For a ringAwe will denoteK∗M(Spec A)also byKM∗(A). Below we will improve the definition of Milnor K-theory in order to get a sensible theory for local rings with finite residue fields.
ThereforeK∗M will usually be called naiveK-theory.
The following facts are standard for Milnor K-groups of local rings with infinite residue fields, see [29], [36] and [40].
Proposition 4.2.1. LetAbe a local ring with infinite residue field. Then we have
4.2. IMPROVED MILNORK-THEORY OF LOCAL RINGS 43 1. The natural map K2M(A) → K2(A) from Milnor K-theory to algebraic K-theory is an
isomorphism.
2. The relation{a,−a}= 0holds inK2M(A)fora∈A×. 3. The ringK∗M(A)is skew-symmetric.
4. Fora1, . . . , an∈A×witha1+· · ·+an= 1the relation 0 ={a1, . . . , an} ∈KnM(A) holds.
Moreover there exists a transfer for MilnorK-groups which is constructed in Chapter 3 and whose main properties we recall in the next proposition. The transfer will be essential for the constructions of this paper.
Let i :A→B be a finite étale extension of local rings with infinite residue fields. Fix an explicit presentationB∼=A[t]/(f)which exists by EGA IV 18.4.5.
Proposition 4.2.2. For a fixed presentation of B over A there exists a canonical transfer homomorphisms
NB/A:KnM(B)−→KnM(A) satisfying:
1. NB/A:K1M(B)→K1M(A)is the usual norm map on unit groups.
NB/A:K0M(B)→K0M(A)is multiplication bydeg(B/A).
2. The projection formula holds, i.e. forx ∈KnM(A),y ∈KmM(B)we have x NB/A(y) =NB/A(i∗(x)y)∈Kn+mM (A).
3. IfAcontains a field the transfer does not depend on the presentation ofBoverAchosen.
4. Letj :A→A0 be a homomorphism of local rings and leti0:A0→B0=B⊗AA0 be the induced inclusion, for which we fix the induced presentation. AssumeB0 is local. Then the diagram
KnM(B) //
NB/A
KnM(B0)
NB0/A0
KnM(A)
j∗
//KMn (A0)
commutes.
Proof. This is proved in Section 3.5.
Next we consider general abelian sheaves with a weak form of a transfer. In fact we will construct the improved MilnorK-groups axiomatically such that they have a transfer map.
44 CHAPTER 4. FINITE RESIDUE FIELDS Let S be the category of abelian sheaves on the big Zariski site of all schemes. Let ST be the full subcategory ofS such that a sheafF is in STif for every finite étale extension of local ringsi :A⊂B there exists a compatible system of norms
NB0/A0 :F(B0)→F(A0)
A0
where A0 runs over all local A-algebras for which B0 = B⊗AA0 is also local. Compatibility means that ifA0 →A00 are both local A-algebras such thatB0 =B⊗AA0 andB00=B⊗AA00
commutes. Furthermore we assume that if i0 : A0 → B0 is the induced inclusion our norm NB0/A0 satisfies
NB0/A0◦i∗0 = deg(B/A)i dF(A0).
Let ST∞ be the full subcategory of sheaves inS which have such norms if we restrict to the system of localA-algebrasA0 with infinite residue fields.
Proposition 4.2.3. The functorKnM is an object ofST∞ for alln≥0.
Proof. Immediate from Proposition 4.2.2.
Actually the MilnorK-functor should have some more global canonical transfer but at the moment we can define it only in the case of equicharacteristic schemes. We shall not be concerned with this problem here.
Proposition 4.2.4. The MilnorK-sheafK∗M is continuous.
Proof. It is clear from the definition thatK∗M is continuous. Furthermore a simple calculation shows that if a presheaf is continuous the associated sheaf in the Zariski topology is so too.
Our existence and uniqueness result, which is motivated by a construction of improved MilnorK-theory due to Gabber [7], reads now:
Theorem 4.2.5. For every continuousF ∈ST∞ there exists a universal continuous Fˆ∈ST and a natural transformation F →Fˆ. That is for arbitrary continuous G ∈ST and natural transformation F → G there exists a unique natural transformation Fˆ → G such that the diagram commutes.
