Milnor’s conjecture on quadratic forms and <span class="mathjax-formula">$~mod \ ; 2$</span> motivic complexes

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

FABIENMOREL(*)

ABSTRACT- LetFbe a field of characteristic different from2. 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 overF, firstproven by Orlov, Vishik and Voevodsky. Our approach also relies on Voevodsky's affir- mation of Milnor's conjecture on the mod 2 Galois cohomology of fields of characteristic different from2, but, besides this fact, we only use some ele- mentary homological algebra in the abelian category of Zariski sheaves on the category of smoothk-varieties, involving classical results on sheaves of Witt groups, Rost's cycle modules and sheaves of 0-equidimensional cycles.

1. Introduction.

LetFbe a field of characteristic6ˆ2. In this paper we give a new proof of Milnor's conjecture identifying the mod 2 Milnor K-theory ofFto the graded ring associated to the filtration of the Witt ringW(F) of anisotropic quadratic forms overF 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 operationsQ_i.

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 the previous ones. Fix a perfectbase fieldk, letSmkdenote the category of smoothk-schemes and letAbk denote the abelian category of sheaves of abelian groups in the Zariski topology on Sm_k. Besides Voe- vodsky's resultwe will only use some elementary homological algebra in

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

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 denote byW(F) the Witt ring of anisotropic quadratic forms overF[14, 25]. The kernel of the mod 2 rank homomorphismW(F)!Z=2 is denoted by I(F) and calledthe fundamental ideal of W(F). For each integern, t hen-th power ofI(F) is denoted byIⁿ(F).

For any unitu2F, the symbolhuiwill denote the class inW(F) of t he quadratic form of rank oneu:X²and the symbolhhuiiwill denote the class of thePfister form1‡ h ui ˆ1 hui 2W(F), indeed an elementinI(F).

The followingSteinberg relation:

hhuii:hh1 uii ˆ0 …1†

which holds inI²(F) foru2F f1g, is a reformulation of the well-known relationhui ‡ h1 ui ˆ1‡ hu:(1 u)i(see [25] for instance). One also has the following equality for any pair (u;v)2(F)²

hhuii ‡ hhvii hhu:vii ˆ hhuiihhvii 2I²(F) which shows thatu7! hhuiiinduces a group homomorphism

F=(F)²!I(F)=I²(F) …2†

The Milnor K-theory K^M(F) of the field F introduced in [15] is the quotient of the tensor algebraTensZFon the abelian group of unitsFby the two sided ideal generated by tensors of the form u(1 u), for u2F f1g. The mod 2 Milnor K-theory ofFis the quotient

k(F):ˆK^M(F)=2

As observed in [15] the above computations give a canonical graded ring homomorphism extending (2)

sF:k(F)! nIⁿ(F)=I^n‡1(F)

which we call theMilnor homomorphismand which was shown inloc: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:

THEOREM1.1. For any field F of characteristic not2and any integer n the Milnor homomorphism

s_n(F):k_n(F)!Iⁿ(F)=I^n‡1(F) is an isomorphism.

REMARK 1.2. The powers of the fundamental ideal of F form alto- gether a commutative graded ring denoted byI(F), in facta gradedW(F)- algebra becauseI⁰(F)ˆW(F). We have a canonical morphism of graded W(F)-algebras TensW(F)(I(F))!I(F) where the left hand side denotes the tensor algebra over W(F) of t heW(F)-moduleI(F). The relation (1) above, which holds inI²(F), shows that this morphism factors trough the quotient algebra K^W(F):ˆTens_W(F)(I(F))=hhuii hh1 uii by the two- sided ideal generated by the tensorshhuii hh1 uii 2I(F)_W(F)I(F). It is quite natural to call K^W(F) t he Witt K-theory of F. In [17] we have proven that the induced morphism ofW(F)-algebras

K^W(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 con- versely that Theorem 1.1 can be recovered from the isomorphism K^W (F)I(F) by tensorization byZ=2 overW(F).

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 tos_n(F), the so-calledn-th invariant of quadratic forms

e_n(F):Iⁿ(F)=I^n‡1(F)!k_n(F)

The statement in the Theorem corresponding to a fixed integernwill be called in the sequelthe Milnor conjecture on the Witt ring of F in weight n.

Denote by H(F;Z=2(n)) the mod 2 motivic cohomology groups ofF in weight n as defined by Suslin-Voevodsky in [27]. These groups H(F;Z=2()) altogether form a bigraded commutative ring. We have the particular elementt:ˆ 12m₂(F)ˆH⁰(F;Z=2(1)).

Our main resultis:

THEOREM 1.3. Let k be a perfect field of characteristic 6ˆ2 and let N>0be an integer.Assume that the following assumptions hold:

H₁(N):Milnor's conjecture on the Witt ring of k holds in weightsN.

H2(N 1): For any finite type field extension Fjk and for any integer 1nN 1, the cup product byt

H^{n 1}(F;Z=2(n 1))!^t[H^{n 1}(F;Z=2(n)) is onto.

Then Milnor's conjecture on the Witt ring holds for any field extension Fjk in weightsN.

