. But then X ′ would be proper and f : X ′ → Y should beconstant, a contradiction. It can be shown that the localisation functor

(Spb )−1 Vardom P(F )→ S−1b Vardom P(F )

has a right adjoint-right inverse given by (Spb )−1 Varpropdom P(F ) →

(Spb )−1 Vardom P(F ) via the equivalence (S

pb )−1 Varpropdom P(F )

∼−−→S−1b Vardom P(F ) given by Theorem 3.1.2. The proof is similar to thatof Theorem 3.2.4 (iii) below.

3.2. The coarse birational category.

l8a 3.2.1. Proposition. The category VarP(F ) admits a calculus of rightfractions with respect to Spb . This is also true in Var

qp P(F ). In par-ticular, in these categories, any morphism may be written in the formfp−1 with p ∈ Spb .Proof. Let

eq3.4 (3.4)

Y ′

s

yX

u−−−→ Ybe a diagram in VarP(F ), with s ∈ Spb . Let λ : F (Y ) 99K F (X) bethe place compatible with u which is implicit in the statement. ByProposition 2.0.6, λ has centre z = u(ηX) on Y . Since s is proper, λtherefore has also a centre z′ on Y ′. By Lemma 2.0.14 a), s(z′) = z.By Lemma 2.0.12, there exists a unique rational map ϕ : X Ã Y ′compatible with λ, and s ◦ ϕ = u by Lemma 2.0.14 b). By the graphtrick, we get a commutative diagram

eq3.5 (3.5)

X ′ u′−−−→ Y ′

s′y s

yX

u−−−→ Yin which X ′ ⊂ X ×Y Y ′ is the closure of the graph of ϕ, s′ ∈ Sb andu′ is compatible with λ. Moreover, since s is proper, the projection


X ×Y Y ′ → Y ′ is proper and s′ is proper. This proves that Condition(2) of calculus of right fractions is verified.

For Condition (3), let (X, Y, Y ′, f, g, s) be as in this condition. ByCorollary 2.0.7 c), the place underlying s is the identity. Hence thetwo places underlying f and g must be equal. But then f = g byProposition 2.0.6.

For Varqp P(F ) the proof is the same, because the graph trick pre-serves quasi-projectiveness. 2

p4 3.2.2. Proposition. Consider a diagram in Var(F )

Zf

~~}}}}

}}}} p

ÃÃAAA

AAAA

A

X Y

Z ′f ′

``AAAAAAAA p′

>>}}}}}}}

where p, p′ ∈ Spb . Let K = F (Z) = F (Z ′) = F (Y ), L = F (X) andsuppose given a place v : L 99K K compatible both with f and f ′.Then (v, fp−1) = (v, f ′p′−1) in (Spb )

−1 VarP(F ). This is also true inVarqp P(F ).

Proof. By the graph trick, complete the diagram as follows:

Zf

~~||||

|||| p

ÃÃAAA

AAAA

A

X Z ′′

p1

OO

p′1²²

Y

Z ′f ′

``BBBBBBBB p′

>>}}}}}}}}

with p1, p′1 ∈ Spb . Then we have

pp1 = p′p′1, fp1 = f

′p′1

(the latter by Lemma 2.0.14 b)), hence the claim. Exactly the sameproof works for quasi-projective varieties. 2

l3.2 3.2.3. Proposition. Any morphism ϕ in S−1b VarP(F ) can be writtenas j−1fq−1, with j ∈ So and q ∈ Spb . This is also true in Varqp P(F ).


Proof. Consider a commutative diagram in S−1b VarP(F )

Xj1 //

ϕ

²²

X̄

ψ²²

Yj // Ȳ

with j, j1 ∈ So, X̄, Ȳ proper and where ψ is defined by the commuta-tivity of the diagram. (In the quasi-projective case, we of course takeX̄ and Ȳ projective.) By Proposition 3.2.1, we may write ψ = gq−1

with q ∈ Spb , hence ϕ = j−1ψj1 = j−1gq−1j1. By Proposition 3.2.1again, q−1j1 = j′p−1 for p ∈ Spb , which proves the claim. 2

t3.1 3.2.4. Theorem. Consider the string of functors induced by the func-tor Φ2 of Diagram (2.1)

(Spb )−1 Varprop P(F )

S→ (Spb )−1 VarP(F ) T→ S−1b VarP(F ) U→ place(F )op

and let V = TS. Then

(i) U and V are equivalences of categories.(ii) S is fully faithful and T is faithful.(iii) Through V , T is left adjoint to S: for X ∈ (Spb )−1 VarP(F )

and Y ∈ (Spb )−1 Varprop P(F ), T : Hom(X,S(Y )) →Hom(T (X), V (Y )) is an isomorphism.

(iv) Let X,Y ∈ (Spb )−1 VarP(F ). Then the image of Hom(X,Y ) inHom(UT (Y ), UT (X)) via UT is contained in the set of placeswhich are finite on Y . In particular, T is not full.

This is also true in Varqp P(F ). Moreover, the functor(Spb )

−1 Varqp P(F ) → (Spb )−1 VarP(F ) is an equivalence of cate-gories.