Moreover for a local ringAwith infinite residue field we haveF(A) = ˆF(A).
4.2. IMPROVED MILNORK-THEORY OF LOCAL RINGS 45 Before we can give the proof we have to recall the construction of the rational function ring A(t) over a ring A and some of its basic properties. For a commutative ring A we let A(t1, . . . , tn) be the ring A[t1, . . . , tn]S where S is the multiplicative set consisting of all polynomials P
Ia(I)tI such that the ideal in A generated by all the coefficientsa(I)∈ A is A itself. IfA is local with maximal ideal m the ringA(t1, . . . , tn) is local too, in fact it is easy to see that for A local S = A[t1, . . . , tn]−mt where mt is the prime ideal m A[t1, . . . , tn].
Denote byi :A→A(t)the natural ring homomorphism. Denote byi1resp.i2the natural ring homomorphismA(t)→A(t1, t2) which sendst tot1resp. t2.
Lemma 4.2.6. IfA⊂Bis a finite étale extension of local rings there is a canonical isomorphism B⊗AA(t1, . . . , tn) ˜→B(t1, . . . , tn)
Proof. For simplicity we restrict to n = 1. Let m be the maximal ideal ofA. Consider the finite extension of ringsA[t]→B[t]. LetmAt be the prime idealm A[t] andmBt be the prime idealm B[t]. The latter ideal is indeed a prime ideal, because m B is the maximal ideal ofB by assumption. This also implies that
B(t) =B[t]mB
t .
Moreover the finiteness of A ⊂ B implies that mBt is the only prime ideal over mtA. Recall that according to standard facts B[t]⊗A[t]A(t) is the semi-local ring whose maximal ideals correspond to the finite set of prime ideals inB[t]which lie overmAt. But as we saw there is only one prime ideal overmtA, namelymBt, so
B⊗AA(t) =B[t]⊗A[t]A(t) and B(t) =B[t]mB
t
must be isomorphic.
Proof of Theorem 4.2.5. For an arbitrary Zariski sheafG we letGˆbe the Zariski sheafification of the following presheaf defined on affine schemes:
Spec(A) //k er[G(A(t)) i1∗−i2∗ //G(A(t1, t2))]
We claim that if G is an object in ST∞ then Gˆ is an object in ST. The continuity of Gˆ follows because the presheaf on affine schemes just defined is clearly continuous and the Zariski sheafification of a continuous presheaf is continuous. For a finite étale extension of local rings A⊂B Lemma 4.2.6 and the existence of a compatible system of norms allow us to write down the commutative diagram
So there exists a norm mapG(B)ˆ →G(A)ˆ for which one easily verifies the compatibility with base change. This shows thatGˆis an object inST.
The next proposition will be essential for the proof of the universal property ofFˆ.
46 CHAPTER 4. FINITE RESIDUE FIELDS Proposition 4.2.7. LetG∈ST(resp.G∈S) be continuous. Then for a local ringA(resp. a local ring with infinite residue field) we haveG(A) = ˆG(A).
Proof. First we prove the statement in parenthesis. So letAhave infinite residue field and let G ∈ S be continuous. We will prove the injectivity of G(A) →Gˆ(A) first. In the following S0⊂S will always be some finitely generated submonoid whereS ⊂A[t] is defined as above.
So by continuity we clearly have
G(A)ˆ ⊂G(A[t]S) = lim
−→
S0
G(A[t]S0)
So it is enough to show that G(A)→G(A[t]S0)is injective for every S0. For fixedS0 we will explain how to choose an element α ∈A such that for all p ∈S0 we havep(α) ∈A×. Let p1, . . . , pr ∈S0 be generators of the finitely generated monoidS0. Because the residue field of Ais infinite, it is possible to findα∈A withp1(α)· · ·pr(α)∈A×. This is the elementαwe were looking for. Letπ:A[t]S0 →Abe the ring homomorphism such thattmaps toα. As
G(A)−→G(A[t]S0)−→π∗ G(A) is the identity the injectivity of the first arrow follows.
