Milnor's Conjecture on Quadratic Forms and Mod 2 Motivic Complexes

REND.SEM.MAT.UNIV.PADOVA, Vol. 114 2005)

Milnor's conjecture on quadratic forms and mod 2 motivic complexes.

FABIEN MOREL *)

ABSTRACT - Let F be a field of characteristic different from 2. In this paper we give a new proof of Milnor's conjecture on the graded ring associated to the powers of the fundamental ideal of the Witt ring of quadratic forms over F, firstproven by Orlov, Vishik and Voevodsky. Our approach also relies on Voevodsky's affirmation of Milnor's conjecture on the mod 2 Galois cohomology of fields of characteristic different from 2, but, besides this fact, we only use some elementary homological algebra in the abelian category of Zariski sheaves on the category of smooth k-varieties, involving classical results on sheaves of Witt groups, Rost's cycle modules and sheaves of 0-equidimensional cycles.

1. Introduction.

Let F be a field of characteristic 6 2. In this paper we give a new proof of Milnor's conjecture identifying the mod 2 Milnor K-theory of F to the graded ring associated to the filtration of the Witt ring WF) of anisotropic quadratic forms over F by the powers of its fundamental ideal [15]. This result was first obtained by Orlov, Vishik and Voevodsky in [23] where they proved more. This also appears in [11, Remark 3.3]. In both cases however, sophisticated techniques and results from Voevodsky's proof of Milnor conjecture on the mod 2 Galois cohomology of fields [32, 33] are used, such as triangles involving the Rost motives, the Milnor operations Qi. Our approach uses Voevodsky's affirmation of Milnor's conjecture on the mod 2 Galois cohomology of fields, but, however, it is quite different in spiritfrom theprevious ones. Fix a perfectbase field k, let Smk denote the category of smooth k-schemes and let Abk denote the abelian category of sheaves of abelian groups in the Zariski topology on Smk. Besides Voe- vodsky's resultwe will only use some elementaryhomological algebra in

*) Indirizzo dell'A.: Mathematisches Institut der LMU, Theresienstr. 39 - 80333 Muenchen Germany). 64 Fabien Morel

the abelian category Abk involving sheaves of Witt groups, Rost's cycle modules [24] and sheaves of 0-equidimensional cycles [28]. Itis also rather differentfrom our original proof announced in [16]. There we were using the Adams spectral sequence based on mod 2 motivic cohomology and the computation by Voevodsky of the whole corresponding Steenrod algebra. In the approach taken here we don't use these anymore. We will explain the relationship between these two approaches in [19]. Letus denoteby WF) the Witt ring of anisotropic quadratic forms over F [14, 25]. The kernel of the mod 2 rank homomorphism WF) ! Z=2 is denoted by IF) and called the fundamental ideal of WF). For each integer n,then-th power of IF) is denoted by InF). For any unit u 2 FÂ, the symbol hui will denote the class in WF)ofthe quadratic form of rank one u:X2 and the symbol hhuii will denote the class of the Pfister form 1 hÀui1 Àhui2WF), indeed an elementin IF). The following Steinberg relation:

1 hhuii:hh1 À uii 0 which holds in I2F) for u 2 FÂ Àf1g, is a reformulation of the well-known relation huih1 À ui1 hu:1 À u)i see [25] for instance). One also has the following equality for any pair u; v) 2 FÂ)2 hhuii hhviiÀhhu:vii hhuiihhvii 2 I2F) which shows that u 7!hhuii induces a group homomorphism

2 FÂ=FÂ)2 ! IF)=I2F)

M The Milnor K-theory KÃ F) of the field F introduced in [15] is the Â Â quotient of the tensor algebra TensZF on the abelian group of units F by the two sided ideal generated by tensors of the form u 1 À u), for u 2 FÂ Àf1g. The mod 2 Milnor K-theory of F is the quotient

M kÃF) : KÃ F)=2 As observed in [15] the above computations give a canonical graded ring homomorphism extending 2)

n n1 sF : kÃF) !ÈnI F)=I F) which we call the Milnor homomorphism and which was shown in loc: cit: to be surjective in any degree and an isomorphism in degrees 2. The Milnor conjecture on the Witt ring stated as question 4.3 on page 332 in loc: cit:, is the content of the following statement: Milnor's conjecture on quadratic forms and mod 2 motivic complexes 65

THEOREM 1.1. For any field F of characteristic not 2 and any integer n the Milnor homomorphism

n n1 snF) : knF) ! I F)=I F) is an isomorphism.

REMARK 1.2. The powers of the fundamental ideal of F form altogether a commutative graded ring denoted by IÃF), in facta graded WF)- algebra because I0F) WF). We have a canonical morphism of graded Ã WF)-algebras TensWF)IF)) ! I F) where the left hand side denotes the tensor algebra over WF)oftheWF)-module IF). The relation 1) above, which holds in I2F), shows that this morphism factors trough the W quotient algebra KÃ F) : TensWF)IF))=hhuii hh1 À uii by the two- sided ideal generated by the tensors hhuii hh1 À uii 2 IF) WF) IF). It W is quite natural to call KÃ F)theWitt K-theory of F. In [17] we have proven that the induced morphism of WF)-algebras

W Ã KÃ F) ! I F) is an isomorphism. This is in fact a reformulation of the main result of [2], which relies on Voevodsky's results and on Theorem 1.1. Observe conversely that Theorem 1.1 can be recovered from the isomorphism W Ã KÃ F) I F) by tensorization by Z=2 over WF). We already mentioned that the surjectivity of the Milnor morphism holds, so that the main point of Theorem 1.1 is the injectivity. Our proof will consist in constructing inductively a left inverse to snF), the so-called n-th invariant of quadratic forms

n n1 enF) : I F)=I F) ! knF) The statement in the Theorem corresponding to a fixed integer n will be called in the sequel the Milnor conjecture on the Witt ring of F in weight n. Denote by HÃF; Z=2n)) the mod 2 motivic cohomology groups of F in weight n as defined by Suslin-Voevodsky in [27]. These groups HÃF; Z=2Ã)) altogether form a bigraded commutative ring. We have 0 the particular element t :À1 2 m2F) H F; Z=21)). Our main resultis:

THEOREM 1.3. Let k be a perfect field of characteristic 6 2 and let N > 0 be an integer.Assume that the following assumptions hold:

H1N): Milnor's conjecture on the Witt ring of k holds in weights N. 66 Fabien Morel

H2N À 1): For any finite type field extension Fjk and for any integer 1 n N À 1, the cup product by t

HnÀ1F; Z=2n À 1)) !t [ HnÀ1F; Z=2n)) is onto. Then Milnor's conjecture on the Witt ring holds for any field extension Fjk in weights N.

PROOF that THEOREM 1.3 ) THEOREM 1.1. The group HiF; Z=2n)) is the group of sections on F of the i-th cohomology sheaf HiZ=2n)) of an 1 explicitchain complex Z=2n)inAbk, the mod 2 motivic complex in weight n defined in [27]. The ring structure is induced by explicit morphisms of complexes Z=2n) Z=2m) ! Z=2n m)byloc.cit. The cup productby t is thus induced by a morphism of the form 3 t [ : Z=2n À 1) ! Z=2n) Voevodsky's main theorem [32, 33] implies the Beilinson-Lichtenbaum conjecture on mod 2 motivic cohomology, which identifies, for i 2f0; ...; ng, the sheaf HiZ=2n)) to the sheaf associated in the Zariski i n topology to the presheaf X 7! HetX; m2 ), see [27, 32]. This identification being compatible to the cup-product and because the cup-product by 0 t 2 m2F) HetSpeck); m2)ineÂtale cohomology induces isomorphisms i n i n1) HetX; m2 ) HetX; m2 ), this implies that the morphism 3) induces isomorphisms on cohomology sheaves of degrees n À 1, for any n. This establishes a fortiori H2N À 1) for any N. Now choose for the field k a prime field of characteristic 6 2. The

Milnor conjecture on the Witt ring holds for k by [15], which proves H1N) for any N. Theorem 1.3 now implies Milnor's conjecture on the Witt ring of field extensions Fjk in any weight, establishing Theorem 1.1. p

REMARK 1.4. It is possible to prove Theorem 1.3 without H1N) but this would make the exposition more complicated.

REMARK 1.5. Our method clearly emphasizes that Voevodsky's proof of Milnor's conjecture on mod 2 Galois cohomology in weights N À 1 implies the Milnor conjecture on the Witt ring in weights N.