Proof. In 7 steps:A) Thanks to Propositions 3.2.1 and 3.2.2, UT and UTS are faith-

ful, hence S and T are faithful. Let X, Y ∈ (Spb )−1 Varprop P(F ) andϕ : S(X) → S(Y ). By Proposition 3.2.1 we may write ϕ = fp−1with p ∈ Spb . But then the source of p is proper and ϕ comes from(Spb )

−1 Varprop P(F ). So S is full, which proves (ii).B) To prove (iii) it is sufficient to show surjectivity. Let ϕ ∈

Hom(T (X), V (Y )). By Proposition 3.2.3, ϕ = j−1fp−1 with j ∈ Soand p ∈ Spb . Since Y is proper, j is necessarily an isomorphism, whichproves the claim.


C) It follows from A) and B) that V is fully faithful. Since it isessentially surjective by Theorem 1.7.1 (or tautologically in the case ofVarproj P), it is an equivalence of categories.

D) U is also full. Let X,Y ∈ VarP(F ) and λ : F (Y ) 99K F (X) aplace. Let Y → Ȳ be a compactification of Y (Theorem 1.7.1): in thequasi-projective case we choose Ȳ projective. Then λ is finite over Ȳ .By Lemma 2.0.12, there is a rational map X Ã Ȳ compatible to λ.

E) UV is faithful as seen in A), full by C) and D) and it is obviouslyessentially surjective. Hence it is an equivalence of categories. Togetherwith C), this proves (i).

F) (iv) is obvious.G) For the last statement, the functor is faithful by (ii). It is full

and essentially surjective thanks to Proposition 3.2.1 and the relativeChow lemma (Theorem 1.7.2). 2

If we replace varieties by smooth varieties, we get some general re-sults over a perfect field but we have not been able to avoid resolutionof singularities to obtain the best possible result, which is thereforeonly proven in characteristic 0.

t3.1sm 3.2.5. *Theorem. Consider the same string of functors as in Theorem3.2.4 for smooth varieties:

(Spb )−1 Smprop P(F ) S→ (Spb )−1 SmP(F ) T→ S−1b SmP(F ) U→ place(F )op.

Thena) If F is perfect, U is full and essentially surjective.b) Under resolution of singularities, items (i)–(iv) of Theorem 3.2.4are valid. Moreover, the functor (Spb )

−1 SmP(F ) → (Spb )−1 VarP(F ) isan equivalence of categories.The same holds by replacing “smooth” by “smooth and quasi-projective”.

Proof. This is proven exactly as in Theorem 3.2.4 by using Hironaka’stheorem (Theorem 1.7.3) everywhere. 2

r3.1 3.2.6. Remarks. 1) Even in (Spb )−1 VarP(F ), So does not admit a cal-

culus of right fractions. In fact, even property (2) (the Ore condition)fails. Indeed, consider a diagram in (Spb )

−1 VarP(F )

Y ′

j²²

Xf // Y


where j ∈ So and, for simplicity, f comes from VarP(F ). Supposethat we can complete this diagram into a commutative diagram in(Spb )

−1 VarP(F )

X̃ ′

p

²²

g

ÃÃAAA

AAAA

A

X ′ϕ //

j′²²

Y ′

j²²

Xf // Y

with p ∈ Spb and g comes from VarP(F ). By Proposition 2.0.6 the local-isation functor VarP(F )→ (Spb )−1 VarP(F ) is faithful, so the diagram(without ϕ) must already commute in VarP(F ). If f(X)∩Y ′ = ∅, thisis impossible.

2) The category (Spb )−1 VarP(F ) is equivalent to a nonfull subcate-

gory of place(F )op via the localisation functor UT of Theorem 3.2.4,but it seems difficult to describe exactly the image of UT . For example,if X and Y are proper, then the image is all of Hom(UT (Y ), UT (X)).On the other hand, if X is proper and Y is affine, then for any mapϕ = fp−1 : X → Y , the source X ′ of p is proper hence f(X ′) is aclosed point of Y , so that Hom(UT (Y ), UT (X)) is contained in theset of places from F (Y ) to F (X) whose centre on Y is a closed point(and one sees easily that this inclusion is an equality). In general, thedescription seems to depend heavily on the nature of X and Y .

Let us summarise what we have done so far. We have realisedplace(F )op as the localisation of VarP(F ) with respect to Sb in twosteps: first localising VarP(F ) with respect to Spb and then localisingthe resulting category with respect to So. The first step enjoys cal-culus of right fractions but the second one does not by Remark 3.2.6.However, the localisation functor

(Spb )−1 VarP(F )→ S−1b VarP(F ) ∼−−→ place(F )op

has a right adjoint-right inverse as in Appendix A.7. The correspondinglocal objects of (Spb )

−1 VarP(F ) are proper varieties. We get the samelocalisations when restricting to quasi-projective varieties, and also tosmooth or smooth quasi-projective varieties but only under resolutionof singularities.

