aa r X i v : . [ m a t h . K T ] A ug SEMILOCAL MILNOR K-THEORY
GRIGORY GARKUSHAA
BSTRACT . In this paper, semilocal Milnor K -theory of fields is introduced and studied. A stronglyconvergent spectral sequence relating semilocal Milnor K -theory to semilocal motivic cohomologyis constructed. In weight 2, the motivic cohomology groups H p Zar ( k , Z ( )) , p
1, are computed assemilocal Milnor K -theory groups b K M , − p ( k ) . The following applications are given: (i) several criteriafor the Beilinson–Soul´e Vanishing Conjecture; (ii) computation of K of a field; (iii) the Beilinsonconjecture for rational K -theory of fields of prime characteristic is shown to be equivalent to vanishingof rational semilocal Milnor K -theory. C ONTENTS
1. Introduction 12. Preliminaries 33. Semilocal Milnor K -theory 54. Some criteria for the Beilinson–Soul´e Vanishing Conjecture 95. K of a field 116. On conjectures of Beilinson and Parshin 14References 151. I NTRODUCTION
It is a classical fact of algebraic K -theory of fields that Milnor K -groups K M , K M , K M agree withQuillen’s K , K , K . However, in higher degrees K Mn , n >
3, is only a small piece of Quillen’s K n . Akey technical tool to make computations in algebraic K -theory is the motivic spectral sequence E p , q = H q − p Zar ( k , Z ( q )) = ⇒ K p + q ( k ) relating algebraic K -theory to motivic cohomology (see, e.g., [5]).By well-known theorems of Nesterenko–Suslin [16] and Totaro [21] the Milnor K -theory ring K M ∗ ( k ) is isomorphic to the ring L H n Zar ( k , Z ( n )) . The computation of the other motivic cohomol-ogy groups, and hence algebraic K -theory, is one of the hardest problems in the field and severaloutstanding conjectures are related to this problem. For instance, the celebrated Beilinson–Soul´eVanishing Conjecture states that all motivic cohomology groups H p Zar ( k , Z ( q )) vanish for p
0. In
Mathematics Subject Classification.
Key words and phrases.
Milnor K -theory, motivic cohomology, algebraic K -theory of fields. ositive characteristic, the Beilinson conjecture states that Milnor K -theory and Quillen K -theoryagree rationally: K Mn ( k ) Q ∼ = −→ K n ( k ) Q . As we have mentioned above, Milnor K -theory is isomorphic to the motivic cohomology diagonal L H n Zar ( k , Z ( n )) . The main purpose of this paper is to introduce and investigate “semilocal Milnor K -theory of fields”. We show that it is precisely related to motivic cohomology outside the diagonal.An advantage of this theory is that it is defined in elementary terms whereas the motivic complexesare sophisticated and enormously hard for computations.By definition, semilocal Milnor K -theory of a field consists of bigraded Abelian groups b K Mn , m ( k ) , m , n >
0. Precisely, let b ∆ • k be the cosimplicial scheme, where each b ∆ ℓ k is the semilocalization of thestandard affine scheme ∆ ℓ k at its vertices v , . . . , v ℓ . Let K Mn be the Zariski sheaf of Milnor K -theoryin degree n >
0. Semilocal Milnor K -theory complex is the chain complex K Mn ( b ∆ • k ) and b K Mn , m ( k ) : = H m ( K Mn ( k )) . The main result of the paper, Theorem 3.4, says that there is a strongly convergent spectral se-quence relating semilocal Milnor K -theory to semilocal motivic cohomology E pq = H p ( H n − − q Zar ( b ∆ • k , τ < n Z ( n ))) = ⇒ b K Mn , p + q + ( k ) , where τ < n Z ( n ) is the truncation complex of Z ( n ) for degrees smaller than n . Moreover, if n =
2, then the spectral sequence above collapses, and hence for any p H p Zar ( k , Z ( )) = b K M , − p ( k ) . Thus semilocal Milnor K -theory is related to motivic cohomology exactlyoutside the diagonal ⊕ H n ( k , Z ( n )) in contrast to the classical Milnor K -theory.Various applications of semilocal Milnor K -theory are given in the paper. First, several criteriafor the Beilinson–Soul´e Vanishing Conjecture are established in Theorem 4.1. We next pass tocomputation of K of a field. The group K ( k ) was actively investigated in the 80-s – see Levine [14],Merkurjev and Suslin [15, 17] (it is worth mentioning that semilocal PID-s play an important role intheir analysis). Recall that