PROOFthat THEOREM1.3)THEOREM1.1. The groupHⁱ(F;Z=2(n)) is the group of sections onF of thei-th cohomology sheafHⁱ(Z=2(n)) of an explicitchain complex¹Z=2(n) inAb_k, the mod 2 motivic complex in weight ndefined in [27]. The ring structure is induced by explicit morphisms of complexesZ=2(n)Z=2(m)!Z=2(n‡m) byloc.cit.The cup productby tis thus induced by a morphism of the form

t[:Z=2(n 1)!Z=2(n) …3†

Voevodsky's main theorem [32, 33] implies the Beilinson-Lichtenbaum conjecture on mod 2 motivic cohomology, which identifies, for i2 f0;. . .;ng, the sheaf Hⁱ(Z=2(n)) to the sheaf associated in the Zariski topology to the presheafX7!Hⁱ_et(X;mⁿ₂ ), see [27, 32]. This identification being compatible to the cup-product and because the cup-product by t2m₂(F)ˆH⁰_et(Spec(k);m₂) in eÂtale cohomology induces isomorphisms Hⁱ_et(X;mⁿ₂ )Hⁱ_et(X;m^(n‡1)₂ ), this implies that the morphism (3) induces isomorphisms on cohomology sheaves of degreesn 1, for anyn. This establishesa fortioriH2(N 1) for anyN.

Now choose for the field k a prime field of characteristic 6ˆ2. The Milnor conjecture on the Witt ring holds forkby [15], which provesH₁(N) for anyN. Theorem 1.3 now implies Milnor's conjecture on the Witt ring of field extensionsFjkin any weight, establishing Theorem 1.1. p REMARK1.4. It is possible to prove Theorem 1.3 withoutH₁(N) but this would make the exposition more complicated.

REMARK1.5. Our method clearly emphasizes that Voevodsky's proof of Milnor's conjecture on mod 2 Galois cohomology in weights N 1 implies the Milnor conjecture on the Witt ring in weightsN.

(¹) for us ``chain complex'' means that the differential is of degree 1

In the rest of the paper, we will concentrate on the proof of Theorem 1.3. We now fix once for all a perfectfieldkof characteristic6ˆ2. Letus describe our strategy. Recall that we denote byAb_kthe abelian category of sheaves of abelian groups in the Zariski topology on the categorySm_k of smoothk-schemes. We denote by

K^M_n 2 Abk

the sheaf ofunramified Milnor K-theoryin weightnconstructed in [13, 24]. Its fiber on a fieldFjkisK_n^M(F); see section 2.2 for a recollection. We will denote by

k_n:ˆK^M_n=2

the cokernel inAb_k of the multiplication by 2 onK^M_n. We will denote by W2 Ab_k

the associated sheaf to the presheaf of Witt groups on Sm_k: X7!W(X) constructed in [14] for instance.

We will then show how the filtration of the Witt ringW(F)ˆW(F) of each field extensionFjkby the powers of their fundamental ideal naturally arises from a decreasing filtration by sub-sheaves

. . .Iⁿ. . .W

For each n, we seti_n:ˆIⁿ=I^n‡1 2 Ab_k. The Milnor homomorphism for fields then arises for eachnfrom an epimorphism of sheaves:

s_n:k_n!i_n

called theMilnor morphismin weightnwhose 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 1, so that j_n ˆ0 for 0nN 1. Then:

(1) Using H₁(N) and H₂(N 1) prove the vanishing of the groups Extⁱ_Abk(k_n;j_N) for 0nN 1 andi2 f0;1;2g.

(2) Using elementary homological algebra in Ab_k, deduce the Milnor conjecture on the Witt ring in weightNfor all field extensionsFjkfrom(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

the Suslin-Voevodsky motivic complexes Z=2(n). The firstone is the vanishing forn>0 of the groups of morphisms in the derived category of Ab_k of the form

Hom_D(Abk)(Z=2(n);M[])

for some type of sheavesM, like j_N, which are birational invariant. This is done by adapting a simple geometric argument due to Voevodsky [31]. The second type of information concerns the cohomology sheavesHⁱ(Z=2(n)).

These vanish for i>n, for iˆn we have the Suslin-Voevodsky iso- morphism²:k_n Hⁿ(Z=2(n)) and hypothesisH2(N 1) provides an epi- morphism k_{n 1}! H^{n 1}(Z=2(n)) 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):

E^p;q₂ ˆExt^p_Abk(H^q(Z=2(n));M))Hom_D(Abk)(Z=2(n);M[p q]) p PROOF OF STEP(2)will be given in section 3.4. Here is a sketch. Assume the Milnor conjecture on the Witt ring in weightsN 1 for fieldsFjk and the vanishing in(1)are both established.

For each integern, setW_n :ˆW=I^n‡1. The sheafW_{N 1}thus admits a filtration with associated subquotients of the formIⁿ=I^n‡1ˆi_n, for some 0nN 1. The Milnor conjecture on the Witt ring in weightsN 1 gives isomorphismsk_ni_n, and one deduces from the long exact sequence ofExtgroups and(1)the vanishing

Extⁱ_Abk(W_{N 1};j_N)ˆ0 fori2 f0;1;2g. Using the short exact sequence 0! j_N!k_N!i_N!0 …4†

one deduces that the morphismExt¹_Abk(W_{N 1};k_N)!Ext¹_Abk(W_{N 1};i_N) is an isomorphism. There exists thus a commutative diagram

0 ! k_N ! G_N ! W_{N 1} ! 0

# # #

0 ! i_N ! W_N ! W_{N 1} ! 0

…5†

with exact horizontal rows where the bottom horizontal row is the obvious

(²) see [33, 7] and section A.3

one and where the left vertical map is the Milnor epimorphism. Next we show that the canonical epimorphism

W!W_N

canonically lifts trough the epimorphismG_N!W_Nto a morphism W!G_N

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 sub- schemes of productofG_m as relations''; taking into account the inductive assumption that j_{N 1}ˆ0 which implies j_N is a birational invariant sheaf, the existence of the lifting follows easily (see section 3.4).

To finish the proof one observes that the composition I^NW!