3.2.7. Definition. We denote the category S−1b SmP(F ) by coarse(F )and call it the coarse birational category.


3.3. The fine birational categories.

p3.5 3.3.1. Proposition. In the commutative diagram

S−1b Varprop(F )

A // S−1b Var(F )

S−1b Varproj(F )

B //

C

OO

S−1b Varqp(F )

D

OO

all functors are equivalences of categories. The same holds by addingthe index sm everywhere.

Proof. We first prove that A and B are equivalences of categories. Forthis, we apply Theorem A.1.2 with C = Varprop(F ) (resp. Varproj(F )),D = Var(F ) (resp. Varqp(F )), T the obvious inclusion, S = Sb andS ′ = Sb. We have to check that conditions (i)–(vii) hold:

• (i) is clear and (ii) holds because any birational morphism isdominant (see Lemma 2.0.4).• (iii) and (iv) are clear.• Condition b) of Lemma A.1.4 is verified so it remains to check

(vii). Take its notation. Pick a compactification X → T (X̄1)of X. We get X̄, s and ḡ by applying the graph trick.

We now prove that D is an equivalence of categories, which willalso imply that C is an equivalence of categories. This time we applyTheorem A.1.8 with C = Varqp(F )op, D = Var(F )op, T the obviousinclusion and S = So, S

′ = So. Let us check that its conditions areverified:

• (i) is clear; (ii) holds because open immersions are monomor-phisms.• (iii) and (iv) are clear.• (v’): for f : Y → X, take a quasi-projective open neighbour-

hood U of f(ηY ) in X. Then f−1(U) 6= ∅ and we can pick a

nonempty quasi-projective open V in f−1(U).• (vi’): the intersection of two quasi-projective open subsets is

quasi-projective.• (vii”): we have the situation

X Yfoo Z.

goo

Pick a quasi-projective open neighbourhood U of fg(ηZ). Then(fg)−1(U) 6= ∅ and in particular g−1(U) 6= ∅. This is an openneighbourhood of g(ηZ): in it, pick an open quasi-projective Vcontaining g(ηZ).


• (viii’): this boils down to the fact that, if a given morphismlands into two open subsets, then it lands into their intersection.

The proofs with indices sm is the same. 2

p3.6 3.3.2. *Proposition. In the commutative diagram

S−1b Smprop(F )

A′ // S−1b Sm(F )

S−1b Smproj(F )

B′ //

C′OO

S−1b Smqp(F )

D′OO

D′ is an equivalence of categories. Under resolution of singularities,this is true of the three other functors.

Proof. The same as that of Proposition 3.3.1, except that for A′ andB′, we need to desingularise a compactification of a smooth varietyusing Theorem 1.7.3. 2

p3.7 3.3.3. Proposition. If F is perfect, in the commutative diagram

S−1b Sm(F )E // S−1b Varsm(F )

S−1b Smqp(F )

F //

G

OO

S−1b Varqpsm(F )

H

OO

all functors are equivalences of categories.

Proof. The case of H has been seen in Proposition 3.3.1, and the caseof G = D′ has been seen in Proposition 3.3.2. We now prove that E andF are equivalences of categories. Here we apply Theorem A.1.2 withC = Sm(F )op (resp. Smqp(F )op), D = Varsm(F )op (resp. Varqpsm(F )op),T the obvious inclusion and S = S ′ = So. Let us check the conditions:

• (i) is clear and (ii) holds because open immersions are monomor-phisms.• (iii) and (iv) are clear.• (v): since F is perfect, any variety contains a smooth nonempty

open subset.• (vi): the union of two smooth open subsets is smooth.• (vii): with its notation, Ȳ is smooth. Since by assumption g is

good, it lands into the smooth locus Xsm of X and we may takeX̄ = Xsm.

2


p3.8 3.3.4. *Proposition. Under resolution of singularities, all functors inthe commutative diagram

S−1b Smprop(F )

I // S−1b Varpropsm (F )

S−1b Smproj(F )

J //

K

OO

S−1b Varprojsm (F )

L

OO

are equivalences of categories.

Proof. The case of K = C ′ has been seen in Proposition 3.3.2 and thecase of L in Proposition 3.3.1. The case of the other functors is thenimplied by the previous propositions (the reader should draw a com-mutative cube of categories in order to check that enough equivalencesof categories have been proven). 2

r3.2 3.3.5. Remark. The functor S−1b Sm(F ) → S−1b Var(F ) is neither fullnor faithful, even under resolution of singularities. In fact, even thefunctor S−1r Sm(F ) → S−1r Var(F ) is not faithful. Indeed, take F ofcharacteristic 0. Let X be a proper irreducible curve of geometric

genus > 0 with one nodal singular point p. Let π̄ : X̃ → X be itsnormalisation, U = X − {p}, Ũ = p̄−1(U), p = p̄|Ũ and j : U → X,̃ : Ũ → X̃ the two inclusions. We assume that p̄−1(p) consists of tworational points p1, p2. Finally, let fi : SpecF → X̃ be the map givenby pi.