For the surjectivity of G(A) → G(A)ˆ we argue similarly. Let S0 ⊂ A(t)×∩A[t] and S00⊂A(t1, t2)×∩A[t1, t2]be some finitely generated submonoids andx∈G(A[t]S0)such that the arrow
G(A[t]S0)i1∗−→−i2∗G(A[t1, t2]S00)
is well defined and kills x. SinceS0 andS00 are finitely generated and the residue field of Ais infinite, we can construct an elementα∈Asuch that for allp∈S0we havep(α)∈A×and for allp∈S00we havep(t, α)∈A(t)×. Denote byπ:A[t]S0 →Aresp.π0:A[t1, t2]S00 →A(t)the ring homomorphisms sendingttoαresp. t1 totandt2toα. Now the sequence of equalities
i∗◦π(x) =π0◦i2∗(x) =π0◦i1∗(x) =i m(x)∈G(A(t)) proves the surjectivity ofG(A)→G(A).ˆ
Next we prove that for G ∈ST continuous andA localG(A)→G(A)ˆ is an isomorphism.
We prove injectivity first. Fix an arbitrary primep. Consider a tower of finite étale extensions ofA
A⊂A1⊂A2⊂ · · · ⊂A∞
such thatAm is local,[Am:Am−1] =pand∪mAm=A∞. NowG(A∞) = ˆG(A∞)according to the first part of the proof. Considerx∈k er[G(A)→G(A)]. So by continuityˆ x ∈k er[G(A)→ G(Am)]for somem >0. The existence of a transfer homomorphismN:G(Am)→G(A)with
G(A)−→G(Am)−→N G(A)
equal to multiplication by pm shows that pmx = 0. As this holds for all primes p we have proved injectivity.
In order to prove surjectivity ofG(A)→G(A)ˆ considerx ∈G(A)ˆ and fix a primepand a tower of finite étale extensions as in the injectivity proof. Again observe thatG(A∞) = ˆG(A∞).
4.2. IMPROVED MILNORK-THEORY OF LOCAL RINGS 47
Now we can finish the proof. LetF andG be as in the statement of the theorem. Define the homomorphismFˆ→G such that the following diagram is commutative
F //
whereαis in isomorphism by Proposition 4.2.7. The uniqueness of the homomorphismFˆ→G follows form the commutative diagram
whereAis a local ring, since by Proposition 4.2.7β is an isomorphism andγis injective.
The next proposition comprises basic information about our improved MilnorK-theoryKˆM∗. We will only sketch the proofs.
Proposition 4.2.8. Let(A, m)be a local ring. Then:
1. Kˆ1M(A) =A×.
2. Kˆ∗M(A)has a natural graded commutative ring structure.
3. Proposition 4.2.1 and 4.2.2 remain true for any any local ringAif we replaceK∗MbyKˆM∗. 4. If F is a field we haveKMn (F) = ˆKMn (F).
5. For everyn≥0there exists a universal natural numberMnsuch that if|A/m|> Mnthe natural homomorphism
KMn (A)−→KˆMn(A) is an isomorphism.
48 CHAPTER 4. FINITE RESIDUE FIELDS 6. There exists a homomorphism
Kn(A)−→KˆMn(A) such that the composition
KˆMn (A)−→Kn(A)−→KˆnM(A) is multiplication by(n−1)!and the composition
Kn(A)−→KˆMn (A)−→Kn(A) is the Chern classcn,n.
7. If(A, I)is a Henselian pair in the sense of [9] and s∈Nis invertible inA/I the map KˆMn(A)/s −→KˆMn (A/I)/s
is an isomorphism.
8. Let A be regular and equicharacteristic, F = Q(A) and X = Spec A. The Gersten conjecture holds for MilnorK-theory, i.e. the Gersten complex
0 //KˆMn (A) //KnM(F) //⊕x∈X(1)Kn−1M (k(x)) //· · · is exact.
9. LetX be a regular scheme containing a field. There is a natural isomorphism HZarn (X,KˆMn )∼=CHn(X).
10. IfAis equicharacteristic of characteristic prime to2the map iA: ˆKM3 (A)−→K3(A) is injective.
11. LetAbe regular and equicharacteristic. There is a natural isomorphism KˆMn(A) ˜−→Hnmot(Spec(A),Z(n))
onto motivic cohomology.