K ( k ) fits into an exact sequence0 → K M ( k ) → K ( k ) → K ind3 ( k ) → , where K ind3 ( k ) is the indecomposable K -theory of k . We show in Corollary 3.9 that K ind3 ( k ) = b K M , ( k ) ,so that K ( k ) is fully determined by Milnor K -theory and semilocal Milnor K -theory. Similarly to K ( k ) we show in Theorem 5.4 that K ( k ) is also fully determined by Milnor K -theory and semilocalMilnor K -theory.Another application is given for the Beilinson conjecture on the rational K -theory of fields ofprime characteristic. Namely, it is shown in Theorem 6.1 that this conjecture is equivalent to vanish-ing of rational semilocal Milnor K -theory, which is much more accessible for computations. Also,vanishing of rational semilocal Milnor K -theory is shown to be a necessary condition for Parshin’sconjecture (see Theorem 6.3).The author would like to thank Daniil Rudenko and Matthias Wendt for numerous discussions onthe Beilinson–Soul´e Vanishing Conjecture. He also thanks Jean Fasel for helpful comments.Throughout the paper we denote by Sm / k the category of smooth separated schemes of finite typeover a field k . By a smooth semilocal scheme over k we shall mean a k -scheme W for which there xists a smooth affine scheme X ∈ Sm / k and a finite set x , . . . , x n of points of X such that W isthe inverse limit of open neighborhoods of this set. We will deal with both complexes for whichthe differential has degree − + A we mean the chain complex A [ ] with A [ ] n = A n + and differential − d A .2. P RELIMINARIES
For any presheaf F : Sm / k → Ab, let e C F denote the following presheaf: e C F ( X ) = lim −→ X ×{ , }⊂ U ⊂ X × A F ( U ) . There are two obvious presheaf homomorphisms (given by restrictions to X × X × i ∗ , i ∗ : e C F → F .2.1. Definition ([18, 20]) . A presheaf F is said to be rationally contractible if there exists a presheafhomomorphism s : F → e C F such that i ∗ s = i ∗ s = Example. (1) Given n , l >
0, the Zariski sheaves with transfers Z tr ( G ∧ nm ) : = Cor ( − , G ∧ nm ) and Z tr ( G ∧ nm ) / l defined in [20, Section 3] are rationally contractible by [20, 9.6].(2) Let k be a perfect field of characteristic not 2. Then the presheaf with Milnor–Witt correspon-dences e Z ( G ∧ nm ) : = g Cor ( − , G ∧ nm ) in the sense of [3] is rationally contractible by [2, Section 2]. It is aZariski sheaf by [3, 5.2.4].Given a field k and ℓ >
0, let O ( ℓ ) k , v denote the semilocal ring of the set v of vertices of ∆ ℓ k = Spec ( k [ t , . . . , t ℓ ] / ( t + · · · + t ℓ − )) and set b ∆ ℓ k : = Spec O ( ℓ ) k , v . Then ℓ b ∆ ℓ k is a cosimplicial semilocal subscheme of ∆ • k .2.3. Proposition (Suslin [18]) . The following statements are true: ( ) Let F : Sm / k → Ab be a rationally contractible presheaf. Then the presheaf C n ( F ) = Hom ( ∆ nk , F ) is also rationally contractible. ( ) Assume that the presheaf F is rationally contractible. Then the complex F ( b ∆ • k ) is con-tractible, and hence acyclic. Given a cochain complex F • , the canonical truncation τ < F • of F • has the property that H i ( τ < F • ) = H i ( F • ) for i < H i ( τ < F • ) = i > b ∆ • k is recovered,up to homology, from negative cohomology evaluated at b ∆ • k .2.4. Theorem.
Suppose F • · · · d − −−→ F − d − −−→ F − d − −−→ F → → · · · (1) is a cochain complex of rationally contractible presheaves concentrated in non-positive degrees. Let K − n : = Ker d − n , n > , and L : = Coker d − . Then the chain complex of Abelian groups L ( b ∆ • k ) is uasi-isomorphic to the chain complex K − ( b ∆ • k )[ − ] (the shift is homological). Moreover, there isa tower in the derived category D ( Ab ) of chain complexes of Abelian groups which are concentratedin non-positive degrees · · · α − −→ K − ( b ∆ • k )[ − ] α − −→ K − ( b ∆ • k )[ − ] α − −→ K − ( b ∆ • k ) (2) with q-th layer, q > , being the complex H − − q ( b ∆ • k )[ − q ] . Here H − q stands for the − q th coho-mology presheaf of the complex (1) . In particular, the tower (2) gives rise to a strongly convergentspectral sequenceE pq = H p ( H − − q ( b ∆ • k )) : = H p ( H − − q ( τ < F • )( b ∆ • k )) ⇒ H p + q + ( L ( b ∆ • k )) . Proof.