1) for us ``chain complex'' means that the differential is of degree À1 Milnor's conjecture on quadratic forms and mod 2 motivic complexes 67

In the rest of the paper, we will concentrate on the proof of Theorem 1.3. We now fix once for all a perfectfield k of characteristic 6 2. Letus describe our strategy. Recall that we denote by Abk the abelian category of sheaves of abelian groups in the Zariski topology on the category Smk of smooth k-schemes. We denote by

M Kn 2Abk the sheaf of unramified Milnor K-theory in weight n constructed in [13, M 24]. Its fiber on a field Fjk is Kn F); see section 2.2 for a recollection. We will denote by

M kn : Kn =2 M the cokernel in Abk of the multiplication by 2 on Kn . We will denote by

W 2Abk the associated sheaf to the presheaf of Witt groups on Smk: X 7! WX) constructed in [14] for instance. We will then show how the filtration of the Witt ring WF) WF)of each field extension Fjk by the powers of their fundamental ideal naturally arises from a decreasing filtration by sub-sheaves ... In ... W

n n1 For each n, we set in : I =I 2Abk. The Milnor homomorphism for fields then arises for each n from an epimorphism of sheaves:

sn : kn ! in called the Milnor morphism in weight n whose kernel is denoted by j . n Our strategy to prove Theorem 1.3 can now be decomposed as follows. First by induction we may assume the Milnor conjecture on the Witt ring is proven for all fields Fjk in weights N À 1, so that j 0 for n 0 n N À 1. Then:

1) Using H1N) and H2N À 1) prove the vanishing of the groups Exti k ; j ) for 0 n N À 1 and i 2f0; 1; 2g. Abk n N

2) Using elementary homological algebra in Abk, deduce the Milnor conjecture on the Witt ring in weight N for all field extensions Fjk from 1).

PROOF OF STEP 1) will be given in section 3.3 below in a more general form; see Theorem 3.10. The idea is to use two types of information about 68 Fabien Morel the Suslin-Voevodsky motivic complexes Z=2n). The firstone is the vanishing for n > 0 of the groups of morphisms in the derived category of

Abk of the form Hom Z=2n); M[Ã]) DAbk) for some type of sheaves M, like j , which are birational invariant. This is N done by adapting a simple geometric argument due to Voevodsky [31]. The second type of information concerns the cohomology sheaves HiZ=2n)). These vanish for i > n, for i n we have the Suslin-Voevodsky iso- 2 n morphism : kn H Z=2n)) and hypothesis H2N À 1) provides an epi- nÀ1 morphism knÀ1 !H Z=2n)) for n < N. The vanishing in step 1) is then deduced from these facts combined to the universal coefficient spectral sequence see Lemma 3.9):

p;q p q E Ext H Z=2n)); M) ) HomDAb )Z=2n); M[p À q]) p 2 Abk k

PROOF OF STEP 2) will be given in section 3.4. Here is a sketch. Assume the Milnor conjecture on the Witt ring in weights N À 1 for fields Fjk and the vanishing in 1) are both established.

n1 For each integer n, set Wn : W=I . The sheaf WNÀ1 thus admits a n n1 filtration with associated subquotients of the form I =I in, for some 0 n N À 1. The Milnor conjecture on the Witt ring in weights N À 1 gives isomorphisms kn in, and one deduces from the long exact sequence of Ext groups and 1) the vanishing

Exti W ; j ) 0 Abk NÀ1 N for i 2f0; 1; 2g. Using the short exact sequence

4 0 ! j ! k ! i ! 0 N N N one deduces that the morphism Ext1 W ; k ) ! Ext1 W ; i )is Abk NÀ1 N Abk NÀ1 N an isomorphism. There exists thus a commutative diagram

0 ! kN ! GN ! WNÀ1 ! 0 5 ## #

0 ! iN ! WN ! WNÀ1 ! 0 with exact horizontal rows where the bottom horizontal row is the obvious

2) see [33, 7] and section A.3 Milnor's conjecture on quadratic forms and mod 2 motivic complexes 69 one and where the left vertical map is the Milnor epimorphism. Next we show that the canonical epimorphism

W ! WN canonically lifts trough the epimorphism GN ! WN to a morphism

W ! GN This is one of the main arguments: it uses the fact that the presentation of the Witt ring or rather sheaf) uses units as generators and ``open subschemes of productof Gm as relations''; taking into account the inductive assumption that j 0 which implies j is a birational invariant sheaf, NÀ1 N the existence of the lifting follows easily see section 3.4). To finish the proof one observes that the composition IN W !

! GN ! WNÀ1 is trivial by construction, and that the induced morphism N I ! kN Ker : GN ! WNÀ1) induces a left inverse to the Milnor morphism in weight N

N N1 eN : I =I ! kN p

REMARK 1.6. Let F k be the category of finite type field extensions Fjk. It is tempting to work in the abelian category of functors from F k to the category of abelian groups implicitly considered by Serre in [9], instead of the more elaborated category Abk. If one could prove an analogue of the step 1) or Theorem 3.10) there, then our strategy could be simplified further.

NOTATIONS. For a scheme X and an integer i 2 N, we will denote by Xi) the set of points of codimension i of the scheme X.

Given x 2 X 2 Smk,wewilldenotebyOX;x the local ring of X at x. For a given presheaf of sets M on Smk we will denote by MOX;x), or simply by Mx,thefiberofM at x 2 X, in the Zariski topology. Im- portant examples for us will be that of a finite type field extension

Fjk 2Fk, considered as the local ring of the generic points of its models in Smk or that of a geometric discrete valuation ring. Such a discrete valuation ring Ov is one with field of fractions F of finite type over k and which is isomorphic to the local ring of some X 2 Smk at some point x of codimension 1. The associated valuation v on F will be called a geometric discrete valuation on F; its residue field will be denoted by kv). 70 Fabien Morel

2. The Milnor morphism as a morphism of sheaves

2.1 ± Unramified Witt groups and related sub-sheaves

DEFINITION 2.1. A sheaf of sets M on Smk in the Zariski topology is said to be 0-pure if for any irreducible X 2 Smk with field of functions F the canonical map MX) ! MF) is injective and induces a bijection

MX) \y2X1) MOX;y) MF)

Letus denoteby W : Smk !Ab; X 7! WX) the presheaf of Witt groups on Smk; see [14] for instance, or [22] for a quick recollection. We will denote by W the associated sheaf in the Zariski topology. The following result is a reformulation of some results of [22]:

THEOREM 2.2. 1Ojanguren-Panin) The sheaf W is 0-pure.

PROOF. Fix an irreducible X 2 Smk with field of functions F. Clearly, because W is a sheaf in the Zariski topology, the morphism

WX) ! Px2XWOX;x) is injective. Now by loc.cit. for each x 2 X the morphism

WOX;x) WOX;x) ! WF) WF) is injective. This implies that the canonical morphism WX) ! WF) is injective. Now again because W is a sheaf in the Zariski topology, the previous injection identifies WX) with \x2XWOX;x) \x2XWOX;x) WF). By Theorem A of loc.cit. , for each point x 2 X one has

WOX;x) \yWOX;y) where y runs over the set of all prime ideal in OX;x of height one, that is to say points y 2 X1) whose closure contains x. Itfollows then that

WX) \x2X WOX;x) \y2X1) WOX;y) WF) which establishes the Theorem. p

Let n 2 N be an integer. For any irreducible X 2 Smk with function Milnor's conjecture on quadratic forms and mod 2 motivic complexes 71 field F we set InX) InF) \ WX) WF)

We extend the definition to any X 2 Smk by setting n n I X) :Èx2X0) I Xx) Èx2X0) WXx) WX) 0) where Xx X denotes the irreducible component containing x 2 X .

THEOREM 2.3. Given any morphism f : X ! YinSmk and any n 2 N the morphism W f ) : WY) ! WX) maps the subgroup InY) WY) into InX) WX).Thus the correspondence X 7! InX) admits a unique structure of presheaf of abelian groups, denoted by In, such that the inclusions InX) WX) define a monomorphism of presheaves In W.Moreover, In is a 0-pure sheaf.