!G_N!W_{N 1}is trivial by construction, and that the induced morphism I^N!k_Nˆ(Ker:G_N!W_{N 1})

induces a left inverse to the Milnor morphism in weightN e_N:I^N=I^N‡1!k_N 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 fromF_kto the category of abelian groups implicitly considered by Serre in [9], instead of the more elaborated categoryAb_k. 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 schemeX and an integeri2N, we will denote by X⁽ⁱ⁾the set of points of codimensioniof the schemeX.

Given x2X2Sm_k, we will denot e byO_X;x the local ring ofX atx.

For a given presheaf of sets M on Sm_k we will denote by M(O_X;x), or simply by M_x, t he fiber of M at x2X, in the Zariski topology. Im- portant examples for us will be that of a finite type field extension Fjk2 Fk, considered as the local ring of the generic points of its models in Sm_k or that of a geometric discrete valuation ring. Such a discrete valuation ringO_v is one with field of fractions Fof finite type over k and which is isomorphic to the local ring of some X2Sm_k at some pointxof codimension 1. The associated valuationv onF will be called a geometric discrete valuation on F; its residue field will be denoted by k(v).

2. The Milnor morphism as a morphism of sheaves 2.1 ±Unramified Witt groups and related sub-sheaves

DEFINITION2.1. A sheaf of sets M on Sm_kin the Zariski topology is said to be0-pure if for any irreducible X2Smk with field of functions F the canonical map

M(X)!M(F) is injective and induces a bijection

M(X)ˆ \_y2X(1)M(O_X;y)M(F)

Letus denote by W:Smk ! Ab;X7!W(X) the presheaf of Witt groups onSm_k; 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]:

THEOREM2.2. (Ojanguren-Panin) The sheafWis0-pure.

PROOF. Fix an irreducibleX2Sm_k with field of functionsF. Clearly, becauseW is a sheaf in the Zariski topology, the morphism

W(X)!P_x2XW(O_X;x)

is injective. Now byloc.cit.for eachx2Xthe morphism W(OX;x)ˆW(OX;x)!W(F)ˆW(F) is injective. This implies that the canonical morphism

W(X)!W(F)

is injective. Now again because W is a sheaf in the Zariski topology, the previous injection identifies W(X) with \_x2XW(O_X;x)ˆ \_x2XW(O_X;x) W(F). By Theorem A of loc.cit., for each point x2X one has W(OX;x)ˆ \yW(OX;y) whereyruns over the set of all prime ideal inOX;xof height one, that is to say pointsy2X⁽¹⁾whose closure containsx. Itfollows then that

W(X)ˆ \_x2XW(O_X;x)ˆ \_y2X⁽¹⁾W(O_X;y)W(F)

which establishes the Theorem. p

Let n2N be an integer. For any irreducibleX2Smk with function

fieldF we set

Iⁿ(X)ˆIⁿ(F)\W(X)W(F) We extend the definition to anyX2Smk by setting

Iⁿ(X):ˆ _x2X⁽⁰⁾Iⁿ(Xx) _x2X⁽⁰⁾W(Xx)ˆW(X)

whereXxXdenotes the irreducible component containingx2X⁽⁰⁾. THEOREM2.3. Given any morphism f :X!Y in Sm_kand any n2N the morphism

W(f):W(Y)!W(X)

maps the subgroup Iⁿ(Y)W(Y) into Iⁿ(X)W(X).Thus the corre- spondence X7!Iⁿ(X) admits a unique structure of presheaf of abelian groups, denoted by Iⁿ, such that the inclusions Iⁿ(X)W(X) define a monomorphism of presheavesIⁿ W.Moreover,Iⁿ is a0-pure sheaf.

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

r_v:W(Ov)ˆW(Ov)!W(k(v))ˆW(k(v))

mapsIⁿ(Ov)ˆIⁿ(F)\W(Ov) intoIⁿ(k(v)). LetO^h_vdenote the henselization ofO_v,F_v^hits fraction field. By naturality we have the following commutative diagram of Witt rings

W(O_v) ! W(O^h_v)

# #

W(k(v)) ˆ W(k(v))

andIⁿ(Ov) clearly maps toIⁿ(O^h_v):ˆIⁿ(F_v^h)\W(O^h_v)W(F^h_v). Lemma 2.4

(4) below implies the result. p

LEMMA2.4. Let v be a discrete valuation on a field F, with residue field k(v)of characteristic not2, and letpbe a uniformizing element for v.Then 1) The ring homomorphism W(O_v)!W(F)is injective and the dia- gram0!W(O_v)!W(F)!^@vp W(k(v))!0is a short exact sequence, where

@_v^p is the residue morphism of[15, 25].

2) For each n>0 the residue morphism @_v^p :W(F)!W(k(v)) maps Iⁿ(F)onto I^{n 1}(k(v)).

3)LetO^h_v denote the henselization ofOv, F_v^hits fraction field.Then the ring homomorphism r:W(O^h_v)!W(k(v)) is an isomorphism and the canonical W(O^h_v)-algebra homomorphism

W(O^h_v)[T]=(T² 1)!W(F^h_v); T7! hpi an isomorphism (Springer).

4)For each n>0the intersection Iⁿ(F_v^h)\W(O^h_v)is equal to the n-th power Iⁿ(O^h_v)of I(O^h_v)and the exact sequence of1) induces an exact se- quence

0!Iⁿ(O^h_v)!Iⁿ(F^h_v)!^@vp I^{n 1}(k(v))!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 mapsIⁿ(F) intoI^{n 1}(k(v)). The surjectivity is easy.

The first isomorphism in statement 3) isloc.cit.Theorem 2.4 and the second one is Corollary 2.6, due to Springer.