Consider a short exact sequence of presheaves0 → Im d − → F → L → . It induces a short exact sequence of Abelian groups in each degree n > → colim U ∋ v ,..., v n ( Im d − )( U ) → colim U ∋ v ,..., v n ( F )( U ) → colim U ∋ v ,..., v n L ( U ) → , where v , . . . , v n are the vertices of ∆ nk . We also use here the fact that the direct limit functor is exact.The latter is nothing but the short exact sequence0 → ( Im d − )( b ∆ nk ) → ( F )( b ∆ nk ) → L ( b ∆ nk ) → . Since F is rationally contractible by assumption, it follows that the complex of Abelian groups ( F )( b ∆ • k ) is contractible by Proposition 2.3(2). Now the induced triangle in D ( Ab )( Im d − )( b ∆ • k ) → ( F )( b ∆ • k ) → L ( b ∆ • k ) τ −→ ( Im d − )( b ∆ • k )[ − ] yields a quasi-isomorphism of chain complexes τ : L ( b ∆ • k ) ∼ −→ ( Im d − )( b ∆ • k )[ − ] . For the same rea-sons, ( Im d − )( b ∆ • k ) is quasi-isomorphic to K − ( b ∆ • k )[ − ] . For this, one uses the short exact sequenceof presheaves K − ֒ → F − ։ Im d − . So L ( b ∆ • k ) is quasi-isomorphic to K − ( b ∆ • k )[ − ] .Similarly, each short exact sequence of presheaves0 → K − n → F − n → Im d − n → ( Im d − n )( b ∆ • k ) ≃ K − n ( b ∆ • k )[ − ] .Next, each short exact sequence of presheaves0 → Im d − n − → K − n → H − n → D ( Ab ) K − n − ( b ∆ • k )[ − ] −→ K − n ( b ∆ • k ) → H − n ( b ∆ • k ) → K − n − ( b ∆ • k ) . In this way we obtain the desired tower (2) with layers as stated. Note that the n th complex of thetower K − n − ( b ∆ • k )[ − n ] is ( n − ) -connected, and hence the tower gives rise to a strongly convergentspectral sequence E pq = H p ( H − − q ( b ∆ • k )) ⇒ H p + q + ( L ( b ∆ • k )) after applying [5, 6.1.1] to it. This completes the proof. (cid:3) et A be a V -category of correspondences on Sm / k in the sense of [8] ( V -categories are justa formal abstraction of basic properties for the category of finite correspondences Cor ). We saythat A is nice if for any smooth semilocal scheme W and any A -invariant presheaf F with A -correspondences the canonical morphism of presheaves F → F Zar induces an isomorphism ofAbelian groups F ( W ) ∼ = −→ F Zar ( W ) . Here F Zar is the Zariski sheaf associated to the presheaf F . For example, the category of finite correspondences Cor is nice by [22, 4.24]. If the base field k is infinite perfect of characteristic different from 2, then the category of finite MW -correspondencesin the sense of [3] is nice by [2, 3.5].2.5. Corollary.
Under the conditions of Theorem 2.4 suppose that (1) is a cochain complex of Zariskisheaves with nice correspondences such that its presheaves L and H − q -s are A -invariant. Thenthe chain complex of Abelian groups L Zar ( b ∆ • k ) is quasi-isomorphic to the chain complex of Abeliangroups K − ( b ∆ • k )[ − ] = K − ( b ∆ • k )[ − ] (the shift is homological). Moreover, the q-th layer of thetower (2) equals the complex H − − q Zar ( b ∆ • k )[ − q ] . Here H − q Zar stands for the − q th cohomology Zariskisheaf of the complex of Zariski sheaves (1) . In particular, the tower (2) gives rise to a stronglyconvergent spectral sequenceE pq = H p ( H − − q Zar ( b ∆ • k )) : = H p ( H − − q Zar ( τ < F • )( b ∆ • k )) ⇒ H p + q + ( L Zar ( b ∆ • k )) .
3. S
EMILOCAL M ILNOR K - THEORY
Let Z ( n ) be Suslin–Voevodsky’s [20, Definition 3.1] motivic complex of Zariski sheaves of weight n > Sm / k . By definition, it is concentrated in cohomological degrees m n . More precisely,it equals the cochain complex with differential (of degree +
1) equal to the alternating sum of faceoperations · · · →
Cor ( ∆ k × − , G ∧ nm ) → Cor ( ∆ k × − , G ∧ nm ) → Cor ( − , G ∧ nm ) → → · · · (3)Here the Zariski sheaf Cor ( − , G ∧ nm ) is in cohomological degree n . Denote by K Mn the Zariski sheaf H n Zar ( Z ( n )) . We shall also refer to K Mn as the n-th Milnor K-theory sheaf .Similarly, let e Z ( n ) be Calm`es–Fasel’s [3] Milnor–Witt motivic complex of Zariski sheaves ofweight n > Sm / k . More precisely, it equals the cochain complex with differential (of degree +
1) equal to the alternating sum of face operations · · · → g Cor ( ∆ k × − , G ∧ nm ) → g Cor ( ∆ k × − , G ∧ nm ) → g Cor ( − , G ∧ nm ) → → · · · (4)Denote by K MWn the Zariski sheaf H n Zar ( e Z ( n )) . We shall also refer to K MWn as the n-th Milnor–WittK-theory sheaf .3.1.
Definition.
Let k be any field and n >
0. The n-th semilocal Milnor K-theory complex of thefield k is the chain complex of Abelian groups K Mn ( b ∆ • k ) .The ( n , q ) -th semilocal Milnor K-theory group b K Mn , q ( k ) of k is defined as the q -th homology group H q ( K Mn ( b ∆ • k )) of the n -th semilocal Milnor K -theory complex of k . By definition, b K Mn , q ( k ) = q < k be an infinite perfect field of characteristic not 2 and n >
0. The n-th semilocal Milnor–WittK-theory complex of the field k is the chain complex of Abelian groups K MWn ( b ∆ • k ) . he ( n , q ) -th semilocal Milnor–Witt K-theory group b K MWn , q ( k ) of k is defined as the q -th homologygroup H q ( K MWn ( b ∆ • k )) of the n -th semilocal Milnor K -theory complex of k . By definition, b K MWn , q ( k ) = q < A is an Abelian group then the same definitions are given “with A -coefficients”, in which case wejust tensor the relevant complexes by A to get K Mn ( b ∆ • k ) ⊗ A and K MWn ( b ∆ • k ) ⊗ A and then semilocalMilnor and Milnor–Witt K -theory groups with A -coefficients b K Mn , ∗ ( k , A ) , b K MWn , ∗ ( k , A ) are homologygroups of these complexes. In what follows we mostly deal with the case A = Q , in which case wewrite the subscript Q .All statements which are proven below with integer coefficients will automatically be true with Q -coefficients. The interested reader will always be able to repeat the relevant proofs rationally (wedo not write them for brevity). Also, many statements are valid with any coefficients, say, when A isfinite. Since motivic cohomology with finite coefficients is well studied, we do not discuss this caseeither, assuming that the interested reader will do this easily.We also recall from [4, 13, 16] that the n th Milnor K -group K Mn ( R ) of a commutative ring R isthe abelian group generated by symbols { a , . . . , a n } , a i ∈ R × , i = , . . . , n , subject to the followingrelations:(1) for any i , { a , . . . , a i a ′ i , . . . , a n } = { a , . . . , a i , . . . , a n } + { a , . . . , a ′ i , . . . , a n } ;(2) { a , . . . , a n } = i , j , i = j , such that a i + a j = Remark.