PROOF. By Lemma A.1 of the Appendix below, it is sufficient to prove that for any geometric discrete valuation v on a finite type field extension Fjk, the morphism

rv : WOv) WOv) ! Wkv)) Wkv)) n n n h maps I Ov) I F) \ WOv) into I kv)). Let Ov denote the henselization h of Ov, Fv its fraction field. By naturality we have the following commutative diagram of Witt rings

h WOv) ! WOv ) ## Wkv)) Wkv))

n n h n h h h and I Ov) clearly maps to I Ov ) : I Fv ) \ WOv ) WFv ). Lemma 2.4 4) below implies the result. p

LEMMA 2.4. Let v be a discrete valuation on a field F, with residue field kv) of characteristic not 2, and let p be a uniformizing element for v.Then

1) The ring homomorphism WOv) ! WF) is injective and the dia- p @v gram 0 ! WOv) ! WF) ! Wkv)) ! 0 is a short exact sequence, where p @v is the residue morphism of [15, 25]. p 2) For each n > 0 the residue morphism @v : WF)!Wkv)) maps InF) onto InÀ1kv)). 72 Fabien Morel

h h 3) Let Ov denote the henselization of Ov,Fv its fraction field.Then the h ring homomorphism r : WOv) ! Wkv)) is an isomorphism and the h canonical WOv )-algebra homomorphism h 2 h WOv )[T]=T À 1) ! WFv ) ; T 7!hpi an isomorphism 1Springer).

n h h 4) For each n > 0 the intersection I Fv ) \ WOv ) is equal to the n-th n h h power I Ov ) of IOv ) and the exact sequence of 1) induces an exact sequence @p n h n h v nÀ1 0 ! I Ov ) ! I Fv ) ! I kv)) ! 0

PROOF. Statement 1) is proven in [25, Theorems 2.1, 2.2]. It follows from the proof of [15] Corollary 5.2 that the residue morphism maps InF) into InÀ1kv)). The surjectivity is easy. The first isomorphism in statement 3) is loc.cit. Theorem 2.4 and the second one is Corollary 2.6, due to Springer. The last statement is proven in loc.cit. §5 in the case of complete discrete valuation rings but using 1), 2) and 3) the proof carries over to our case: one proceeds as in loc.cit. , proof of Corollary 5.2, establishing in- n h À1 n À1 nÀ1 ductively that I Fv ) r I kv))) Èhhpii:r I kv))). p

REMARK 2.5. Let A be a regular local ring in which 2 is invertible with fraction field F. Assume that WA) ! WF) is injective for instance if it contains a field of characteristic 6 2 by [22]). We don'tknow in general whether or not for any n 2 N the group InA) : InF) \ WA) WA) is always the n-th power of the ideal IA) : IF) \ WA), though the previous Lemma establishes it for an henselian discrete valuation ring.

2.2 ± Unramified Milnor K-theory and Rost's cycle modules

Let n be an integer. For any field F and any n-tuple u1; ...; un) 2 Â n M 2 F ) of units we will denote by fu1g ...fung2Kn F) its image through Â n M the obvious map F ) ! Kn F). Recall from [15] that for any discrete valuation v on the field F, with residue field kv) there exists one and only one homomorphism:

M M 6) @v : Kn F) ! KnÀ1kv)) Milnor's conjecture on quadratic forms and mod 2 motivic complexes 73 satisfying

@vfpg:fu2g ...fung) fu2g ...fung Â if vp) 1 and vui) 0 for each i 2. For u 2Ov the notation u 2 kv) means the image of u in kv)Â. This homomorphism is called the residue homomorphism associated to v. M For any smooth k-variety X, we let Kn X) denote the kernel, introduced in [13], of all the residue homomorphisms associated to points in X of codimension 1: P @ M M x M Kn X) : Ker Èx2X0) Kn kx)) ! Èy2X1) KnÀ1ky))

M The correspondence X 7! Kn X) is turned into a Zariski sheaf on Smk by M Rostin [24]. This sheaf will be denoted by Kn and called the sheaf of unramified Milnor K-theory in weight n.

More generally, let us recall briefly the notion of cycle module over k Ã from [24]. We justremind thata cycle module MÃ is a triple MÃ; W ;@v) consisting of a functor

MÃ : F k !AbÃ with AbÃ being the category of graded abelian groups, a transfer morphism Ã W : MÃF) ! MÃE) of degree 0 for each finite extension E F in F k and a residue morphism @v : MÃF) ! MÃkv)) of degree À1 for any geometric discrete valuation v on Fjk 2Fk. These data satisfy some axioms which we will notrecall here; see loc.cit. p. 329 and p. 337. M The Milnor K-theory groups KÃ , endowed with the transfer morphisms of Kato [12], and the above residue morphisms, form the fundamental ex- ample of cycle module, as itfollows from [15, 5, 12, 24]. The mod m Milnor K- M theory groups KÃ =m also form a cycle module for any integer m. In fact, the category of cycle modules, with the obvious notion of morphisms, is abelian: the kernel and cokernel of a morphism are performed termwise on each

Fjk 2Fk, and the axioms of [24, p. 329 and p. 337] follow formally. In the sequel,wewillsimplydenotebykÃ thecycle moduleofmod2 MilnorK-theory. 0 Given a cycle module MÃ and X 2 Smk we will denote by A X; MÃ)the 3 group of unramified sections of M0 on X, by which we mean the kernel of

3) Here we slightly differ from [24] where Rost considers rather the direct sum 0 Èn2ZA X; MÃn)), see below. 74 Fabien Morel the sum of residue morphism at points of codimension 1: P @x Èx2X0) M0kx)) ! Èy2X1) MÀ1ky)) In [24, §12] these groups are turned canonically into a Zariski sheaf on

Smk which we denote by M0. A sheaf of abelian groups in the Zariski topology on Smk will be said to come from a cycle module, or to have a cycle module structure, if it is isomorphic to a sheaf of the form M0.It is easy to check that such sheaves are 0-pure in the sense of Defini- tion 2.1.

REMARK 2.6. By DeÂglise [7] the sheaf M0 admits a canonical structure of homotopy invariant sheaf with transfers in the sense of Voevodsky [28], and these two notions of homotopy invariant sheaves with transfers and of cycle modules are essentially the same.

0 LEMMA 2.7. For any 1termwise) short exact sequence 0 ! MÃ ! MÃ ! 00 ! MÃ ! 0 of cycle modules, the diagram 0 00 0 ! M 0 ! M0 ! M 0 ! 0 is a short exact sequence of sheaves in the Zariski topology.

PROOF. We will freely use the notations from [24]. For any Y 2 Smk,or any localization Y of a smooth k-scheme, and any cycle module NÃ is defined Ã a cochain complex C Y; NÃ), see loc: cit: p. 355 and p. 359. A shortexact 0 00 sequence of cycle modules 0 ! MÃ ! MÃ ! MÃ ! 0 then induces a short exactsequence of cochain complexes

Ã 0 Ã Ã 00 0 ! C Y; MÃ) ! C Y; MÃ) ! C Y; MÃ ) ! 0 and a corresponding long exactsequence of associated cohomology groups.

ButTheorem 6.1) of loc: cit: establishes that for any x 2 X 2 Smk and any Ã cycle module NÃ, the cochain complex C SpecOX;x); NÃ) has trivial cohomology in > 0 degrees. The shortexactsequence

Ã 0 Ã Ã 00 0 ! C SpecOX;x); MÃ) ! C SpecOX;x); MÃ) ! C SpecOX;x); MÃ ) ! 0 thusproduces a shortexactsequence

0 0 0 0 00 0 ! A SpecOX;x); MÃ) ! A SpecOX;x); MÃ) ! A SpecOX;x); MÃ ) ! 0 0 00 in other words, of the form 0 ! M 0OX;x) ! M0OX;x) ! M 0OX;x) ! 0, which gives the result. p Milnor's conjecture on quadratic forms and mod 2 motivic complexes 75

For any cycle module MÃ and any integer n 2 Z, we denote by MÃn) the cycle module obtained in the obvious way by setting MÃn))m Mmn and endowed with the corresponding shifted data. For instance, for each n 2 N we have KM KMn) and we define the sheaf of unramified mod 2 n Ã 0 Milnor K-theory in weight n as

k : k n) KM=2 n Ã 0 n More generally for a cycle module M we will simply set M : M n) . Ã n Ã 0

DEFINITION 2.8. Let n 2 Z be an integer.We will say that a cycle module MÃ is of weights n if and only if for any Fjk 2Fk the group MiF) vanishes for i < Àn.

Observe that if MÃ is in weights n, MÃm) is in weights n m. Also MÃ is in weights n if and only if for any Fjk 2Fk, MÀnÀ1F) 0. In that M case the sheaves MÀm all vanish for m n 1. The cycle modules KÃ and kÃ are in weights 0.

LEMMA 2.9. Let X 2 Smk and let Z X be a closed subscheme everywhere of codimension n, then for any sheaf M coming from a cycle module of weights n À 1 the groups HÃX; X À Z); M) vanish for any Ã.

PROOF. This follows from the results of [24]: for any cycle module NÃ Ã one can compute H X; X À Z; N0) as the cohomology of the complex Ã Ã C X; NÃ)=C X À Z; NÃ) which is of the form

Èx2X0)=xZN0kx)) ! ...!Èx2Xi)=xZNÀikx)) ! ... Now because the codimension of Z is n, that complex is trivial up to codimension n.ButifNÃ is of weights n À 1, it is trivial from there as well, so itvanishes. p

We will also need the following lemma whose first part is proven in [24, 12] and whose second part is due to Bass and Tate [5, Prop. 4.5 b)].

LEMMA 2.10. Let v be a geometric discrete valuation on a finite type field Fjk.Then:

M M @v M 1) The diagram 0 ! Kn Ov) ! Kn F) ! KnÀ1kv)) ! 0 is a short 76 Fabien Morel

exact sequence.Moreover, given a uniformizing element p 2Ov the following diagram is commutative

M M Kn Ov) Kn F)

rv ##@vfpg[ À )) M M Kn kv)) Kn kv)) Â M M 2) For any unit u 2Ov the symbol fug2K1 F) lies in K1 Ov) and M moreover the group Kn Ov) is generated by symbols of the form

fu1g ...fung Â with the ui's in Ov .