The last statement is proven inloc.cit.§5 in the case of complete dis- crete 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- ductively thatIⁿ(F^h_v)ˆr ¹(Iⁿ(k(v))) hhpii:r ¹(I^{n 1}(k(v))). p REMARK2.5. LetAbe a regular local ring in which 2 is invertible with fraction fieldF. Assume thatW(A)!W(F) is injective (for instance if it contains a field of characteristic 6ˆ2 by [22]). We don'tknow in general whether or not for anyn2Nthe group

Iⁿ(A):ˆIⁿ(F)\W(A)W(A)

is always the n-th power of the ideal I(A):ˆI(F)\W(A), 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;. . .;u_n)2 2(F)ⁿof units we will denote byfu₁g. . .fu_ng 2K^M_n(F) its image through the obvious map (F)ⁿ !K_n^M(F). Recall from [15] that for any discrete valuationvon the fieldF, with residue fieldk(v) there exists one and only one homomorphism:

@v:K_n^M(F)!K^M_{n 1}(k(v)) (6)

satisfying

@v(fpg:fu2g. . .fung)ˆ fu2g. . .fung

ifv(p)ˆ1 andv(u_i)ˆ0 for each i2. Foru2 O_v the notationu2k(v) means the image of uink(v). This homomorphism is called the residue homomorphismassociated tov.

For any smooth k-variety X, we let K^M_n(X) denote the kernel, in- troduced in [13], of all the residue homomorphisms associated to points in Xof codimension 1:

K^M_n(X):ˆKer

_x2X⁽⁰⁾K_n^M(k(x)) ! P@x

_y2X⁽¹⁾K^M_{n 1}(k(y))

The correspondenceX7!K^M_n(X) is turned into a Zariski sheaf onSm_k by Rostin [24]. This sheaf will be denoted by K^M_n and called the sheaf of unramified Milnor K-theory in weight n.

More generally, let us recall briefly the notion ofcycle moduleoverk from [24]. We justremind thata cycle moduleM is a triple (M;W; @v) consisting of a functor

M:F_k! Ab

withAbbeing the category of graded abelian groups, a transfer morphism W :M(F)!M(E) of degree 0 for eachfiniteextensionEFinFkand a residue morphism@v:M(F)!M(k(v)) of degree 1 for anygeometric discrete valuation vonFjk2 F_k. These data satisfy some axioms which we will notrecall here; seeloc.cit.p. 329 and p. 337.

The Milnor K-theory groupsK^M, 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 modmMilnor K- theory groupsK^M=malso form a cycle module for any integerm. 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 Fjk2 Fk, and the axioms of [24, p. 329 and p. 337] follow formally. In the sequel,wewillsimplydenotebykthecycle moduleofmod2 MilnorK-theory.

Given a cycle moduleMandX2Sm_kwe will denote byA⁰(X;M) t he group of unramified sections ofM₀onX, by which we mean³the kernel of

(³) Here we slightly differ from [24] where Rost considers rather the direct sum n2ZA⁰(X;M(n)), see below.

the sum of residue morphism at points of codimension 1:

_x2X⁽⁰⁾M₀(k(x)) ! P@x

_y2X⁽¹⁾M ₁(k(y))

In [24, §12] these groups are turned canonically into a Zariski sheaf on Sm_k which we denote by M₀. A sheaf of abelian groups in the Zariski topology onSmk 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 formM₀. It is easy to check that such sheaves are 0-pure in the sense of Defini- tion 2.1.

REMARK2.6. By DeÂglise [7] the sheafM₀admits 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.

LEMMA2.7. For any (termwise) short exact sequence0!M⁰!M!

!M⁰⁰!0of cycle modules, the diagram

0!M⁰₀!M₀!M⁰⁰₀!0

is a short exact sequence of sheaves in the Zariski topology.

PROOF. We will freely use the notations from [24]. For anyY2Sm_k, or any localizationYof a smoothk-scheme, and any cycle moduleNis defined a cochain complexC(Y;N), seeloc:cit:p. 355 and p. 359. A shortexact sequence of cycle modules 0!M⁰ !M!M⁰⁰!0 then induces a short exactsequence of cochain complexes

0!C(Y;M⁰)!C(Y;M)!C(Y;M⁰⁰)!0

and a corresponding long exactsequence of associated cohomology groups.

ButTheorem (6.1) ofloc:cit:establishes that for anyx2X2Sm_kand any cycle moduleN, the cochain complexC(Spec(O_X;x);N) has trivial coho- mology in>0 degrees. The shortexactsequence

0!C(Spec(O_X;x);M⁰)!C(Spec(O_X;x);M)!C(Spec(O_X;x);M⁰⁰)!0 thus produces a shortexactsequence

0!A⁰(Spec(OX;x);M⁰)!A⁰(Spec(OX;x);M)!A⁰(Spec(OX;x);M⁰⁰)!0 in other words, of the form 0!M⁰₀(OX;x)!M₀(OX;x)!M⁰⁰₀(OX;x)!0,

which gives the result. p

For any cycle moduleM and any integern2Z, we denote byM(n) the cycle module obtained in the obvious way by setting (M(n))_mˆMm‡n

and endowed with the corresponding shifted data. For instance, for each n2Nwe haveK^M_n ˆK^M (n)₀and we define the sheaf ofunramified mod2 Milnor K-theory in weight nas

k_n :ˆk(n)₀ˆK^M_n=2

More generally for a cycle moduleMwe will simply setM_n:ˆM(n)₀. DEFINITION 2.8. Let n2Z be an integer.We will say that a cycle module Mis of weightsn if and only if for any Fjk2 Fkthe group Mi(F) vanishes for i< n.

Observe that ifMis in weightsn,M(m) is in weightsn‡m. Also Mis in weightsnif and only if for anyFjk2 F_k,M _{n 1}(F)ˆ0. In that case the sheavesM _mall vanish formn‡1. The cycle modulesK^Mand k are in weights0.