SpecFf1 //f2

// X̃p̄ // X

Ũ

̃

OO

p // U

j

OO

In S−1b Var(F ), p̄ is an isomorphism so that f1 = f2. We claimthat f1 6= f2 in S−1r Smprop(F ) ∼−−→ S−1r Sm(F ). Otherwise, since R-equivalence is a stable birational invariant of smooth proper varieties

[8, Prop. 10], we would have p1 = p2 ∈ X̃(F )/R. But this is falsesince X̃ has nonzero genus. We thank A. Chambert-Loir for his helpin finding this example.

More generally, it is well-known that for any integral curve C and anytwo closed points x, y ∈ C there exists a proper birational morphisms : C → C ′ such that s(x) = s(y) (cf. [58, Ch. IV, §1, no 3] whenF is algebraically closed). This shows that any two morphisms f, g :X ⇒C such that f(ηX) and g(ηX) are closed points become equal in


S−1b Var(F ). This can be used to show that the functor S−1r Sm(F )→

S−1r Var(F ) does not have a right adjoint.Nevertheless, the category S−1r Var(F ) is not trivial as it enjoys at

least one nontrivial functor to an interesting category: the “birational”Albanese functor, cf. Lang [37, p. 46].

We shall go back to R-equivalence in Section 4.

Summarising what we have done so far, under resolution of singu-larities we have defined two fine birational categories: S−1b Var(F ) andS−1b Sm(F ). These categories have various incarnations, summarisedin Propositions 3.3.1 to 3.3.4. The natural functor S−1b Sm(F ) →S−1b Var(F ) is neither full nor faithful by Remark 3.3.5. We formalisethis in the following definition:

3.3.6. Definition. We denote the category S−1b Sm(F ) by fine(F ) andthe category S−1b Var(F ) by ffine(F ). We call them the fine birationalcategories.

Note that, unlike for the coarse birational category, we have notused any calculus of fractions here. In fact, Spb does not admit weakcalculus of fractions within Var(F ), contrary to the case of VarP(F ) (cf.Proposition 3.2.1). This is shown by the following simple examples:

3.3.7. Examples. Let Y be a singular (integral) curve, and s : Y ′ → Yits normalisation. Suppose that y ∈ Y is a closed point such that allthe points in s−1(y) have larger residue field than y. Let X = y andu : X → Y be the closed immersion. Then the pair (s, u) does notenjoy left calculus of fractions.

For an example with Y normal, take for Y the affine cone x2 + y2 +z2 = 0 over R as in Remark 2.0.11, and for Y ′ the blow-up of thesingular point (0, 0, 0).

If we restrict to Sm(F ), we can use Proposition 3.2.1 and Lemma2.0.10 to prove weak calculus of fractions, but we have then to resolvethe singularities of the variety X ′ in the proof of Proposition 7.1.4. Thefollowing lemma will yield a slightly different proof in Proposition 3.3.9below. It does not require resolution of singularities, but the proof ofProposition 3.3.9 will require it.

l3.5 3.3.8. Lemma (cf. Kollár-Szabó [36, Prop. 6]). Let s : Y → X be inSpr , with X smooth. Then s is an envelope [19]: for any extensionK/F , the map Y (K)→ X(K) is surjective.Proof. It suffices to deal with K = F .

a) We first handle the case where s ∈ Spb . Let x ∈ X(F ). Considerthe blow-up Blx(X) of X. By the graph trick we have a commutative


diagram

Y ′ //

s′²²

Y

s

²²Blx(X)

p // X

with s′ ∈ Spb . Let E be the exceptional locus of s′ (see page 13). SinceBlx(X) is smooth, s

′(E) has codimension at least 2 (cf. [11, 1.40]),hence it does not contain p−1(x). If F is infinite we stop here sincep−1(x) ' Pd−1, where d = dimX. Otherwise we conclude by inductionon d, applying the induction hypothesis to the restriction of s′ to asuitable irreducible component of s′−1(p−1(x)).

b) Suppose now that s ∈ Spr . Then Y is birational to X × (P1)nfor some n. Consider the same commutative diagram as in the proofof Lemma 1.9.1 b). By a) applied to u in this diagram, we get that

Ỹ (F ) → X(F ) is surjective and therefore that Y (F ) → X(F ) is sur-jective. 2

p3.4 3.3.9. *Proposition. Under resolution of singularities, the multiplica-tive set Spb admits weak calculus of right fractions within Sm(F ).

Proof. We consider a diagram (3.4) in Sm(F ), with s ∈ Spb . ByLemma 3.3.8, z = u(ηX) has a preimage z

′ ∈ Y ′ with same residuefield. Let Z = {z} and Z ′ = {z′}: the map Z ′ → Z is birational. Sincethe map ū : X → Z factoring u is dominant, we get in the same wayas in the proof of Lemma 3.3.8 a commutative diagram like (3.5), withs′ proper birational. We then need to resolve the singularities of X ′ tofinish the proof. 2

c3.2 3.3.10. *Corollary. Under resolution of singularities, any morphismin S−1b Sm(F ) may be represented as j

−1fp−1, where j ∈ So and p ∈ Spb .Proof. As that of Proposition 3.2.3. 2

3.3.11. Remark. Even under resolution of singularities, Spb is far fromadmitting a calculus of right fractions within Sm(F ). Indeed, let s :Y → X be a proper birational morphism that contracts some closedsubvariety i : Z ⊂ Y to a point. Then, given any two morphismsf, g : Y ′ ⇒Z, we have sif = sig. But if ift = igt for some t ∈ Spb ,then if = ig (hence f = g) since t is dominant.