2.3 ± Sheafifying the Milnor homomorphism

For each integer n 2 N and any field Fjk 2Fk we set n n1 inF) : I F)=I F)

We denote by iÃF) the corresponding graded abelian group.

THEOREM 2.11. There exists one and only one structure of cycle module on the correspondence

F k !AbÃ; F 7! iÃF) such that the Milnor homomorphisms

sÃF) : kÃF) ! iÃF) altogether define a morphism of cycle modules, an epimorphism indeed.

PROOF. We will show below that the transfers morphisms

Ã W : kÃF) ! kÃE) for finite extensions E F in F k and the residue morphisms

@v : kÃF) ! kÃÀ1kv)) for geometric discrete valuations v on Fjk 2Fk, descend to morphisms on iÃ. Endowed with these induced morphism, iÃ becomes a cycle module because the axioms are automatic consequences of the corresponding ones for mod 2-Milnor K-theory. Moreover, by construction, the Milnor homomorphism preserves the cycle module structures, defining an epimorphism Milnor's conjecture on quadratic forms and mod 2 motivic complexes 77

of cycle modules of the form kÃ ! iÃ. The uniqueness of the structure is clear. To prove our claim for the residue morphisms, let v be any geometric discrete valuation on Fjk 2Fk, and p be a uniformizing elementfor v. Let us denote by

p @v : inF)!inÀ1kv)) the homomorphism induced by the one on InF) given in Lemma 2.4 2) above. Clearly from the explicit description of residues in Milnor K-theory and in Witt theory [15, 25] the diagram

@v knF) ! knÀ1k) ## p @v inF) ! inÀ1k)

p is commutative. This also shows that @v doesn'tdepend on p. Let E F be any finite extension of fields and n be an integer. Assume first this extension is purely unseparable. Then in that case, because the p p characteristic p is odd, one has hx ihxi in WF) and fx gfxg2k1F); for any x 2 F, some power xpn is an E. Thus the extension of scalars

WE) ! WF) and kÃE) ! kÃF) are both epimorphisms. But the same formula shows that the Frobenius induces the identity morphism WE) WE); as some iterated of the Frobenius on F maps to E F the factorization of the identity as WE) ! WF) ! WE) this shows the extension of scalars is in fact an isomorphism. The degree [F : E], being a power of p, is odd so that the standard property of the transfer morphism implies it is a monomorphism on mod 2 Milnor K-theory, showing that kÃE) ! kÃF) is an isomorphism as well. This clearly imply our claim on the transfers in that case. Assume now the extension E F is separable. Let t : F ! E denote the trace morphism and tÃ : WF) ! WE), q 7! t q the corresponding n n Scharlau transfer [25, §2.5]. By Arason [1, Satz 3.3] tÃ maps I F) into I E) and thus induces a natural morphism tÃ : inF) ! inE). To check that the diagram

WÃ knF) ! knE) 7) ## tÃ inF) ! inE) is commutative one proceeds as follows. Let E E0 be a separable al- 0 0 gebraic extension of odd degree, and F : E E F, a finite separable 78 Fabien Morel algebra over E0. Clearly the square above maps by extension of scalars to the corresponding one involving E0 and F0.ButE ! E0 and F ! F0 being 0 of odd degree, the extension morphisms inE) ! inE )isamono- morphism: this is proven after the proof of [1, Satz 3.3]. Thus to prove the commutativity of our square 7) it suffices to prove it for the one obtained by extending the scalars to E0. If we choose for E E0 separable algebraic extension such that the absolute Galois group of E0 is a 2-Sylow of that of E,wemayassumefurtherthatE0 ! F0 admits a finite increasing 0 0 0 0 filtration E E1 ... Er F by quadratic extensions. Thus we reduced our claim to the commutativity of 7) in case E F is a quadratic extension. By [5, Corollary 5.3 p. 29], kÃF) is generated as a module over Ã kÃE)byk1F). So it suffices by the projection formula both for W and tÃ) to prove it for n 1, which is an easy computation use for instance [25, Lemma 5.8]). p

For each n 2 N, in denotes the associated sheaf to the cycle module iÃn). The Milnor morphism of cycle modules

sÃ : kÃ ! iÃ being an epimorphism itdefines for each n 2 N an epimorphism of sheaves in the Zariski topology

sn : kn ! in by Lemma 2.7. This morphism is called the Milnor morphism in weight n.

n n1 2.4 ± The isomorphism I =I in

The following resultjustifies a posteriori our quick definition of in in the introduction as In=In1.

THEOREM 2.12. Let n 0 be an integer.

n 1) The canonical transformation between functors on F k: I F) ! n ! inF) arises from a unique morphism of sheaves I ! in. 2) The kernel of this morphism is the subsheaf In1 In.

3) This morphism is an epimorphism 1in the Zariski topology) and n n1 thus induces a canonical isomorphism I =I in. Milnor's conjecture on quadratic forms and mod 2 motivic complexes 79

PROOF. 1) We use Lemma A.2. The firstpropertyfollows easily from the fact that the morphisms

n I F) ! inF) commute to residue morphisms see the proof of Theorem 2.11). The second property means that for any geometric discrete valuation v on Fjk 2Fk the following diagram commutes

n I Ov) ! inOv) ## n I kv)) ! inkv)) h This follows again from Lemma 2.4 by mapping Ov to Ov . This defines the n morphism I ! in for each n.

2) It is sufficient to prove that for each x 2 X 2 Smk the diagram n1 n 8) 0 ! I OX;x) ! I OX;x) ! inOX;x) is an exactsequence.

Choose for each point y of codimension 1 in SpecOX;x) a uniformizing element py of the associated discrete valuation. From the fact that py WOX;x) WOX;x) is the kernel of all the residue morphisms @y , from Theorem 2.2 and from Lemma 2.4 one constructs the commutative diagram in which F is the fraction field of OX;x and the right horizontal maps are residues:

n1 n1 n 0 ! I OX;x) ! I F) !È 1) I ky)) y2SpecOX;x) ## # n n nÀ1 0 ! I OX;x) ! I F) !È 1) I ky)) y2SpecOX;x) ## #

0 ! i OX;x) ! inF) !È 1) inÀ1ky)) n y2SpecOX;x) ## # 00 0 The horizontal rows and the last two vertical rows being each exact sequences, the claim follows. 3) We want now to establish the surjectivity of the right homomorphism in 8). Assume firstthat k is infinite. By [8, Proposition 4.3], the group M Kn OX;x) is generated by symbols fu1g ...fung with the ui's units in OX;x. Thus so is the quotient) group inOX;x). Now we conclude because these n symbols fu1g ...fung in inOX;x) are clearly in the image of I OX;x) ! ! inOX;x) which is thus onto. 80 Fabien Morel

Assume now that k is finite. Let k K be an algebraic extension with K infinite, perfect and of odd degree. By Lemma 2.14 below we see that the n n cokernel of I OX;x) ! inOX;x) injects into the cokernel of I OX;xjK) ! ! inOX;xjK), which is zero by whatwe have seen above. The theoremis now proven in any case. p

REMARK 2.13. It is possible to give a more elementary and more natural proof of the previous Theorem which doesn't use the result of [8] quoted above. It relies on the work [26] and on [24]: for any irreducible

X 2 Smk with function field F and any n 2 N one constructs as in [26] an explicitand canonical complex CÃX; In) of the form see also [4]):

n nÀ1 nÀ2 2 I F) !È 1) I ky); ty) !È 2) I kz); L tz)) ! ... y2SpecOX;x) z2SpecOX;x) in which ty means the tangent vector space a point y 2 X, the dual of the 2 ky)-vector space my=my) . The technique from Rost [24] shows that this n complex gives for a smooth local ring OX;x a resolution of I OX;x). This complex can be shown to extends on the right the horizontal lines of the diagram above used in the proof of part 2); from this it is quite easy to n prove the surjectivity of I OX;x) ! inOX;x). The following Lemma is directly inspired by [3, 22].

LEMMA 2.14. Let k L be a finite extension and x 2 X 2 Smk.Let F be the fraction field of OX;x.

1) For each n 2 N the Scharlau transfer sÃ : WF k L) ! WF) with respect to a non-zero k-linear map s : L ! k [25, §2.5] maps WOX;x k L) n n WF k L) to WOX;x) WF) and maps I F k L) to I F). Thus it induces a natural morphism

n n I OX;x k L) ! I OX;x) kk n n WOX;x k L) \ I F k L) ! WOX;x) \ I F)

2) For any non-trivial s the natural morphism of 1) is compatible to the morphism

inOX;x k L) ! inOX;x) induced by the transfer inF k L) ! inF) of the cycle module structure on iÃ see 2.11). Milnor's conjecture on quadratic forms and mod 2 motivic complexes 81

n 3) If [L : k] is odd, the morphism I OX;x) ! inOX;x) is a direct sum- n mand of the morphism I OX;x k L) ! inOX;x k L).