Proof. (1): Since K1M is an objects inST, the isomorphismKˆM1 (A) =K1M(A) =A× follows from Proposition 4.2.7.
(2): This follows immediately from the hat construction in the proof of Theorem 4.2.5.
(3): It is well known that the sheaf associated to X 7→ K2(X) is an object in ST, so that Kˆ2(A) =K2(A)by Proposition 4.2.7. But ifAis a local ring with infinite residue field we have K2M(A) =K2(A)according to Proposition 4.2.1 (1) and the isomorphism of sheavesKˆM2 =K2
4.2. IMPROVED MILNORK-THEORY OF LOCAL RINGS 49 follows from the definition of the ‘hat’ in the proof of Theorem 4.2.5. The rest follows from the injectivity of
KˆnM(A)−→KˆMn (A(t)) =KnM(A(t)).
(4): If F is infinite this follows from Proposition 4.2.7, if F is finite then KnM(F) = 0 and so it suffices to show that KnM(F) → KˆMn(F) is surjective. Let si : KnM(F(t1, . . . , ti)) →
(5): It was shown in Section 3.5 that there exists an Mn ∈ N such that the statement of Proposition 4.2.2 remains true if the local ringAhas more thanMnelements in its residue field and ifdeg(B/A)≤ 3. Now an analog of the second part of proof of Proposition 4.2.7 with p1= 2andp2= 3gives (5).
(6): This follows immediately from [29].
(7): An elementary calculation shows that
KnM(A)/s−→KnM(A/I)/s
is an isomorphism for every local ringAwiths invertible inA/I. Now a norm argument shows the analogous result for the improved MilnorK-groups. This is accomplished by choosing an étale local extensionA⊂A0of some prime power degreeq, coprime tos, such that the residue field ofA0 has more thanMnelements. HereMnis the natural number from part (5). Observe that(A0, A0I)is again a Henselian pair. Consider the commutative diagram
KˆMn(A0)/s //
N
KˆnM(A0/I)/s
N
KˆMn(A)/s //KˆMn (A/I)/s
where the upper horizontal arrow is an isomorphism by what has been said so far. A simple diagram chase shows that the kernel and cokernel ofKˆMn (A)/s −→KˆMn(A/I)/s have exponent qand must therefore vanish.
(8): This complex was constructed in [16]. Again ifAhas more thanMnelements in its residue field the result was proven in Section 3.7. If not one uses a norm trick as in (7).
(9): Immediate from (8).
(10): IfAis a field this was shown by Kahn using Voevodsky’s proof of the Milnor conjecture [14]. IfA is regular it follows from the field case and (8). IfAis not regular we first use the norm trick and can and will assume that A has more than M3 elements in its residue field.
Next we use Hoobler’s trick [11] and choose some regular equicharacteristic local ringA0such that there exists an exact sequence
0−→I−→A0−→A−→0
50 CHAPTER 4. FINITE RESIDUE FIELDS such that(A0, I)is a Henselian pair. Letx be ink er(iA). Then2x= 0according to (6). Next choose x0∈K3M(A0) = ˆKM3(A0) which maps tox. An elementary argument left to the reader shows that we can choosex0such that2x0= 0. Now remember that the two torsion inK3(A0) is isomorphic to the two torsion inK3(A)by the rigidity of algebraicK-theory, see [9], so that iA0(x0) = 0and finallyx0= 0by the regular case proved above.
(11): Using Corollary 1.0.3 we are reduced to the caseAessentially smooth over a field. Then improved MilnorK-theory fulfills the Gersten conjecture by (8) and motivic cohomology does so by Proposition 1.0.9, so the result follows from an easy diagram chase of Gersten complexes using the fact that (11) is known ifAis a field, Proposition 2.2.1. The details can be found in Section 3.7.
Remark 4.2.9. In general the map
KnM(A)−→KˆMn(A)
is not an isomorphism. For example forn= 2we haveKˆ2M(A) =K2(A)according to Proposi-tion 4.2.8(3) and it was shown in the appendix to [13] that forA=F2[t](t)the groupsK2M(A) andK2(A)are not isomorphic.