3.4. From coarse to fine. We now put the results of the last twosections together and get, via the functor Φ1 of Diagram (2.1), a stringof functors

coarse(F )→ fine(F )→ ffine(F ).These two functors are full and essentially surjective. Under resolu-

tion of singularities, they yield a corresponding string of functors

eq3.3 (3.6) place(F )op → S−1b Smproj(F )→ S−1b Varproj(F ).Note that, unfortunately, even the functor place(F ) → S−1b Sm(F )

is only defined under resolution of singularities.

3.4.1. Remark. For X, Y two smooth F -varieties, let Rat(X, Y ) bethe set of rational maps from X to Y : by Lemma 3.1.1 this set is inone-to-one correspondence with Y (K) via f 7→ f|ηX , where K = F (X).If f ∈ Rat(X,Y ) and jU : U ↪→ X is an open subset of definition off , then the element f|U ◦ j−1U ∈ S−1b Sm(X,Y ) does not depend on thechoice of U . In other words, we have constructed a map

eq3.2 (3.7) Y (F (X)) = Rat(X, Y )→ S−1b Sm(X, Y ).Under resolution of singularities, (3.7) is surjective by Corollary

3.3.10.Assuming further Y proper, we have another map c :

place(F (Y ), F (X)) → Y (F (X)) (cf. Subsection 1.4), which is sur-jective by Lemma 2.0.10. The composition of c with (3.7) is clearlyequal to the map induced by the composite functor of (3.6). So, whenX, Y are smooth and Y is proper, we have factored this functor into acomposition of two surjections

place(F (Y ), F (X)) −→→ Y (F (X)) −→→ S−1b Sm(X,Y ).3.5. The stable birational categories. By localising further withrespect to Sr, we get results corresponding to those of the two previ-ous subsections, replacing Sb by Sr everywhere, and replacing place(F )by T−1 place(F ) where T is the multiplicative system of purely tran-scendental extensions. We define as above the stable coarse and finebirational categories

scoarse(F ) = S−1r coarse(F )

sfine(F ) = S−1r fine(F )

sffine(F ) = S−1r ffine(F ).

and we have full essentially surjective functors between them.


In the next section, we shall give alternate descriptions of the cate-gories T−1 place(F ) and S−1r Sm

proj(F ), involving Manin’sR-equivalenceand an analogous notion of “homotopy” between places.

4. Stable birationality and R-equivalencesbr

4.1. scoarse(F ) and homotopy of places.

deq1 4.1.1. Definition. Let K,L ∈ place(F ). Two places λ0, λ1 : K 99K Lare elementarily homotopic if there exists a place µ : K 99K L(t) suchthat si ◦ µ = λi, i = 0, 1, where si : L(t) 99K L denotes the placecorresponding to specialisation at i.

The property of two places being elementarily homotopic is preservedunder composition. Indeed if λ0 and λ1 are elementarily homotopic andif µ : M 99K K is another place, then obviously so are λ0 ◦µ and λ1 ◦µ.If on the other hand τ : L 99K M is another place, then τ ◦λ0 and τ ◦λ1are also elementarily homotopic as one sees by considering the uniqueplace τ̃ : L(t) 99K M(t) extending τ that sends t to t (correspondingto the valuation τ̃(

∑ait

i) = inf τ(ai)).Consider the equivalence relation h generated by elementary homo-

topy. So λ hµ iff v and w are linked by a series of elementary ho-motopies i.e. if and only if there exist finitely many places λ0, · · · , λrsuch that λ0 = λ, λr = µ and λi is elementarily homotopic to λi+1 for0 ≤ i < r. We say that λ and µ are homotopic if λ hµ. The homotopyrelation remains compatible with composition of places.

deq2 4.1.2. Definition. We denote by place(F )/h the factor category ofplace(F ) by the homotopy relation h.

Thus the objects of place(F )/h are function fields, while the set ofmorphisms consists of equivalence classes of homotopic places betweenthe function fields. There is an obvious full surjective functor

Π : place(F )→ place(F )/h.

The following lemma provides a first, more elementary, descriptionof T−1 place(F ) and of the localisation functor. It is valid in all char-acteristics.

leq1 4.1.3. Proposition. There is a unique isomorphism of categoriesplace(F )/h→ T−1 place(F ) which makes the diagram of categories and


functors

place(F )Π

wwpppppp

ppppp T−1

''OOOOO

OOOOOO

place(F )/h∼ // T−1 place(F )

commutative. In particular, the localisation functor T−1 is full and itsfibres are the equivalence classes for h.