LEMMA2.9. Let X2Smk and let ZX be a closed subscheme every- where of codimension n, then for any sheaf M coming from a cycle module of weightsn 1the groups

H(X;(X Z);M) vanish for any.

PROOF. This follows from the results of [24]: for any cycle moduleN

one can compute H(X;X Z;N₀) as the cohomology of the complex C(X;N)=C(X Z;N) which is of the form

_x2X(0)=xZN₀(k(x))!. . .! _x2X(i)=xZN _i(k(x))!. . .

Now because the codimension ofZisn, that complex is trivial up to co- dimensionn. But ifNis of weightsn 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:

1) The diagram 0!K^M_n(Ov)!K^M_n(F)!^@v K^M_{n 1}(k(v))!0 is a short

exact sequence.Moreover, given a uniformizing element p2 Ov the fol- lowing diagram is commutative

K^M_n(Ov) K^M_n(F)

r_v # # @v(fpg [( ))

K^M_n(k(v)) ˆ K^M_n(k(v))

2)For any unit u2 O_v the symbol fug 2K^M₁ (F)lies inK^M₁ (O_v)and moreover the groupK^M_n(O_v)is generated by symbols of the form

fu1g. . .fung with the u_i's inO_v.

2.3 ±Sheafifying the Milnor homomorphism

For each integern2Nand any fieldFjk2 F_k we set in(F):ˆIⁿ(F)=I^n‡1(F)

We denote byi(F) the corresponding graded abelian group.

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

Fk ! Ab;F7!i(F) such that the Milnor homomorphisms

s(F):k(F)!i(F)

altogether define a morphism of cycle modules, an epimorphism indeed.

PROOF. We will show below that the transfers morphisms W:k(F)!k(E)

for finite extensionsEFinF_kand the residue morphisms

@v:k(F)!k 1(k(v))

for geometric discrete valuationsvonFjk2 F_k, descend to morphisms on i. Endowed with these induced morphism,ibecomes a cycle module be- cause the axioms are automatic consequences of the corresponding ones for mod 2-Milnor K-theory. Moreover, by construction, the Milnor homo- morphism preserves the cycle module structures, defining an epimorphism

of cycle modules of the formk!i. The uniqueness of the structure is clear.

To prove our claim for the residue morphisms, letvbe any geometric discrete valuation onFjk2 F_k, andpbe a uniformizing elementforv. Let us denote by

@^p_v :in(F)!in 1(k(v))

the homomorphism induced by the one on Iⁿ(F) 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

k_n(F) !^@v k_{n 1}(k)

# #

in(F) !^@vp in 1(k)

is commutative. This also shows that@^p_v doesn'tdepend onp.

LetEFbe any finite extension of fields andnbe an integer. Assume first this extension is purely unseparable. Then in that case, because the characteristicpis odd, one hashx^pi ˆ hxiinW(F) andfx^pg ˆ fxg 2k₁(F);

for any x2F, some power x^pn is an E. Thus the extension of scalars W(E)!W(F) and k(E)!k(F) are both epimorphisms. But the same formula shows that the Frobenius induces the identity morphism W(E)ˆW(E); as some iterated of the Frobenius onFmaps toEFthe factorization of the identity asW(E)!W(F)!W(E) this shows the ex- tension of scalars is in fact an isomorphism. The degree [F:E], being a power ofp, is odd so that the standard property of the transfer morphism implies it is a monomorphism on mod 2 Milnor K-theory, showing that k(E)!k(F) is an isomorphism as well. This clearly imply our claim on the transfers in that case.

Assume now the extension EF is separable. Let t:F!Edenote the trace morphism and t:W(F)!W(E), q7!tq the corresponding Scharlau transfer [25, §2.5]. By Arason [1, Satz 3.3]tmapsIⁿ(F) intoIⁿ(E) and thus induces a natural morphismt:i_n(F)!i_n(E).

To check that the diagram

k_n(F) !^W k_n(E)

# #

in(F) !^t in(E) (7)

is commutative one proceeds as follows. Let EE⁰ be a separable al- gebraic extension of odd degree, and F⁰:ˆE⁰_EF, a finite separable

algebra overE⁰. Clearly the square above maps by extension of scalars to the corresponding one involvingE⁰andF⁰. ButE!E⁰andF!F⁰being of odd degree, the extension morphisms in(E)!in(E⁰) is a mono- 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 toE⁰. If we choose for EE⁰ separable alge- braic extension such that the absolute Galois group ofE⁰is a 2-Sylow of that ofE, we may assume furt her t hat E⁰!F⁰admits a finite increasing filtrationE⁰E⁰₁. . .E⁰_rˆF⁰ by quadratic extensions. Thus we re- duced our claim to the commutativity of (7) in caseEFis a quadratic extension. By [5, Corollary 5.3 p. 29],k(F) is generated as a module over k(E) byk1(F). So it suffices (by the projection formula both forW and t) to prove it fornˆ1, which is an easy computation (use for instance

[25, Lemma 5.8]). p

For each n2N,i_n denotes the associated sheaf to the cycle module i(n). The Milnor morphism of cycle modules

s:k!i

being an epimorphism itdefines for eachn2Nan epimorphism of sheaves in the Zariski topology

s_n:k_n!i_n

by Lemma 2.7. This morphism is called theMilnor morphismin weightn.

2.4 ±The isomorphism Iⁿ=I^n‡1ˆi_n

The following resultjustifies a posterioriour quick definition ofi_n in the introduction asIⁿ=I^n‡1.

THEOREM2.12. Let n0be an integer.