If the field k is infinite, it follows from [4, 13] that K Mn ( b ∆ ℓ k ) = K Mn ( O ( ℓ ) k , v ) . Wesee that K Mn ( b ∆ ℓ k ) is defined naively in terms of generators and relations. The n -th semilocal Milnor K -theory chain complex K Mn ( b ∆ • k ) is therefore isomorphic to the chain complex K Mn ( O ( • ) k , v ) . Inparticular, b K Mn , q ( k ) = H q ( K Mn ( O ( • ) k , v )) for all n , q > Lemma.
Given any field k, the complex K M ( b ∆ • k ) has only one non-zero homology group indegree zero, b K M , ( k ) , which is isomorphic to Z .Proof. This is straightforward. (cid:3)
We are now in a position to prove the main result of the paper.3.4.
Theorem.
Suppose k is any field. The following statements are true: ( ) For any n > , b K Mn , ( k ) = b K Mn , ( k ) = . ( ) For any n > , there is a strongly convergent spectral sequenceE pq = H p ( H n − − q Zar ( τ < n Z ( n ))( b ∆ • k )) = ⇒ b K Mn , p + q + ( k ) , where τ < n Z ( n ) is the truncation complex of Z ( n ) for degrees smaller than n. ( ) If n = , then the spectral sequence above collapses, and hence for any p there is anisomorphism H p Zar ( k , Z ( )) = b K M , − p ( k ) . ( ) If the field k is infinite perfect of characteristic different from 2 and n > , then the naturalmorphism of chain complexes of Abelian groups K MWn ( b ∆ • k ) → K Mn ( b ∆ • k ) is a quasi-isomorphism. Inparticular, it induces isomorphisms of Abelian groups b K MWn , q ( k ) ∼ = −→ b K Mn , q ( k ) for all q ∈ Z . roof. (1)-(2). By [20, 9.6] the Zariski sheaf with transfers Cor ( − , G ∧ nm ) , n >
1, is rationally con-tractible. It follows from [18, 2.4] that the Zariski sheaf with transfers
Cor ( ∆ ℓ k × − , G ∧ nm ) is rationallycontractible for every ℓ >
0. Now the desired spectral sequence of the second assertion follows fromCorollary 2.5 if we apply it to the cochain complex of rationally contractible Zariski sheaves withtransfers (3).By Theorem 2.4 and Corollary 2.5 there is a quasi-isomorphism K Mn ( b ∆ • k ) ≃ K − ( b ∆ • k )[ − ] ofchain complexes for some presheaf with transfers K − (the shift is homological). Assertion (1) nowfollows.(3). Suppose k is perfect. Since the motivic complex of weight one Z ( ) is acyclic in non-positivecohomological degrees by [20, 3.2], Voevodsky’s Cancellation Theorem [25] implies thatHom ( G ∧ m , H p Zar ( Z ( ))) = ( H p ( Hom ( G ∧ m , Z ( )))) Zar = p
1. We also use here the proof of [22, 4.34]. It follows from [12, 2.5.2] that each H p Zar ( Z ( )) , p
1, is a birational (Nisnevich) sheaf with transfers in the sense of [12, 2.3.1].The proof of [11, 4.2.1] shows that the natural map of chain complexes H p Zar ( Z ( ))( k ) → H p Zar ( Z ( ))( b ∆ • k ) , p , is a quasi-isomorphism. Thus H i ( H p Zar ( Z ( ))( b ∆ • k )) = i = H ( H p Zar ( Z ( ))( b ∆ • k )) = H p Zar ( Z ( ))( k ) . We see that the strongly convergent spectral sequence from the second assertioncollapses for any perfect field k .Next, suppose K / k is a finitely generated field extension of the perfect field k . Then K = k ( U ) forsome U ∈ Sm / k . Each scheme b ∆ ℓ K is the semilocalization of ∆ ℓ k × U at the points ( v , η ) , . . . , ( v n , η ) ,where η is the generic point of U (see the proof of [2, 3.10]). The proof of [11, 4.2.1] shows that thenatural map of chain complexes H p Zar ( Z ( ))( K ) → H p Zar ( Z ( ))( b ∆ • K ) , p , is a quasi-isomorphism. Thus H i ( H p Zar ( Z ( ))( b ∆ • K )) = i = H ( H p Zar ( Z ( ))( b ∆ • K )) = H p Zar ( Z ( ))( K ) . We see that the strongly convergent spectral sequence from the second assertionalso collapses for any finitely generated field extension K / k .Finally, for any field of characteristic p we use the fact that it can be written as a direct limit offields finitely generated over Z / p , and the fact that the above homology/cohomology groups obvi-ously commute with direct limits. We also use here the fact that the cohomology groups H ∗ Zar ( K , Z ( n )) are defined intrinsically in terms of the field K and are independent of the choice of the base field.(4). The proof is based on Bachmann’s results on the MW -motivic cohomology [1]. It followsfrom [2, 2.5] that the Zariski sheaf g Cor ( − , G ∧ nm ) is rationally contractible for every n >