Proof. We first note that any two homotopic places become equal inT−1 place(F ). Clearly it suffices to prove this when they are elemen-tarily homotopic. But then s0 and s1 are left inverses of the natural in-clusion i : L → L(t), which becomes an isomorphism in T−1 place(F ).Thus s0 and s1 become equal in T

−1 place(F ). So the localisationfunctor place(F )→ T−1 place(F ) canonically factors through Π into afunctor place(F )/h −→ T−1 place(F ).

On the other hand we claim that, with the above notation, i ◦ s0 :L(t) 99K L(t) is homotopic to 1L(t) in place(F ). Indeed they are el-ementarily homotopic via the place L(t) 99K L(t, s) (in this case aninclusion) that is the identity on L and maps t to st. Hence the pro-jection functor Π factors as T−1 place(F )→ place(F )/h, and it is plainthat this functor is inverse to the previous one. 2

4.2. sfine(F ) and R-equivalence. We now review Manin’s R-equiva-lence, referring to the exposition in [8, §4]. Recall that if X is a varietyover F , then two points x0 and x1 in X(F ) are directly R-equivalentif there is an F -rational map h : P1 Ã X defined at 0 and 1 andsuch that h(0) = x0 and h(1) = x1. (When X is proper, h extendsto a morphism.) The points are said to be R-equivalent if they areequivalent in the equivalence relation generated by the above. If F ′

is any extension field of F , then we shall write X(F ′)/R for the setXF ′(F

′)/R, where XF ′ denotes base change of X to F ′. We shall usethe following results about R-equivalence:

• If X and Y are two varieties over F , then (X ×F Y )(F )/R 'X(F )/R× Y (F )/R.• (Colliot-Thélène–Sansuc [8, Prop. 10]) If X and Y are smooth

projective varieties over F , any F ′-rational map f : X Ã Yinduces a map f̃ : X(F ′)/R → Y (F ′)/R. If f is a stable bira-tional morphism, then f̃ is bijective.

We use this to introduce a pre-category (or diagram) Smproj(F )/Rwith a more geometric flavour, and extend it to a category when F isof characteristic 0.


deq3 4.2.1. Definition. The pre-category Smproj(F )/R is defined as follows:

• Objects are (connected) smooth projective F -varieties.• Let X, Y ∈ Smproj(F )/R, and let K = F (X). Then

(Smproj(F )/R)(X, Y ) = Y (K)/R.

Forgetting composition of morphisms, we define a pre-functor R :Smproj(F ) → Smproj(F )/R as follows: map a variety to itself; iff ∈ Smproj(F )(X,Y ), map f to f(ηX) ∈ Y (F (X))/R.

peq1 4.2.2. *Proposition. Assume resolution of singularities. Then thereis a unique structure of a category on Smproj(F )/R such that R factorsthrough an isomorphism of categories

Smproj(F )

R

wwoooooo

ooooo S−1r

''PPPPP

PPPPPP

P

Smproj(F )/R S−1r Smproj(F )

∼oo

Proof. 1) For the existence of the category structure, we define thecomposition in Smproj(F )/R as follows. Let X,Y, Z be objects inSmproj(F )/R whose function fields are respectively K,L and M . Givenelements [P ] in Y (K)/R and [Q] in Z(L)/R, we need to define [Q]◦ [P ]as an element in Z(K)/R. Choose a representative Q of [Q] in Z(L).This gives a rational map θQ : Y Ã Z over F . As recalled above, thisrational map induces a map

θ̃Q : Y (K)/R→ Z(K)/R.We define [Q] ◦ [P ] := θ̃Q([P ]).Let us check that this map is independent of the chosen representa-

tiveQ. LetQ′ ∈ Z(L) be R-equivalent toQ: we need to show that θ̃Q =θ̃Q′ . For this we may assume that Q and Q

′ are directly R-equivalent.Then there is an L-morphism g : P1L → ZL such that g(0) = Q andg(1) = Q′. Spread this to an F -rational map g̃ : P1 × Y Ã Z × Y .Letting i0, i1 denote respectively the 0-section and the 1-section of theprojection P1 × Y → Y and π denote the projection Z × Y → Z, wethen have

θQ = π ◦ g̃ ◦ i0, θQ′ = π ◦ g̃ ◦ i1(in the sense that the underlying morphisms coincide on suitable opensets). Since P1(K)/R = ∗, i0 and i1 induce the same map mod R andwe do get θ̃Q = θ̃Q′ .

2) Let X, Y and Z be in Smproj(F ) with morphisms f : X → Y andg : Y → Z. Let us check that R(g ◦ f) = R(g) ◦ R(f). Let ηX and ηYbe the generic points of X and Y respectively. We need to show that


gf(ηX) = g(ηY ) ◦ f(ηX) in Z(F (X))/R. But this is obvious as in thiscase θg(ηY ) = g.

3) We now verify that composition is associative. Let Xi, i = 1, . . . , 4be smooth projective varieties with respective function fields Ki. Sup-pose we are given points Pi ∈ Xi+1(Ki) for 1 ≤ i ≤ 3. We need to showthat

([P3 ◦ P2]) ◦ [P1] = [P3] ◦ ([P2 ◦ P1]).By birational invariance of R-equivalence, we may assume up to

blowing up that Pi = fi(ηXi) for morphisms fi : Xi → Xi+1 (modifyX3, then X2, then X1). Then associativity follows via 2) from theassociativity of the composition in Smproj(F ).