1) The canonical transformation between functors on F_k: Iⁿ(F)!

!in(F)arises from a unique morphism of sheavesIⁿ!i_n. 2)The kernel of this morphism is the subsheafI^n‡1Iⁿ.

3) This morphism is an epimorphism (in the Zariski topology) and thus induces a canonical isomorphismIⁿ=I^n‡1i_n.

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

Iⁿ(F)!i_n(F)

commute to residue morphisms (see the proof of Theorem 2.11). The second property means that for any geometric discrete valuationvonFjk2 Fkthe following diagram commutes

Iⁿ(Ov) ! i_n(Ov)

# #

Iⁿ(k(v)) ! in(k(v))

This follows again from Lemma 2.4 by mappingOvtoO^h_v. This defines the morphismIⁿ!i_n for eachn.

2) It is sufficient to prove that for eachx2X2Sm_k the diagram 0!I^n‡1(O_X;x)!Iⁿ(O_X;x)!i_n(O_X;x)

(8)

is an exactsequence.

Choose for each pointyof codimension 1 inSpec(O_X;x) a uniformizing element p_y of the associated discrete valuation. From the fact that W(O_X;x)ˆW(O_X;x) is the kernel of all the residue morphisms @_y^py, from Theorem 2.2 and from Lemma 2.4 one constructs the commutative diagram in whichF is the fraction field ofOX;xand the right horizontal maps are residues:

0 ! I^n‡1(O_X;x) ! I^n‡1(F) ! _y2Spec(OX;x)⁽¹⁾Iⁿ(k(y))

# # #

0 ! Iⁿ(O_X;x) ! Iⁿ(F) ! _y2Spec(OX;x)⁽¹⁾I^{n 1}(k(y))

# # #

0 ! i_n(O_X;x) ! in(F) ! _y2Spec(OX;x)⁽¹⁾in 1(k(y))

# # #

0 0 0

The horizontal rows and the last two vertical rows being each exact se- quences, the claim follows.

3) We want now to establish the surjectivity of the right homomorphism in (8). Assume firstthatk is infinite. By [8, Proposition 4.3], the group K^M_n(O_X;x) is generated by symbolsfu1g. . .fungwith theu_i's units inO_X;x. Thus so is the (quotient) groupi_n(O_X;x). Now we conclude because these symbols fu₁g. . .fu_ng in i_n(O_X;x) are clearly in the image of Iⁿ(O_X;x)!

!i_n(O_X;x) which is thus onto.

Assume now thatkis finite. LetkKbe an algebraic extension withK infinite, perfect and of odd degree. By Lemma 2.14 below we see that the cokernel ofIⁿ(O_X;x)!i_n(O_X;x) injects into the cokernel ofIⁿ(O_X;xjK)!

!i_n(O_X;xjK), which is zero by whatwe have seen above. The theorem is

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 X2Sm_kwith function fieldFand anyn2None constructs as in [26] an explicitand canonical complexC(X;Iⁿ) of the form (see also [4]):

Iⁿ(F)! _y2Spec(OX;x)⁽¹⁾I^{n 1}(k(y);ty)! _z2Spec(OX;x)⁽²⁾I^{n 2}(k(z);L²(tz))!. . . in whicht_ymeans the tangent vector space a pointy2X, the dual of the k(y)-vector spacemy=(my)². The technique from Rost [24] shows that this complex gives for a smooth local ringOX;x a resolution ofIⁿ(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 prove the surjectivity ofIⁿ(O_X;x)!i_n(O_X;x).

The following Lemma is directly inspired by [3, 22].

LEMMA2.14. Let kL be a finite extension and x2X2Sm_k.Let F be the fraction field ofO_X;x.

1)For each n2Nthe Scharlau transfer s:W(F_kL)!W(F)with respect to a non-zero k-linear map s:L!k[25, §2.5]maps W(OX;xkL) W(FkL)to W(OX;x)W(F)and mapsIⁿ(FkL)toIⁿ(F).

Thus it induces a natural morphism

Iⁿ(O_X;xkL) ! Iⁿ(O_X;x)

k k

W(O_X;xkL)\Iⁿ(F_kL) ! W(O_X;x)\Iⁿ(F)

2)For any non-trivial s the natural morphism of1)is compatible to the morphism

i_n(OX;xkL)!i_n(OX;x)

induced by the transferi_n(FkL)!i_n(F)of the cycle module structure on i(see2.11).

3)If[L:k]is odd, the morphismIⁿ(OX;x)!i_n(OX;x)is a direct sum- mand of the morphismIⁿ(OX;xkL)!i_n(OX;xkL).

PROOF. The fact that the Scharlau transfer maps W(O_X;xkL) W(F_kL) t oW(O_X;x)W(F) follows from [22, §2 & §3]. The factthat the Scharlau transfer (for any choice ofs) mapsIⁿ(F) int oIⁿ(E) is [1, Satz 3.3]. This proves 1).

By the previous result of Arason, the induced transfer t:in(F)!

!i_n(E) doesn'tdepend on the choice 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 forsthe morphism of [25, Lemma 5.8 p. 49]. By loc:cit: the composition Iⁿ(O_X;x)!Iⁿ(O_X;xkL)!Iⁿ(O_X;x) is multi- plication by the classs(1)ˆ1. The same holds fori_nby [1]. The Lemma is

proven. p

3. Proof of the main Theorem 3.1 ±SomeA¹-homological algebra

For any chain complexCinAb_k, any sheafM2 Ab_k, and any integer n2Zwe denote by

Hom_D(Abk)(C;M[n])

the group of morphisms in the derived categoryD(Ab_k) of the abelian ca- tegoryAb_kfromCto then-th shiftM[n] ofM. Thatgroup can be computed as follows: choose an injective resolutionM!IofMinAbk of the form

M!I0!I 1!I 2!I 3!. . .. ThenHom_D(Abk)(C;M[n]) is the group

of morphisms of chain complexesC !I[n] modulo the subgroup of those which are homotopic to zero.