PROOF. The fact that the Scharlau transfer maps WOX;x k L) WF k L)toWOX;x) WF) follows from [22, §2 & §3]. The factthat the Scharlau transfer for any choice of s) maps InF)intoInE) is [1, Satz 3.3]. This proves 1).

By the previous result of Arason, the induced transfer tÃ : inF) ! ! inE) doesn'tdepend on thechoice of s. Choosing for s the trace morphism we thus get the transfer for the cycle module structure, see the proof of Theorem 2.11 above. This proves 2). If [L : k] is odd, choose for s the morphism of [25, Lemma 5.8 p. 49]. By n n n loc: cit: the composition I OX;x) ! I OX;x k L) ! I OX;x) is multiplication by the class sÃ1) 1. The same holds for in by [1]. The Lemma is proven. p

3. Proof of the main Theorem

3.1 ± Some A1-homological algebra

For any chain complex CÃ in Abk, any sheaf M 2Abk, and any integer n 2 Z we denote by Hom C ; M[n]) DAbk) Ã the group of morphisms in the derived category DAbk) of the abelian category Abk from CÃ to the n-th shift M[n]ofM. Thatgroup can be computed as follows: choose an injective resolution M ! IÃ of M in Abk of the form M ! I ! I ! I ! I ! .... Then Hom C ; M[n]) is the group 0 À1 À2 À3 DAbk) Ã of morphisms of chain complexes CÃ ! IÃ[n] modulo the subgroup of those which are homotopic to zero. For technical purposes, we slightly extend the notion of smooth k- 0 scheme. Letus denoteby Smk the full subcategory of that of all k-schemes consisting of k-schemes which are possibly infinite disjoint union of smooth 0 k-schemes. Any such X 2 Smk can be written as qaXa with each Xa irreducible and in Smk;theXa's are called the irreducible components of X. The associated free sheaf of abelian groups on X is the sheaf ZX)

ÈaZXa) 2Abk. A morphism of sheaves of abelian groups of the form f : ZX) ! ZY) 82 Fabien Morel

0 with X and Y in Smk, is said to be elementary if for any irreducible component Xa of X the restriction of that morphism to the summand ZXa)can be written as a finite sum

Sina;i : fa;i with na;i 2 Z and with each fa;i corresponding to a morphism of k-schemes Xa ! Y. An elementary resolution of a sheaf M is a resolution RÃ ! M whose terms Rn, n 2 N are free sheaves on smooth k-schemes possibly in 0 Smk) and whose boundaries are elementary morphisms.

LEMMA 3.1. For any M 2Abk, there exists an elementary resolution of the form

ZXÃ) : ...! ZXn) ! ...! ZX0) ! M ! 0

PROOF. Let Pk be the abelian category of presheaves of abelian groups on Smk. Then for any X 2 Smk, ZX) is just the associated sheaf to the presheaf ZX) 2Pk which maps Y to the free abelian group on the set HomkY; X). These ZX) are projective generators in Pk. One can thus find 0 0 0 a resolution RÃ ! M in Pk with Rn 0 for n < 0, and such that Rn is a directsum of sheaves of theform ZX) with X 2 Smk. Then the sheafifi- cation of that resolution gives a resolution in Abk with the required properties. p

1 DEFINITION 3.2. A sheaf M 2Abk is said to be Zariski strictly A -invariant if for any X 2 Smk the natural homomorphism HÃX; M) ! HÃX Â A1; M) is an isomorphism.

By [24, §9], any sheaf arising from a cycle module is Zariski strictly A1- invariant. Any homotopy invariant sheaf with transfers in the sense of [28] is Zariski strictly A1-invariantby [29]. Recall the construction CÃ from [27, 28] in [28] itis denotedby C ). It associates to a sheaf N 2Abk the complex of sheaves CÃN) with n-th term the sheaf X 7! NDn Â X), with Dn the n-th algebraic simplex4, and with n i differential in degree n, Si0 À 1) @i, with @i induced by the i-th coface DnÀ1 ! Dn.

4 n ) i.e. D Speck[T0; ...; Tn]=SiTi À 1)) Milnor's conjecture on quadratic forms and mod 2 motivic complexes 83

1 LEMMA 3.3. Let M be a Zariski strictly A -invariant sheaf.

1) For any non-negatively graded chain complex CÃ in Abk the morphism

Hom C ; M) ! Hom C ZA1); M) DAbk) Ã DAbk) Ã 1 induced by the projection CÃ ZA ) ! CÃ, is an isomorphism.

2) For any sheaf N 2Abk the morphism N ! CÃN) induces an isomorphism Hom C N); M[Ã]) Hom N; M[Ã]) ExtÃ N; M) DAbk) Ã DAbk) Abk

1 PROOF. 1) By definition, for M to be Zariski strictly A -invariantex- actly means that for any X 2 Smk the homomorphism Hom ZX); M[Ã]) HÃX; M) ! DAbk)

HÃX Â A1; M) Hom ZX) ZA1); M) DAbk) is an isomorphism. The part 1) of the Lemma then follows easily from standard homological algebra and the fact that the sheaves ZX)are generators of Abk. The part 2) of the Lemma follows from 1) exactly in the same way as in [20, Corollary 3.8 p. 89]. p 1 An A -homotopy between morphisms f ; g : CÃ ! DÃ of chain complexes 1 in Abk is a morphism h : CÃ ZA ) ! DÃ which induces f resp. g) through the 0 resp. 1) section Speck) ! A1. As a consequence of 1) of the Lemma, any two A1-homotopic morphisms f ; g : CÃ ! DÃ, with CÃ non-negatively graded induce the same morphism Hom D ; M) ! Hom C ; M), for M Zariski strictly DAbk) Ã DAbk) Ã A1-invariant.

3.2 ± Vanishing of some groups Hom Z=2n); M[Ã]) DAbk)

For X 2 Smk, we let ZtrX) denote the sheaf which maps Y 2 Smk to the group of finite correspondences from Y to X, that is to say the free abelian group cY; X) on the set of irreducible closed subschemes Z Y Â X which are finite on Y and which dominate an irreducible componentof Y [28, 27]. 84 Fabien Morel

For X1 and X2 pointed smooth k-schemes we let

ZtrX1 ^ X2) 2Abk denote the cokernel of the obvious morphism given by the base points

ZtrX1) È ZtrX2) ! ZtrX1 Â X2). Iterating this construction we get for a family X1; ...; Xn) of pointed smooth k-schemes the sheaf [27]:

ZtrX1 ^ ...^ Xn) For any integer n,themotivic chain complex in weight n of Suslin-Voe- ^n vodsky [27, 28] is the chain complex Zn) : CÃZtrGm ))[ À n]. The following resultis a variationon a idea from [31]:

THEOREM 3.4. Let n 1 and let M be a sheaf which comes from a cycle module of weights n À 1.Then one has

9) Hom Zn); M[m]) 0 DAbk) for any integer m 2 Z.

The proof will be given below, after a couple of preliminary Lemmas. Of course the case m < 0 is trivial.

For X 2 Smk, we let zeqX) denote the sheaf which maps Y 2 Smk to the free abelian group zeqX)Y) on the set of irreducible closed subschemes Z Y Â X which are quasi-finite on Y and which dominates an irreducible componentof Y [28]. We have the following geometric lemma:

LEMMA 3.5. Voevodsky [30]) There exist explicit quasi-isomorphisms of chain complexes in Abk of the form

^n n nÀ1 n Zn)[2n] CÃZtrGm ))[ n] CÃZtrP )=ZtrP )) ! CÃzeqA )) The following two Lemmas and their Corollary below are inspired by the proof of [31, Proposition 3.3].

n LEMMA 3.6. Let X 2 Smk and let z 2 zeqA )X).Let us denote by Vz) X Â An the open complement of the support jzjX Â An of z.Let n zjVz) be the pull back of z through the morphism Vz) X Â A ! X.Then there exists a canonical cycle

n 1 hz) 2 zeqA )Vz) Â A ) such that @1hz) zjVz) and @0hz) 0.This cycle is functorial in X. Milnor's conjecture on quadratic forms and mod 2 motivic complexes 85

Moreover, given any 1finite) decomposition z Sjnjzj, with nj 2 Z and n zj 2 zeqA )X), then first jzj[jjzjj, so that \jVzj) Vz) and one has n 1 the following equality in zeqA ) \j Vzj) Â A )

hz) Sjnjhzj) The assertion involving A1-homotopies follows from the explicit construction given in loc.cit. . Beware that a priori itis notclear whetheror f n notgiven a morphism ZX) ! ZY) and an element y 2 zeqA )Y), one has f ZV f Ãy)))) ZVy)), although this is true for f : X ! Y a morphism of schemes. For instance we will be in a situation where f Ãy) 0. This is why we will need the next Lemma.