4) To show that R factors through S−1r , we need to prove that if f :X → Y is a morphism of varieties in Smproj(F ) such that f ∈ Sr, thenR(f) is an isomorphism in Smproj(F )/R. By Yoneda, it suffices to checkthat for any object Z, the induced map Hom(Z, f) in Smproj(F )/Ris an isomorphism. In other words, f should induce an isomorphismX(K)/R ' Y (K)/R for any K = F (Z), which is true as R-equivalenceis a stable birational invariant.

5) We now show that S−1r also factors through R, which will showthat the functor of 4) is an isomorphism of categories. By Theorem3.2.4, it suffices to show that the composite functor

Smproj(F )S−1r−−→ S−1r Smproj(F ) ∼−−→ S−1r Sm(F )

factors through R.Let f, g : X → Y be two morphisms between smooth projective

varieties such that f(ηX) is R-equivalent to g(ηX) in Y (K), whereK = F (X). Choose a chain of directly R-equivalent points x0 =f(ηX), x1, . . . , xn = g(ηX). For each i let fi : X Ã Y be the ratio-nal map corresponding to xi. We may find a smooth projective model

X̃ above X such that all the fi extend to morphisms from X̃: this re-duces us to the case where f(ηX) and g(ηX) are directly R-equivalent.

Let ϕ : P1K → YK be such that ϕ(0) = f(ηX) and ϕ(1) = g(ηX). Wecan spread ϕ into a morphism

ϕ̃ : P1 × U → Y × Uwhere U ⊆ X is a suitable a dense open subset. Let ψ = p1 ◦ ϕ̃, wherep1 is the first projection. If i0 and i1 denote the sections U → P1 × Uof the second projection π2 : P

1×U → U respectively at 0 and 1, thenwe have

ψ ◦ i0 = f ◦ j, ψ ◦ i1 = g ◦ j


where j is the inclusion U → X. But in S−1r Sm(F ) i0 = i1 and j isinvertible, hence f and g become equal in this category.

6) Finally, the uniqueness of the category structure on Smproj(F )/Rimmediately follows from the fact that the functor R is “full up toblow-ups”.

This completes the proof of the proposition. 2

Let X and Y be two smooth projective varieties over F with functionfields K and L respectively. Consider the map

c : place(F )(K,L)→ X(L)λ 7→ c(λ)

where c(λ) is the centre of the place λ considered as an element of X(L)(see subsection 1.4).

leq2 4.2.3. Lemma. a) The above map is surjective and induces a surjection

c̄ : place(F )(K,L)/h −→→ X(L)/Rwhere h denotes homotopy equivalence.b) Let λ0, λ1 ∈ place(F )(K,L) be such that c(λ0) is directly R-equivalentto c(λ1) in X(L). Then there exist µ0, µ1 ∈ place(F )(K,L) such that

(i) µ0 and µ1 are elementarily homotopic;(ii) c(µ0) = c(λ0) and c(µ1) = c(λ1) in X(L).

Proof. a) As X is smooth, the surjectivity follows from Corollary 2.0.7(given P ∈ X(L), spread it to an F -morphism θP : V → X where V isopen in Y ). For the second assertion, it clearly suffices to prove thatif λ0 and λ1 are elementarily homotopic, then their centres in X(L)are R-equivalent. Let ν : K 99K L(t) be a place such that si ◦ ν = λifor i = 0, 1. Let i0 and i1 be the morphisms from Y to Y ×F P1which are respectively 0 and 1 on the second co-ordinate. Then i0and i1 are compatible respectively with s0 and s1, whose centres areclearly R-equivalent in Y ×F P1. By Lemma 2.0.12, the graph trickand Theorem 1.7.3, there is a smooth projective variety Z over F witha birational morphism to Y ×F P1 and an F -morphism g : Z → Xsuch that g is compatible with ν. But R-equivalence is invariant underbirational maps, and birational morphisms being compatible with theidentity place, we see that the centres c0 and c1 of s0 and s1 in Z areR-equivalent. As g is compatible with ν and si ◦ ν = λi, it is clear thatthe centres of λ0 and λ1 are g(c0) and g(c1), hence are R-equivalent inX(L).

b) The centres c(λ0) and c(λ1) induce F -morphisms

θ0 : V0 → X, θ1 : V1 → X,


where V0 and V1 are open sets in Y , such that θi maps the generic pointof Y to c(λi) for i = 1, 2 respectively. Further, by Corollary 2.0.7, θi iscompatible with λi for i = 0, 1 respectively. By R-equivalence, we havean L-morphism g : P1L → XL such that g(0) = c(λ0) and g(1) = c(λ1).Spreading this morphism, we get an F -morphism g′ : V ′ ×F P1 →V ′ ×F X where V ′ is open in Y . Further the images of the genericpoint of V ′ × {i} under g′ for i = 0, 1 are respectively the centres ofλi. By Corollary 2.0.7 we get a place ν : K 99K L(t) compatible withp2 ◦ g′, where p2 : V ′ × X → X is the second projection. Shrinkingthe open sets if necessary, we may assume that V0 = V1 = V