0. If weconsider the cochain complex of Zariski sheaves (4) and repeat the arguments for the proof of thesecond assertion, we shall get a strongly convergent spectral sequence e E pq : = H p ( H n − − q Zar ( e Z ( n ))( b ∆ • k )) = ⇒ b K MWn , p + q + ( k ) . The natural functor of additive categories of correspondences g Cor → Cor induces a map of spectralsequences e E pq → E pq , where E pq is the spectral sequence of the second assertion.It follows from [1, Theorem 17] that the morphism of complexes τ < n e Z ( n ) → τ < n Z ( n ) is a quasi-isomorphism, locally in the Nisnevich topology. This means that each morphism of Nisnevich heaves H p Nis ( e Z ( n ) Nis ) → H p Nis ( Z ( n )) , p = n , is an isomorphism. It follows from [2, 3.5] that H p Nis ( e Z ( n ) Nis )( b ∆ ℓ k ) = H p Zar ( e Z ( n ))( b ∆ ℓ k ) , ℓ > . Using [22, 5.5], one has H p Nis ( Z ( n ))( b ∆ ℓ k ) = H p Zar ( Z ( n ))( b ∆ ℓ k ) , ℓ > . Therefore the morphism of strongly convergent spectral sequences e E pq → E pq is an isomorphism.This isomorphism implies that the map of chain complexes K MWn ( b ∆ • k ) → K Mn ( b ∆ • k ) is a quasi-isomorphism, as was to be proved. (cid:3) Corollary.
Given any field k, semilocal Milnor K-theory complex K M ( b ∆ • k ) is acyclic. In par-ticular, b K M , q ( k ) = for all q ∈ Z .Proof. By [20, 3.2] the complex τ < Z ( ) is acyclic. Therefore the E -page of the strongly convergentspectral sequence of Theorem 3.4(2) for n = (cid:3) Corollary. b K Mn , ( k ) = Coker ( H n − ( b ∆ k , Z ( n )) ∂ − ∂ −−−→ H n − ( k , Z ( n ))) for any field k and n > . If k is a field of positive characteristic p , the Geisser–Levine theorem [10, 1.1] implies that Milnor K -theory groups of k are p -torsionfree. The following statement says that semilocal Milnor K -theorygroups are p -uniquely divisible.3.7. Corollary.
Given any field k of positive characteristic p > , semilocal Milnor K-theory groups b K Mn , m ( k ) are p-uniquely divisible for all n > and m ∈ Z . In particular, b K Mn , m ( k ) = b K Mn , m ( k ) ⊗ Z [ / p ] .Proof. We claim that H s Zar ( Z ( n )) is a sheaf of Z [ / p ] -modules for all n > s < n . The Geisser–Levine theorem [10, 1.1] implies that H s Zar ( Z ( n ))( K ) is p -uniquely divisible for any field extension K / k . It follows that the morphism of sheaves H s Zar ( Z ( n )) → H s Zar ( Z ( n )) ⊗ Z [ / p ] is an isomorphismon field extensions K / k , and hence it is an isomorphism of sheaves by [22, 4.20].We see that the E -term of the strongly convergent spectral sequence of Theorem 3.4(2) consistsof p -uniquely divisible Abelian groups, and hence the semilocal Milnor K -theory groups of thestatement are p -uniquely divisible. (cid:3) Remark.
The preceding theorem implies that the evaluation of the Milnor–Witt sheaf K MWn , n >
1, at b ∆ • k “deletes” the information about quadratic forms.By Remark 3.2 if the base field k is infinite, Milnor K -theory of semilocal schemes like b ∆ ℓ k hasan explicit, naive description, whereas motivic cohomology involves sophisticated constructions.Thus Theorem 3.4 computes some motivic cohomology groups as homology groups of certain naivecomplexes. In particular, we can apply “symbolic” computations to cycles in motivic complexes.Furthermore, a theorem of Kerz [13, 1.2] implies that the norm residue homomorphism inducesan isomorphism of complexes K Mn ( b ∆ • k ) ⊗ Z /ℓ ∼ = −→ H net ( b ∆ • k , µ ⊗ n ℓ ) , n > , if the field k is infinite of characteristic not dividing ℓ . Since the second and the third statement ofTheorem 3.4 are true with finite coefficients, it follows that semilocal Milnor K -theory groups withfinite coefficients can be computed as homology groups of complexes H net ( b ∆ • k , µ ⊗ n ℓ ) . ecall from [17] that the indecomposable K -group of a field k , denoted by K ind3 ( k ) , is defined asthe cokernel of the canonical homomorphism K M ( k ) → K ( k ) .3.9. Corollary.
For any field k, there is an isomorphism K ind3 ( k ) = b K M , ( k ) .Proof. The motivic spectral sequence gives an isomorphism K ind3 ( k ) = H ( k , Z ( )) . Now Theo-rem 3.4 implies the claim. (cid:3) Corollary.