For technical purposes, we slightly extend the notion of smooth k- scheme. Letus denote bySm⁰_kthe full subcategory of that of allk-schemes consisting ofk-schemes which are possibly infinite disjoint union of smooth k-schemes. Any suchX2Sm⁰_k can be written asq_aX_a with eachX_a irre- ducible and inSm_k; t heX_a's are called the irreducible components ofX.

The associated free sheaf of abelian groups on X is the sheaf Z(X)ˆ

ˆ aZ(Xa)2 Abk.

A morphism of sheaves of abelian groups of the form f :Z(X)!Z(Y)

withXandY inSm⁰_k, is said to beelementaryif for any irreducible com- ponentXaofXthe restriction of that morphism to the summandZ(Xa) can be written as a finite sum

S_in_a;i:f_a;i

withn_a;i2Zand with eachf_a;icorresponding to a morphism ofk-schemes X_a!Y. An elementary resolution of a sheafM is a resolution R!M whose termsRn,n2Nare free sheaves on smoothk-schemes (possibly in Sm⁰_k) and whose boundaries are elementary morphisms.

LEMMA3.1. For any M2 Ab_k, there exists an elementary resolution of the form

Z(X):ˆ. . .!Z(Xn)!. . .!Z(X0)!M!0

PROOF. LetP_kbe the abelian category of presheaves of abelian groups onSm_k. Then for any X2Sm_k,Z(X) is just the associated sheaf to the presheaf Z(X)2 P_k which maps Y to the free abelian group on the set Hom_k(Y;X). TheseZ(X) are projective generators inP_k. One can thus find a resolutionR⁰!M in Pk with R⁰_n ˆ0 for n<0, and such thatR⁰_n is a directsum of sheaves of the formZ(X) withX2Sm_k. Then the sheafifi- cation of that resolution gives a resolution inAb_k with the required prop-

erties. p

DEFINITION3.2. A sheaf M2 Abk is said to beZariski strictlyA¹-in- variantif for any X 2Sm_kthe natural homomorphism

H(X;M)!H(XA¹;M) is an isomorphism.

By [24, §9], any sheaf arising from a cycle module is Zariski strictlyA¹- invariant. Any homotopy invariant sheaf with transfers in the sense of [28]

is Zariski strictlyA¹-invariantby [29].

Recall the constructionCfrom [27, 28] (in [28] itis denoted byC). It associates to a sheafN2 Ab_kthe complex of sheavesC(N) withn-th term the sheaf X7!N(DⁿX), withDⁿ then-th algebraic simplex⁴, and with differential in degree n,Sⁿ_iˆ0( 1)ⁱ@_i, with@_i induced by the i-th coface D^{n 1}!Dⁿ.

(⁴) i.e.DⁿˆSpec(k[T0;. . .;Tn]=(SiTi 1))

LEMMA3.3. Let M be a Zariski strictlyA¹-invariant sheaf.

1) For any non-negatively graded chain complex C in Ab_k the morphism

HomD(Abk)(C;M)!HomD(Abk)(CZ(A¹);M) induced by the projection CZ(A¹)!C, is an isomorphism.

2) For any sheaf N2 Ab_k the morphism N!C(N)induces an iso- morphism

Hom_D(Abk)(C(N);M[])Hom_D(Abk)(N;M[])ˆExt_Abk(N;M)

PROOF. 1) By definition, forMto be Zariski strictlyA¹-invariantex- actly means that for anyX2Sm_k the homomorphism

HomD(Abk)(Z(X);M[])ˆH(X;M)! H(XA¹;M)ˆHom_D(Abk)(Z(X)Z(A¹);M)

is an isomorphism. The part 1) of the Lemma then follows easily from standard homological algebra and the fact that the sheaves Z(X) 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

AnA¹-homotopybetween morphismsf;g:C!Dof chain complexes in Abk is a morphism h:CZ(A¹)!D which induces f (resp. g) through the 0 (resp. 1) sectionSpec(k)!A¹.

As a consequence of 1) of the Lemma, any twoA¹-homotopic morph- isms f;g:C!D, with C non-negatively graded induce the same morphism Hom_D(Abk)(D;M)!Hom_D(Abk)(C;M), for M Zariski strictly A¹-invariant.

3.2 ±Vanishing of some groups Hom_D(Abk)(Z=2(n);M[])

For X2Sm_k, we letZ_tr(X) denote the sheaf which mapsY2Sm_k to the group of finite correspondences from Y toX, that is to say the free abelian group c(Y;X) on the set of irreducible closed subschemes ZYXwhich are finite onY and which dominate an irreducible com- ponentofY [28, 27].

ForX1andX2pointed smoothk-schemes we let Ztr(X1^X2)2 Ab_k

denote the cokernel of the obvious morphism given by the base points Z_tr(X₁)Z_tr(X₂)!Z_tr(X₁X₂). Iterating this construction we get for a family (X1;. . .;Xn) of pointed smoothk-schemes the sheaf [27]:

Ztr(X1^. . .^Xn)

For any integern, t hemotivic chain complexin weightnof Suslin-Voe- vodsky [27, 28] is the chain complexZ(n):ˆC(Z_tr(G^{^n}_m))[ n].

The following resultis a variation on a idea from [31]:

THEOREM3.4. Let n1and let M be a sheaf which comes from a cycle module of weightsn 1.Then one has

Hom_D(Abk)(Z(n);M[m])ˆ0 (9)

for any integer m2Z.