LEMMA 3.7. Given an elementary morphism f : ZX) ! ZY)

n n an element y 2 zeqA )Y), considered as a morphism ZY) ! zeqA ), and n an open subscheme VY Y Â A , whose complement ZY is quasi-finite equidimensional over Y and contains jyj 1in other words VY Vy)), then there exits an open subscheme

n VX X Â A whose complement ZX is quasi-finite equidimensional over X and con- Ã tains jxj 1in other words VX Vx)), with x y f f y, and such that f 1 maps ZVX ) into ZVY ).Moreover, the restriction to ZVX ) of the A - homotopy hx) of Lemma 3.6 is compatible with the restriction to ZVY ) of the A1-homotopy hy).

PROOF. We may assume X is irreducible. Write f Sjnjfj, a finite sum. Define Z X Â An to be the union over the finite set of indices j of the Ã n supportof thecycles fj ZY ) 2 zeqX Â A ). By construction one has

Ã Ã Ã jxjjSjnj fj y)jSjjfj y)jSjj fj ZY )jZ

n Clearly VX X Â A À Z satisfies the conclusions. To check that f maps ZVX ) into ZVY ) it suffices to observe that Ã ZVX ) \jZVfj ZY ))) so that each morphism fj separately maps ZVX ) into ZVY ). Following this, one proves easily the assertion on the A1-homotopies using Lemma 3.6 above. p 86 Fabien Morel

' n COROLLARY 3.8. Let rÃ : ZXÃ) ! zeqA ) be an elementary resolution 1given by Lemma 3.1). Then there exists a subcomplex

n n ZVÃ) ZXÃ) ZA ) ZXÃ Â A ) n such that for each q 0, Vq is an open subscheme in Xq Â A whose complement is a closed subscheme Zq quasi-finite equidimensional over Xq , such that the composition

n n rÃ n ZVÃ) ZXÃ) ZA ) ZXÃ Â A ) ! ZXÃ) ! zeqA ) is homotopic to zero.

PROOF. We construct the subcomplex

n ZVÃ) ZXÃ) ZA ) by an induction on the degree q 0 using Lemmas 3.6 and 3.7 above, by imposing that for each q, Vq is an open subsetof Vzq), where zq 0 for n q > 0 and z0 r0 : X0 ! zeqA ), and that the homotopy is the restriction in 1 each degree q of the homotopy on ZVzq) Â A ) given by Lemma 3.6. n To start with, apply Lemma 3.6 to the morphism r0 : ZX0) ! ZA ) n which we see as an element z0 2 zeqX0 Â A ). We getan open subset n V0 Vz0) X0 Â A whose complementis quasi-finiteequidimensional over X0 such that the composition

n n r0 n ZV0) ZX0) ZA ) ZX0 Â A ) ! ZX0) ! zeqA ) is homotopic to zero, through the explicit homotopy of the Lemma 3.6. Now Lemma 3.7 applied to the boundary, an elementary morphism by assumption, d1 : ZX1) ! ZX0), and to z0 allows one to define V1 observe Ã that even if d1z0) Z1 0, Lemma 3.7 works and is non trivial!). The process continues thanks to Lemma 3.6 and the functorial property of the homotopy. p

PROOF OF THEOREM 3.4. Set M : M0. To prove the vanishing of the Theorem, itis clearly sufficient,by Lemmas 3.5 and 3.3, toprove for each m 2 N the vanishing

m n n Ext zeqA ); M) Hom zeqA ); M[m]) 0 Abk DAbk)

We proceed inspired by Voevodsky's proof of [31, Prop. 3.6]. Choose an n 1 elementary resolution ZXÃ) ! zeqA ). As M is Zariski strictly A -in- Milnor's conjecture on quadratic forms and mod 2 motivic complexes 87 variant, Lemma 3.3 implies that the morphisms

n n ZXÃ) ZA ) ! ZXÃ) ! zeqA ) induce isomorphisms Hom z An); M[m]) Hom ZX ); M[m]) DAbk) eq DAbk) Ã Hom ZX ) ZAn); M[m]) DAbk) Ã By Corollary 3.8 and Lemma 3.3 again, the restriction morphism

Hom ZX ) ZAn); M[m]) ! Hom ZV ); M[m]) DAbk) Ã DAbk) Ã is 0. The group Hom ZX ) ZAn); M[m]) is thus a quotient of the DAbk) Ã group Hom ZX ) ZAn)=ZV ); M[m]). By construction the com- DAbk) Ã Ã n plex ZXÃ) ZA )=ZVÃ) is degreewise a directsum of sheaves of the form ZX)=ZX À Z) with Z everywhere of codimension n; the groups Hom ZX)=ZX À Z); M[m]) HmX; X À Z; M) thus vanish by DAbk) Lemma 2.9. The Theorem follows. p

i 3.3 ± Vanishing of some extension groups Ext kn; M) Abk

Let CÃ be a non-negatively graded chain complex in Abk and n 2 Z an integer. We denote by HnCÃ its n-th homology sheaf or in other words its Àn À n)-th cohomology sheaf H CÃ. We have the following well-known construction, which can be derived from standard homological algebra [10]:

LEMMA 3.9. 1Universal coefficient spectral sequence) Let CÃ be a non- negatively graded chain complex in Abk and let M be a sheaf of abelian groups on Smk.Then there exists a natural, strongly convergent spectral sequence of cohomological type of the form

p;q p E Ext HqCÃ; M) ) HomDAb )CÃ; M[p q]) 2 Abk k We are now in position to prove our main result on the vanishing of Ext groups:

THEOREM 3.10. Let n > 0 be an integer and MÃ be a cycle module. Then: 1) If M k) 0 then Hom k ; M ) 0; Àn Abk n 0 1 2) If MÃ is of weights n À 1 then Ext k ; M ) 0; Abk n 0 88 Fabien Morel

3) If MÃ is of weights n À 1 and MÀn1k) 0 and if H2N À 1) holds and 1 n N À 1, then: Ext2 k ; M ) 0 Abk n 0

PROOF. Set M : M0. Let n > 0 be an integer. Let's first prove 1). An easy computation as in [27, Lemma 3.3 p. 23] shows that À Á n n n m MGm) ) Èm0 MÀmk) As a consequence the intersection of the kernels of the morphisms n nÀ1 nÀ1 n Mii) : MGm) ) ! MGm) ), with ii : Gm) ! Gm) the closed subscheme defined by i-th coordinate 1, is exactly MÀnk). Let n f : kn ! M be a morphism and let Gm) ! kn be the morphism of sheaves n of sets corresponding to the obvious symbol in knGm) ). From the pre- n f vious observation we see that the composition Gm) ! kn ! M is zero because MÀnk) 0 and its composition with the ii's is 0 because a symbol of length n containing 1 is trivial. Thus f is the zero morphism on sections over fields, and is thus zero because MX) Èx2X0) M0kx)) for any X 2 Smk. Now let's prove 2). Assume MÃ is of weights n À 1. By Theorem 3.4 the groups Hom Z=2n); M[Ã]) vanish. The universal coefficient DAbk) spectral sequence of Lemma 3.9 thus converges to 0. By definition of the complex Z=2n) [27], see also section 3.2, the homology sheaves HiZ=2n)) vanish for i < Àn and by Theorem A.7 we have a canonical isomorphism:

kn HÀnZ=2n))

Looking atthe E2-term gives the vanishing already known by 1)): Hom k ; M) Hom Z=2n); M[ À n]) 0 Abk n DAbk) and the vanishing: Ext1 k ; M) Hom Z=2n); M[ À n 1]) 0 Abk n DAbk)

To prove 3) we go further into the study of the E2-term which also gives a canonical isomorphism 2 10) Hom HÀn1Z=2n)); M) Ext k ; M) Abk Abk n

By H2N À 1), for any integer 1 n N À 1 the morphism of sheaves

11) knÀ1 HÀn1Z=2n À 1)) !HÀn1Z=2n)) induced by the cup-product by t, induces an epimorphism on sections over any field Fjk 2Fk. Because M is 0-pure, this easily implies that the induced Milnor's conjecture on quadratic forms and mod 2 motivic complexes 89 morphism Hom H Z=2n)); M) ! Hom k ; M) is injective. Abk Àn1 Abk nÀ1 Together with the isomorphism 1) we get an injection Ext2 k ; M) Hom k ; M) Abk n Abk nÀ1 but by the case 1) already proven, the group on the right vanishes because

MÀn1k) 0 by assumption. p

3.4 ± Construction of eN

We can now complete the proof of our main result Theorem 1.3, following the lines of the introduction.

Let N > 0 be a fixed integer. We assume hypothesis H1N) and H2N À 1) hold. Proceeding by increasing induction we may assume the Milnor conjecture on the Witt ring in weights N À 1 for fields Fjk is proven.