For any perfect field k, any connected X ∈ Sm / k and any p , there is an isomor-phism H p Zar ( X , Z ( )) = b K M , − p ( k ( X )) , where k ( X ) is the function field of X .Proof. This follows from Theorem 3.4 and the fact that for any p H p Zar ( X , Z ( )) = H p Zar ( k ( X ) , Z ( )) . (cid:3) Since H p Zar ( k , Z ( q )) is uniquely divisible for p k (see, e.g., [26, Excercise VI.4.6]),Theorem 3.4(3) implies the following3.11. Corollary.
For any field k and any n > , the group b K M , n ( k ) is uniquely divisible. Corollary.
For any field k there are isomorphisms of rational vector spacesK ( k ) ( ) Q = b K M , ( k ) Q , K + p ( k ) ( ) Q = b K M , + p ( k ) , p > . In particular, K ( k ) Q = K M ( k ) Q ⊕ b K M , ( k ) Q .Proof. This follows from Theorem 3.4, Corollary 3.11 and the fact that for any p > − H − p Zar ( k , Q ( )) = K + p ( k ) ( ) Q . (cid:3) Theorem.
Semilocal Milnor K-theory is invariant under purely transcendental extensions.Namely, b K Mn , q ( k ) = b K Mn , q ( k ( x )) for any field k and n , q > .Proof. Suppose the base field k is perfect. Then the Zariski sheaf K Mn on Sm / k is strictly homotopyinvariant. Let K / k is a finitely generated field extension of k . There is X ∈ Sm / k such that K = k ( X ) .It follows from [11, 2.2.6 and 4.2.1] that the natural map of chain complexes K Mn ( b ∆ • K ) → K Mn ( b ∆ • K ( x ) ) induced by the projection X × A → X is a quasi-isomorphism. We tacitly use here [19, 4.7] aswell. We have thus shown the theorem for finitely generated field extensions K / k . For any fieldof characteristic p we use the fact that it can be written as a direct limit of fields finitely generatedover Z / p , and the fact that semilocal Milnor K -theory groups groups obviously commute with directlimits. (cid:3)
4. S
OME CRITERIA FOR THE B EILINSON –S OUL ´ E V ANISHING C ONJECTURE
In this section an application of the technique developed in the previous sections is given. Recallthat the Beilinson–Soul´e Vanishing Conjecture states that each complex Z ( n ) , n >
0, on Sm / k isacyclic outside the interval of cohomological degrees [ , n ] . It follows from [24, p. 352] that itsuffices to verify acyclicity of the complex Z ( n ) on Sm / k outside the interval [ , n ] whenever k isperfect. Therefore the base field k is assumed to be perfect throughout this section. he main result of this section, Theorem 4.1, gives equivalent conditions for the Beilinson–Soul´eVanishing Conjecture. In particular, it says that instead of verifying acyclicity of the sophisticatedcomplexes Z ( n ) in non-positive cohomological degrees, it is enough to verify acyclicity of the chaincomplexes of Abelian groups Z ( n )( b ∆ • K / k ) , where Z ( n ) : = Ker ∂ with ∂ being the zeroth differ-ential of the complex Z ( n ) , and K / k is a finitely generated field extension.4.1. Theorem.
The following conditions are equivalent: ( ) the Beilinson–Soul´e Vanishing Conjecture is true for complexes Z ( n ) , n > , on Sm / k; ( ) for every n > and every finitely generated field extension K / k, Z ( n )( b ∆ • K / k ) is an acyclicchain complex; ( ) for every n > , Z ( n ) has a resolution in the category of Zariski sheaves · · · d − −→ R − d − −→ R → Z ( n ) such that each R i is rationally contractible and cohomology presheaves H i < are trivial onthe semilocal schemes of the form b ∆ ℓ K / k , where K / k is a finitely generated field extension; ( ) for every n > and every finitely generated field extension K / k, the total complex of thebicomplex τ ( Z ( n ))( b ∆ • K / k ) is acyclic, where τ is the truncation in corresponding coho-mological degrees; ( ) for every n > and every finitely generated field extension K / k, the total complex of thebicomplex τ [ , n ] ( Z ( n ))( b ∆ • K / k ) is acyclic.Proof. ( ) ⇒ ( ) . Given ℓ > n >
0, set Z − ℓ ( n ) : = Ker ( ∂ − ℓ : Cor ( ∆ n + ℓ k × − , G ∧ nm ) → Cor ( ∆ n + ℓ − k × − , G ∧ nm )) . The proof of Theorem 2.4 and Corolary 2.5 shows that there is a tower in the derived category D ( Ab ) of chain complexes of Abelian groups · · · α − −→ Z − ( n )( b ∆ • K / k )[ − ] α − −→ Z − ( n )( b ∆ • K / k )[ − ] α −→ Z ( n )( b ∆ • K / k ) (5)with q -th layer, q >
0, being the complex H − q Zar ( b ∆ • K / k )[ − q ] . Here H − q Zar stands for the − q th coho-mology sheaf of the complex Z ( n ) . We use here the fact that Cor ( ∆ n + ℓ k × − , G ∧ nm ) is a rationallycontractible sheaf by [18, 2.2; 2.4]. By [5, 6.1.1] the tower (5) yields a strongly convergent spectralsequence E pq = H p + q ( H − q Zar ( b ∆ • K / k )) ⇒ H p + q ( Z ( n )( b ∆ • K / k )) (6)By assumption, H p Zar ( Z ( n )) = p
0. Therefore the strongly convergent spectral sequence (6)is trivial, and hence Z ( n )( b ∆ • K / k ) is acyclic. ( ) ⇒ ( ) . Suppose Z ( n − ) is acyclic outside the interval [ , n − ] . Voevodsky’s CancellationTheorem [25] together with [22, 4.34] implies thatHom ( G ∧ m , H p Zar ( Z ( n ))) = Hom ( G ∧ m , H p Nis ( Z ( n ))) = H p − ( Z ( n − )) = p