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

ForX2Sm_k, we letzeq(X) denote the sheaf which mapsY 2Sm_kto the free abelian groupz_eq(X)(Y) on the set of irreducible closed subschemes ZYXwhich are quasi-finite onYand which dominates an irreducible componentofY [28]. We have the following geometric lemma:

LEMMA3.5. (Voevodsky [30])There exist explicit quasi-isomorphisms of chain complexes inAb_k of the form

Z(n)[2n]ˆC(Z_tr(G^{^n}_m))[‡n] C(Z_tr(Pⁿ)=Z_tr(P^{n 1}))!C(z_eq(Aⁿ)) The following two Lemmas and their Corollary below are inspired by the proof of [31, Proposition 3.3].

LEMMA 3.6. Let X2Smk and let z2zeq(Aⁿ)(X).Let us denote by V(z)XAⁿthe open complement of the supportjzj XAⁿ of z.Let zj_V(z)be the pull back of z through the morphismV(z)XAⁿ!X.Then there exists a canonical cycle

h(z)2z_eq(Aⁿ)(V(z)A¹)

such that@₁h(z)ˆzj_V(z)and@₀h(z)ˆ0.This cycle is functorial in X.

Moreover, given any (finite) decomposition zˆSjnjzj, with nj 2Zand zj2zeq(Aⁿ)(X), then firstjzj [jjzjj, so that \jV(zj)V(z)and one has the following equality in zeq(Aⁿ)(\_jV(z_j)A¹)

h(z)ˆS_jn_jh(z_j)

The assertion involving A¹-homotopies follows from the explicit con- struction given inloc.cit.. Beware thata prioriitis notclear whether or notgiven a morphism Z(X)!^f Z(Y) and an elementy2zeq(Aⁿ)(Y), one has f(Z(V(f(y))))Z(V(y)), although this is true for f :X!Y a morphism of schemes. For instance we will be in a situation where f(y)ˆ0. This is why we will need the next Lemma.

LEMMA3.7. Given an elementary morphism f :Z(X)!Z(Y)

an element y2z_eq(Aⁿ)(Y), considered as a morphismZ(Y)!z_eq(Aⁿ), and an open subscheme V_Y YAⁿ, whose complement Z_Y is quasi-finite equidimensional over Y and containsjyj(in other wordsVYV(y)), then there exits an open subscheme

VX XAⁿ

whose complement Z_X is quasi-finite equidimensional over X and con- tainsjxj(in other wordsV_X V(x)), with xˆyf ˆfy, and such that f maps Z(V_X) into Z(V_Y).Moreover, the restriction to Z(V_X) of the A¹- homotopy h(x)ofLemma 3.6is compatible with the restriction toZ(VY)of theA¹-homotopy h(y).

PROOF. We may assumeXis irreducible. Writef ˆS_jn_jf_j, a finite sum.

Define ZXAⁿ to be the union over the finite set of indicesjof the supportof the cyclesf_j(ZY)2zeq(XAⁿ).

By construction one has

jxj ˆ jS_jn_jf_j(y)j S_jjf_j(y)j S_jjf_j(Z_Y)j ˆZ ClearlyV_X ˆXAⁿ Zsatisfies the conclusions.

To check that f maps Z(V_X) into Z(V_Y) it suffices to observe that Z(V_X)ˆ \_jZ(V(f_j(Z_Y))) so that each morphismf_jseparately mapsZ(V_X) intoZ(VY).

Following this, one proves easily the assertion on the A¹-homotopies

using Lemma 3.6 above. p

COROLLARY 3.8. Let r :Z(X)!^' zeq(Aⁿ) be an elementary resolu- tion (given byLemma 3.1).Then there exists a subcomplex

Z(V)Z(X)Z(Aⁿ)ˆZ(XAⁿ)

such that for each q0, Vq is an open subscheme in XqAⁿ whose complement is a closed subscheme Z_q quasi-finite equidimensional over X_q, such that the composition

Z(V)Z(X)Z(Aⁿ)ˆZ(XAⁿ)!Z(X)!^r z_eq(Aⁿ) is homotopic to zero.

PROOF. We construct the subcomplex Z(V)Z(X)Z(Aⁿ)

by an induction on the degreeq0 using Lemmas 3.6 and 3.7 above, by imposing that for eachq,V_q is an open subsetofV(z_q), wherez_qˆ0 for q>0 andz0ˆr₀:X0!zeq(Aⁿ), and that the homotopy is the restriction in each degreeqof the homotopy onZ(V(zq)A¹) given by Lemma 3.6.

To start with, apply Lemma 3.6 to the morphismr₀:Z(X0)!Z(Aⁿ) which we see as an element z₀2z_eq(X₀Aⁿ). We getan open subset V₀ˆV(z₀)X₀Aⁿ whose complementis quasi-finite equidimensional overX0such that the composition

Z(V0)Z(X0)Z(Aⁿ)ˆZ(X0Aⁿ)!Z(X0)!^r0 zeq(Aⁿ) 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:Z(X1)!Z(X0), and toz0allows one to defineV1(observe that even if d₁(z₀)ˆZ₁ˆ0, 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 OFTHEOREM3.4. Set M:ˆM₀. To prove the vanishing of the Theorem, itis clearly sufficient, by Lemmas 3.5 and 3.3, to prove for each m2Nthe vanishing

Ext^m_Abk(zeq(Aⁿ);M)ˆHomD(Abk)(zeq(Aⁿ);M[m])ˆ0

We proceed inspired by Voevodsky's proof of [31, Prop. 3.6]. Choose an elementary resolution Z(X)!zeq(Aⁿ). As M is Zariski strictly A¹-in-