Letus denoteby jÃ the kernel in the category of cycle modules of the Milnor epimorphism kÃ ! iÃ constructed in Theorem 2.11. By our inductive assumption, jn 0 for any n N À 1. By H2N À 1) and Theo- rem 3.10, for any integer 1 n N À 1 one has the vanishing Exti k ; j ) 0 Abk n N for i 2f0; 1; 2g. This vanishing also holds for n 0: use the shortexactsequence of sheaves 0 ! Z !2 Z ! k ! 0andH N) which gives j k) 0. 0 1 N n1 For each integer n 2 N, set Wn : W=I . Using the short exact sequences

12) 0 ! in ! Wn ! WnÀ1 ! 0 and the inductive assumption that kn in for 0 n N À 1, we conclude from the above vanishing of Ext groups that for any i 2f0; 1; 2g, the group Exti W ; j ) Abk NÀ1 N vanishes as well. This implies, by the short exact sequence of Lemma 2.7

13) 0 ! j ! k ! i ! 0 N N N that the homomorphism Exti W ; k ) ! Exti W ; i ) Abk NÀ1 N Abk NÀ1 N is an isomorphism for i 2f0; 1g and a monomorphism for i 2. For i 1, 90 Fabien Morel

this implies the existence of a sheaf of abelian groups GN which fits into a commutative square in Abk of the form:

0 ! kN ! GN ! WNÀ1 ! 0 14) ## k

0 ! iN ! WN ! WNÀ1 ! 0 in which the horizontal rows are exact, the bottom one being of the form 12), and which induces the Milnor morphism on the left.

LEMMA 3.11. Let X 2 Sm be such that j X) 0.Then the morphism k N

GNX) ! WNX) is an isomorphism, and thus so is kNX) ! iNX) by 14).

PROOF. The epimorphism G ! W has kernel j . We getfor any N N N X 2 Sm an exactsequence 0 ! j X) ! k X) ! i X) ! H1X; j ). By k N N N N definition for Y 2 Smk one has À Á j Y) : Ker È j ky)) !È j kz)) N y2Y 0) N z2Y 1) NÀ1 see 2.2) and by our assumptions which implies that, for any field Fjk, jNÀ1F) 0), we see that any open immersion U Y induces an isomorphism j Y) j U); this sheaf is thus flasque and H1X; j ) 0 for N N N any X by [10]. This implies the Lemma. p

This is for instance the case for X G because j )G ) m N m j k) È j k) 0 by H N) ). The same observation holds for a pro- N NÀ1 1 n duct Gm) by the formula used in the proof above. This also holds for any open subscheme X G )n because j X) j G )n) 0. m N N m As a consequence there exists a unique lift

Gm ! GN through GN ! WN of the obvious ``symbol''

Gm ! WN ; u 7!hui2WN

We denote by u : ZGm) ! GN the morphism of abelian sheaves induced by this lift.

We denote by À) : Gm ! ZGm) ; x 7! x) the morphism of sheaves of sets given by the ``inclusion of the base'' into the free sheaf of abelian Milnor's conjecture on quadratic forms and mod 2 motivic complexes 91

groups ZGm)onit.WeletF0 : X0 Gm Â Gm ! ZGm)denotethe morphism U; V) 7! U) À U : V 2) where we denote by U : Gm Â Gm ! Gm and V : Gm Â Gm ! Gm the projections to the first and second factor. We will also need the morphism

F1 : X1 Gm ! ZGm) defined by U) 7! U) À U)

Let X2 Gm Â Gm denote the open complement to the closed subscheme of Gm Â Gm defined by the equation U V 0. Finally we let F2 : X2 ! ZGm) denote the morphism U; V) 7! U) V) À U V) À U V):U:V)

LEMMA 3.12. For i 2f0; 1; 2; g, the composition

Fi u Xi ! ZGm) ! GN is trivial, i.e. constant with value the 0 section of GN.

PROOF. By the above observation, the morphisms

GNXi) ! WNXi) are isomorphisms. The Lemma now follows from the fact that the corresponding statements hold for the compositions Xi ! WN. These are indeed classical relations which hold in the Witt ring WF) of any field Fjk see [25, Corollary 9.4 p. 66]) and we conclude by the remark below, which implies that the WN are 0-pure. p

REMARK 3.13. If 0 ! M0 ! M ! M00 ! 0 is a shortexactsequence of sheaves in the Zariski topology, and that both M0 and M00 are strictly A1- invariant in the sense of Appendix A.2, it is clear that M itself is also 1 strictly A -invariant. This implies easily by induction, because the iN are 1 1 each strictly A -invariant, that the WN are strictly A -invariantas well. By Corollary A.5 these are also 0-pure.

COROLLARY 3.14. For each finite type field extension Fjk, the morphism Â uF) : ZF ) ! GNF) 92 Fabien Morel factors through the epimorphism ZFÂ) ! WF) and thus defines a natural transformation on F k:

UF) : WF) ! GNF)

PROOF. This is clear from the previous Lemma and the fact that for each field F of char 6 2 the relations huihu:v2i, huihÀui and huihvihu vihu v)uvi, with u 2 FÂ and v 2 FÂ u v 6 0inthe last case), generate the kernel of the epimorphism ZFÂ) ! WF) by loc: cit: p

LEMMA 3.15. The natural transformation on F k: U : W À ) ! G j N F k obtained above arises from a unique morphism of sheaves of abelian groups

U : W ! GN

PROOF. We observe by Remark 3.13 that the sheaves iN and WNÀ1 1 being strictly A -invariant, the sheaf GN, being an extension between two strictly A1-invariantsheaves is also strictly A1-invariant, thus 0-pure. Both sheaves in the statement of Lemma 3.15 being 0-pure, we reduce by Lemma

A.2 to checking that for any geometric discrete valuation v on Fjk 2Fk:

1) UF) maps WOv) WF) into GNOv) GNF). 2) the following induced diagram is commutative

WOv) ! GNOv) rv ##rv Ukv)) Wkv)) ! GNkv))

Itis well known that WOv) WF) is the subgroup generated by symbols Â hui with u 2Ov , see [25]. This shows that the morphism of sheaves

ZGm) ! W is onto on geometric discrete valuation rings Ov. Using the morphism of sheaves u : ZGm) ! GN this clearly implies 1) and 2). p

We let À)) : Gm ! ZGm) denote the morphism of sheaves of sets u 7! U)) : 1 À U) Milnor's conjecture on quadratic forms and mod 2 motivic complexes 93 and more generally for each n > 0 we let

n Gm) ! ZGm) denote the morphism of sheaves of sets

U1; ...; Un) 7! U1)) [ ...[ Un)) whose definition uses the obvious structure of sheaf of commutative rings on ZGm) with product denoted by [).

LEMMA 3.16. For each n 2 N the composition

n u en : Gm) ! ZGm) ! GN is trivial if n > N and its image is contained in the subsheaf kN GN for n N.In that case the induced morphism

N Gm) ! kN is the obvious symbol: U1; ...; UN) 7!fU1g ...fUNg.

PROOF. By our above observation, the morphisms

n n n n GNGm) ) ! WNGm) ) and kNGm) ) ! iNGm) ) are isomorphisms. Thus it suffices to check each statements on WN and iN respectfully, which is clear, because the composition

n en Gm) ! GN ! WN is the n-fold Pfister symbol. p

We can now easily combine the above results to prove the Milnor conjecture in weight N, finishing the proof of Theorem 1.3:

COROLLARY 3.17. The composition

n U I W ! GN is zero for n>Nand for n N induces a morphism

N N1 eN : iN I =I ! kN kerGN ! WNÀ1) This is a left inverse to the Milnor morphism in weight N which is thus an isomorphism. 94 Fabien Morel

n PROOF. Lemma 3.16 implies that the morphism I W ! GN is zero on fields Fjk 2Fk for n > N and is thus zero. The induced morphism N N1 I =I ! GN composed with GN ! WNÀ1 is zero on fields by Lemma 3.16 again and is thus zero. Thus we get

N N1 eN : iN I =I ! kN kerGN ! WNÀ1) Lemma 3.16 implies that the composition

N N N N1 Gm) ! I ! I =I ! kN is the N-symbol; thus the composition

s N N eN Gm) ! kN ! iN ! kN

sN eN is also the obvious one, so that the composition kN ! iN ! kN is the identity on fields, thus is the identity. p

REMARK 3.18. In fact to prove the Milnor conjecture in weight N itis not necessary to construct U as a morphism of sheaves. The natural transformation U on fields suffices, and one can adapt the end of the proof to that situation.

A. Complement on sheaves.

A.1. Elementary properties of 0-pure sheaves of sets.