0. We use here that fact that F Zar = F Nis for any homotopy invariant presheaf withtransfers (see [22, 5.5]). It follows from [12, 2.5.2] that each H p Zar ( Z ( n )) , p
0, is a birational(Nisnevich) sheaf with transfers in the sense of [12, 2.3.1]. et K / k be a finitely generated field extension. Then K = k ( X ) for some X ∈ Sm / k . The proofof [11, 4.2.1] shows that the natural map of chain complexes H p Zar ( Z ( n ))( K ) → H p Zar ( Z ( n ))( b ∆ • K ) , p , is a quasi-isomorphism. Thus H i ( H p Zar ( Z ( n ))( b ∆ • K )) = i = H ( H p Zar ( Z ( n ))( b ∆ • K )) = H p Zar ( Z ( n ))( K ) . Therefore the strongly convergent spectral sequence (6) collapses, and hence0 = H i ( Z ( n )( b ∆ • K / k )) = H − i Zar ( Z ( n ))( K ) , i > . Each sheaf H − i Zar ( Z ( n )) is homotopy invariant by [22] and trivial on finitely generated field exten-sions. It follows from [22, 4.20] that H − i Zar ( Z ( n )) =
0, hence Z ( n ) is acyclic outside the interval [ , n ] . Using the fact that Z ( ) is a complex concentrated in degree zero, the above arguments showthat Z ( ) is acyclic in non-positive degrees. The implication now follows by induction in n . ( ) ⇒ ( ) . This is straightforward: set R ℓ : = Cor ( ∆ n + ℓ + k × − , G ∧ nm ) with differentials being thoseof Z ( n ) . We use here the facts that Cor ( ∆ n + ℓ k × − , G ∧ nm ) is rationally contractible [18, 2.2; 2.4] andthat for any smooth semilocal scheme W , any A -invariant presheaf F with transfers the canonicalmorphism F ( W ) ∼ = −→ F Zar ( W ) is an isomorphism [22, 4.24]. ( ) ⇒ ( ) . This follows from the spectral sequence of Theorem 2.4. ( ) ⇒ ( ) . This is obvious. ( ) ⇔ ( ) . It is enough to observe that the total complex of the bicomplex ( Z ( n ))( b ∆ • K / k ) is acyclicfor n >
0. The latter easily follows from [18, 2.2; 2.4] (see [2, 2.3] as well). ( ) ⇒ ( ) . The complex τ ( Z ( n )) equals · · · → Cor ( ∆ n + k × − , G ∧ nm ) → Cor ( ∆ n + k × − , G ∧ nm ) → Z ( n ) → → · · · Example 2.2(1), Proposition 2.3 and [19, 4.7] imply that the complex
Cor ( ∆ n + ℓ k × − , G ∧ nm )( b ∆ • K / k ) isacyclic for all n >
0. The spectral sequence for a double complex implies that the homology groupsof the complex τ ( Z ( n ))( b ∆ • K / k ) are those of the complex Z ( n )( b ∆ • K / k ) , and hence the remainingimplication follows. (cid:3) K OF A FIELD
In this section another application of semilocal Milnor K -theory is given. We show that the group K ( k ) is completely determined by extensions involving the classical Milnor K -theory and semilocalMilnor K -theory. If k is algebraically closed then K ( k ) is a direct sum of relevant Milnor K -theoryand semilocal Milnor K -theory groups of k .Recall that the motivic spectral sequence relates algebraic K -theory to motivic cohomology [5] E p , q = H q − p Zar ( k , Z ( q )) ⇒ K p + q ( k ) . (7)It is a strongly convergent spectral sequence concentrated in the first quadrant. It is obtained from atower of connected S -spectra · · · → K → K → K → K : = K ( k ) , (8) here K ( k ) is Quillen’s K -theory spectrum of k . Rationally, the motivic spectral sequence collapsesat E = E ∞ and K n ( k ) Q = M q H q − n Zar ( k , Q ( q )) (9)(see [6] for details).5.1. Proposition.
Let k be a perfect field (respectively any field with char ( k ) = p > ) and let F be ahomotopy invariant Nisnevich sheaf with transfers of Abelian groups (respectively Z [ / p ] -modules).Let b A k be the semilocalization of the affine line at , . Then F ( b A k ) = F ( k ) M M x ∈ A k \{ , } F − ( k ( x )) , where each x in the direct sum is a closed point of A k .Proof. Suppose k is perfect and U is an open subset of A k with Z = A k \ U = { x , . . . , x n } . The Gysintriangle for motives [23] gives a triangle in DM e f f ( k ) n M M ( G ∧ m , k ( x i ) ) → M ( U ) → M ( pt ) + −→ If x ∈ U is a rational point, then this triangle splits, in which case there is a canonical isomorphism F ( U ) = F ( k ) ⊕ ( ⊕ ni = F − ( k ( x i ))) . It follows that F ( b A k ) = colim U ∋{ , } F ( U ) = F ( k ) M M x ∈ A k \{ , } F − ( k ( x )) , as required. Here the splitting onto the first summand is given by x : = ∈ A k .The statement for fields of positive characteristic is the same if we use Suslin’s results [19] sayingthat Voevodsky’s theory for motivic complexes works for non-perfect fields as well provided that wedeal with sheaves with transfers of Z [ / p ] -modules. (cid:3) The following result says that the motivic cohomology groups H n − ( k , Z ( n )) fit into a finite towerof homomorphisms of Abelian groups with subsequent quotients being semilocal Milnor K -theorygroups.5.2. Theorem.