′. Thenatural inclusions i0 : V → V ×F P1 and i1 : V → V ×F P1 which maprespectively to 0 and 1 in the second co-ordinate are clearly compatiblewith the “specialising” places s0 and s1 respectively. Then µi = si ◦ νand λi have the same centre for i = 0, 1. 2

Let X be a smooth projective model of K and λ a place K 99K L. IfY is a smooth projective model of L, then by Lemma 2.0.12, there isa unique rational map Y Ã X compatible with λ. This rational mapis determined by the centre c(λ) ∈ X(L). Let µ : K 99K L be anotherplace. Write

λCX µ

if c(λ) = c(µ) ∈ X(L), or equivalently if λ and µ are compatiblewith the same rational map to X. The relation CX has the followingpermanence properties:

• CX is an equivalence relation.• If λCX µ and p : X → X ′ is in Sb, then λCX′ µ.• If λCX µ and ν : L 99K M is another place, then (ν ◦ λ) CX(ν ◦µ). To see this, choose a model Y of L such that the commonrational map f : Y Ã X compatible to λ and µ is in fact amorphism; if Z is a smooth model of M and g : Z Ã Y isa rational map compatible to ν, then f ◦ g is defined and, byProposition 2.0.8, it is compatible with ν ◦ λ and ν ◦ µ.

On the other hand, if λCX µ and ν : M 99K K is another place, thenit is not true in general that (λ ◦ ν) CZ(µ ◦ ν) for a suitable model Z ofM . (Give a counterexample!)

Under the functor

Φb : place(F )op → S−1b Smproj(F )of Theorem 3.2.4, λCX µ implies Φ

b(λ) = Φb(µ): this is obvious.Hence, if we denote by C the equivalence relation on place(F ) gen-erated by all CX , Φ

b factors through a functor

Φ̄b : place(F )op/C→ S−1b Smproj(F ).


We have the following diagram of categories and functors, in whichall functors are full and [essentially] surjective:

eqs1 (4.1) place(F )op //

T−1²²

place(F )op/CΦ̄b //

²²

S−1b Smproj(F )

S−1rvvmmmmmm

mmmmmm

T−1 place(F )opΦr //

o²²

S−1r Smproj(F )

o²²

place(F )/hc̄ // Smproj(F )/R

where the (unmarked) top left horizontal functor is the natural pro-jection, the (unmarked) top right vertical functor is the compositionS−1r ◦Φb, the two vertical isomorphisms of categories are those of Propo-sitions 4.1.3 and 4.2.2 respectively, and c̄ is induced by the maps ofLemma 4.2.3 a).

teq1 4.2.4. *Theorem. In Diagram (4.1), the top left square is cocarte-sian, that is, the functors induce cocartesian squares of sets on mor-phisms. The same is true for the top trapeze (with vertices place(F )op,S−1b Sm

proj(F ), T−1 place(F )op and S−1r Smproj(F )).

Indeed, the first claim follows directly from Lemma 4.2.3 b) and thesecond claim follows formally from the first plus the fullness of Φ̄b. 2

Here is a reformulation:

4.2.5. *Corollary. Let P : Smproj(F )op → A be a functor to somecategory A.a) Suppose that P is a birational invariant. Then places act on P : anyplace λ : K 99K L induces a map λ∗ : P (K)→ P (L) where P (K) andP (L) respectively denote P (X) and P (Y ) for any smooth projectivemodels X, Y of K and L. Similarly, rational maps act on P .b) Suppose moreover that P is a stable birational invariant. Then fortwo morphisms f, g : X → Y , P (f) = P (g) as soon as f(ηX) andg(ηX) are R-equivalent.

5. An interesting subcategory

5.0.6. Definition. A proper F -variety X is nonrational if it does notcarry any nonconstant rational curve (over the algebraic closure of F ),or equivalently if the map

X(F̄ )→ X(F̄ (t))is bijective.


5.0.7. Lemma. a) Nonrationality is stable by product and by passingto closed subvarieties.b) Curves of genus > 0 and torsors under abelian varieties are nonra-tional.c) Nonrational smooth projective varieties are minimal in the sensethat their canonical bundle is nef.

Proof. a) and b) are obvious; c) follows from the Miyaoka-Mori theo-rem ([45], see also [35, Th. 1.13] or [11, Th. 3.6]). 2

On the other hand, an anisotropic conic is not a nonrational variety.Smooth nonrational varieties are the local objects of Smproj(F ) with

respect to Sr in the sense of Definition A.7.2: the following lemma isvalid in all characteristics.

5.0.8. Lemma. a) A proper variety X is nonrational if and only if, forany morphism f : Y → Z between smooth varieties such that f ∈ Sr,the map

f ∗ : Map(Z,X)→Map(Y,X)i

contents...birational motives, i bruno kahn and r. sujatha preliminary version contents introduction...

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