We will notrepeathere the definition of 0-pure sheaves of setsgiven in 2.1. Let M be a 0-pure sheaf and denote by Mj : F !Ab F k k its restriction to finite type field extensions of k. For any geometric discrete valuation v on Fjk 2Fk, one has an associated subset MOv) MF) and a restriction map

rv : MOv) ! Mkv)) We will not try to describe explicitly the properties satisfied by these data which characterizes exactly the one coming from a 0-pure sheaf 5. We will only use the following Lemma:

5) Though this can be done Milnor's conjecture on quadratic forms and mod 2 motivic complexes 95

LEMMA A.1. Let M be a 0-pure sheaf of sets and N Mj be a sub- F k functor.For any irreducible X 2 Smk with function field F set NX) : NF) \ MX) MF) and for any X 2 Smk set NX) : Pa2X0) NXa) MX). Assume that for any geometric discrete valuation v on a finite type field extension Fjk, the map rv : MOv) ! Mkv)) sends MOv) \ NF) into Nkv)) Mkv)).

Then for any morphism f : Y ! XinSmk, the map M f ) : MX) ! ! MY) maps NX) into NY) and the correspondence X 7! NX) is a 0- pure sheaf of sets.

PROOF. We assume X and Y are irreducible with field of functions E and F. Also it suffices to prove the claim separately for f a smooth morphism and for f a closed immersion. Assume first f : Y ! X is smooth. The map M f ) : MX) ! MY) extends to a map ME) ! MF), and thus maps NX) MX) \ NE) into NY) MY) \ NF). Assume now f : Y ! X is a closed immersion. Let y 2 Y X be the generic point of Y and OX;y its local ring in X; a regular local ring of dimension codimX Y) d. Choose a regular sequence x1; ...; xd) generating the maximal ideal of OX;y. Because k is perfect, we get the existence of an open subscheme U X containing y and a flag

Y \ U Y1 ... Yd1 U of integral closed subschemes, smooth over k, such that Yi is the principal divisor in Yi1 defined by the function xi.We observe that NX) NU) \ MX) and NY) NY1) \ MY). Itis thus sufficient to check that MU) ! MY1) maps NU) into NY1) and using the above flag we reduce to the case f is a closed immersion with Y a principal divisor in X defined by a function. Denoting by v the discrete valuation on E associated to Y we see that M f ) : MX) ! MY) extends to rv : MOv) ! Mkv)) MF); but then by construction and assumption NX) MX) \ NE) MX) \ NOv) maps to MY) \ NF) NY). The factthat X 7! NX) is a 0-pure sheaf is proven as follows. We may assume X irreducible. Let fUig be a finite open covering of X. To check that the obvious diagram ! NX) PiNUi) ! Pi; jNUi \ Uj) is left exact follows easily from the fact that it imbeds into the corresponding diagram for M. The restis easy. p

Now we can describe morphisms between 0-pure sheaves in analogous 96 Fabien Morel terms. Let M and N be 0-pure sheaves of sets and let f : N ! M be a morphism of sheaves. By restriction to F k this defines a natural transformation fj : NjF ! MjF F k k k between functors F ! Sets. Moreover fj has the following two prop- k F k erties, for any geometric discrete valuation v on Fjk 2Fk:

1) fF) maps NOv) into MOv) NF). 2) the following induced diagram is commutative

NOv) ! MOv)

rv ##rv fkv)) Nkv)) ! Mkv)) Conversely:

LEMMA A.2. Given 0-pure sheaves of sets M and N, the above correspondence defines a bijection from the set HomkN; M) of morphisms of sheaves of sets on Smk from N to M to the set CN; M) consisting of natural transformations

f : NjF k ! MjF k satisfying, for any geometric discrete valuation v, properties 1) and 2) above.

PROOF. By the 0-purity property, the injectivity of the map

HomkN; M) !CN; M) is clear. Now let f : NjF k ! MjF k be a natural transformation in CN; M). Then for each irreducible X 2 Smk, with function field F, f induces by property 1) and Definition 2.1 a morphism

fX) : NX) \y2X1) NOX;y) !\y2X1) MOX;y) MX) It only remains to show that the fX) altogether define a morphism of sheaves, that is to say a natural transformation on functors on Smk.To do this one proceeds using the same argument as in the proof of Lemma A.1 above: to check the property for pull-back along smooth morphisms one considers everything embedded in sections over the corresponding function fields and to check the property for pull-back along closed immersions one reduces to the case of a principal divisor using property 2). p Milnor's conjecture on quadratic forms and mod 2 motivic complexes 97

A.2. Strictly A1-invariant sheaves and 0-purity

1 DEFINITION A.3. A sheaf M 2Abk is said to be strictly A -invariant if it is Zariski strictly A1-invariant and if for any smooth k-variety X the comparison homomorphism

Ã Ã H X; M) ! HNisX; M) from Zariski cohomology to Nisnevich [21] cohomology is an isomorphism.

Any homotopy invariant sheaf with transfers [28, Definition 3.1.10] is strictly A1-invariantby [ loc.cit. , Theorem 3.1.12].

1 LEMMA A.4. For any cycle module MÃ the sheaf M0 is a strictly A - invariant sheaf.

1 PROOF. We know that it is Zariski strictly A -invariantby [24, §9]. The fact that it is a sheaf in the Nisnevich topology and that the comparison homomorphism

Ã Ã HZarX; M0) ! HNisX; M0) is an isomorphism follows from [6, Theorem 8.3.1 and §7.3 Ex 5)]; p Lemma 5.5.4 of [18] gives:

1 LEMMA A.5. A strictly A -invariant sheaf is 0-pure.

1 COROLLARY A.6. A morphism M ! N of strictly A -invariant sheaves which induces an isomorphism on fields Fjk is an isomorphism.

PROOF. By Lemma A.5 both M and N are 0-pure. Thus such a morphism M ! N is a monomorphism of sheaves. Let C be its cokernel. It is clearly a strictly A1-invariantsheaf thusa 0-pure sheaf again by Lemma A.5. Moreover itvanishes on each field Fjk. Itis thus0. p

A.3. Motivic complexes and unramified Milnor K-theory

The cohomology sheaves HiZn)) of the Suslin-Voevodsky motivic complex Zn) in weight n vanish by construction for i > n. A standard result of Suslin-Voevodsky [27] gives the computation of HnZn))F) for 98 Fabien Morel each field Fjk. More precisely, we know from [27] that there is a canonical quasi-isomorphism Z1) Gm[ À 1]. This gives in particular for each field Fjk a canonical isomorphism FÂ H1SpecF); Z1)) It is shown in [27, Theorem 3.4] that it induces, using the product 15) Zn) Zm) ! Zn m) a canonical isomorphism of graded rings

M n n FÃF) : KÃ F) ÈnH SpecF); Zn)) ÈnH Zn))F)

Altogether these isomorphisms define an isomorphism of functors on F k

M n Fn : K H Zn))j n F k The following result [33, Corollary 2.4 and proof)] and [7]) extends naturally the previous isomorphism to an isomorphism of sheaves:

THEOREM A.7. Let n 2 N be an integer. 1) There exists a unique isomorphism of sheaves

M n Fn : Kn H Zn)) which induces the natural isomorphism of Suslin-Voevodsky on the field extensions of k. 2) For each integer m > 0, the above isomorphism induces an isomorphism in Abk): M n Kn =m H Z=mn)) We include the proof for the comfort of the reader:

n n PROOF. 1) In the proof we simply denote by H the sheaf of H Zn)).

We firstconstruct Fn using Lemma A.2. We thus have to check Properties 1) and 2) of that Lemma. Uniqueness is clear.

Fix a discrete valuation v on Fjk 2Fk of geometric type. Each element of HnF) of the form

Fnfu1g ...fung)

Â 1 with the ui's in Ov , can be expressed, using the morphism H ...... H1 !Hn induced by 15) as the cup-product Milnor's conjecture on quadratic forms and mod 2 motivic complexes 99

n n F1fu1g) [ ...[ F1fung) so that it lies in H Ov) H F), because each 1 Â symbol F1fuig) lies in H Ov) Ov . Moreover we clearly have the formula

rvFnfu1g ...fung)) rvF1fu1g) [ ...[ F1fung))

F1rvfu1g)) [ ...[ F1rvfung)) Fnrvfu1g ...fung)) Properties 1) and 2) of Lemma A.2 follow immediately from that observation and from the result of Bass-Tate [5, Prop. 4.5 b) p. 22], see also M 2.10, that any x 2 Kn Ov) is a sum of symbols of the previous form. This defines

M n Fn : Kn !H The n-th cohomology sheaf Hn being a homotopy invariant sheaf with transfers by [28, Definition 3.1.9], it is strictly A1-invariantby [ loc.cit. , M Theorem 3.1.12]; so is Kn by Lemma 2.7. We conclude that Fn is an isomorphism by Corollary A.6.

2) is an easy consequence of 1). p

M REMARK A.8. As a consequence, the sheaves Kn and kn have canonical structures of homotopy invariant sheaf with transfers, given by the above isomorphisms. By [7] any sheaf arising from a cycle module has a canonical structure of homotopy invariant sheaf with transfers; it is also proven in loc.cit. that the above isomorphisms of sheaves are compatible with these additional structures.

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Manoscritto pervenuto in redazione il 25 aprile 2005.