There are exact sequences of Abelian groups M x ∈ A k \{ , } b K M , ( k ( x )) u −→ H ( k , Z ( )) → b K M , ( k ) → and M x ∈ A k \{ , } H n − ( k ( x ) , Z ( n − )) u −→ H n − ( k , Z ( n )) → b K Mn , ( k ) → , where n > and each x of the left direct sums is a closed point of A k . roof. The second exact sequence follows from Proposition 5.1, Corollary 3.6 and CancellationTheorem [25]. We also use here the proof of Corollary 3.7 showing that H n − ( Z ( n )) is a sheaf of Z [ / p ] -modules whenever char ( k ) = p >
0. Here the homomorphism u is the restriction of ∂ − ∂ : H n − ( b A k , Z ( n )) → H n − ( k , Z ( n )) to L x ∈ A k \{ , } H n − ( k ( x ) , Z ( n − )) ⊂ H n − ( b A k , Z ( n )) . The first exact sequence is a particular caseof the second one if we apply Theorem 3.4(3). (cid:3) Corollary.
Let n > and X = { b K M ℓ, ( k ( x )) | x is a closed point in A k and ℓ n } . ThenH n − ( k , Z ( n )) belongs to the least localizing Serre subcategory of Ab containing X .Proof. This is a consequence of Theorem 5.2 and [7, Proposition 2]. (cid:3)
The motivic spectral sequence (7) gives a long exact sequence of Abelian groups H − ( k , Z ( )) → π ( K ) → K ( k ) → H ( k , Z ( )) d −→ K M ( k ) → K ( k ) → H ( k , Z ( )) → . Here K is the fourth entry of the tower (8). It follows from [26, VI.4.3.2] that d =
0. By Corol-lary 3.10 the latter long exact sequence can be rewritten as b K M , ( k ) → π ( K ) → K ( k ) → b K M , ( k ) −→ K M ( k ) → K ( k ) → b K M , ( k ) → . Next, by using the motivic spectral sequence we find that π ( K ) fits into an exact sequence K M ( k ) → π ( K ) → H ( k , Z ( )) → . By Theorem 5.2 H ( k , Z ( )) fits into an exact sequence M x ∈ A k \{ , } b K M , ( k ( x )) u −→ H ( k , Z ( )) → b K M , ( k ) → , where each x in the direct sum is a closed point of A k .We see that π ( K ) is expressed in terms of Milnor K -theory and semilocal Milnor K -theorygroups, and hence so is K ( k ) .We are now in a position to prove the main result of the section.5.4. Theorem.
Let k be any field. The following statements are true: ( ) K ( k ) is entirely expressed in terms of Milnor K-theory and semilocal Milnor K-theory groups.Precisely, K ( k ) fits into an exact sequence b K M , ( k ) → A → K ( k ) → b K M , ( k ) → , where A is an Abelian group fitted into an exact sequenceK M ( k ) → A → B → with B fitted into an exact sequence M x ∈ A k \{ , } b K M , ( k ( x )) → B → b K M , ( k ) → . ) There is an isomorphism of Abelian groupsK ( k ) Q ∼ = K M ( k ) Q ⊕ b K M , ( k ) Q ⊕ b K M , ( k ) ⊕ F , where F is a direct summand of L x ∈ A k \{ , } b K M , ( k ( x )) Q . ( ) If k is algebraically closed then there is an isomorphism of Abelian groupsK ( k ) ∼ = K M ( k ) ⊕ b K M , ( k ) Q ⊕ b K M , ( k ) ⊕ F , where F is a direct summand of L k × \{ } b K M , ( k ) Q .Proof. The first statement follows from the arguments above the theorem. The second statement isa consequence of the first statement and isomorphism (9). We also use here the fact that the group b K M , ( k ) is uniquely divisible by Corollary 3.11. Finally, the third statement follows from the secondone and the fact that K M ( k ) and K ( k ) are uniquely divisible Abelian groups if k is algebraicallyclosed (see, e.g., [26, pp. 267, 511, 514]). (cid:3)
6. O
N CONJECTURES OF B EILINSON AND P ARSHIN
Let k be a field of characteristic p >
0. A conjecture of Beilinson says that Milnor K -theory andQuillen K -theory agree rationally: K Mn ( k ) Q ∼ = −→ K n ( k ) Q . The purpose of this section is to show that the Beilinson conjecture is equivalent to vanishing ofthe rational semilocal Milnor K -theory. Since semilocal Milnor K -theory is defined in elementaryterms, its vanishing with rational coefficients should be much easier for verification than the originalBeilinson conjecture. We shall also show in this section that vanishing of the rational semilocalMilnor K -theory is a necessary condition for Parshin’s conjecture.6.1. Theorem.
The Beilinson conjecture for rational algebraic K-theory of fields of positive charac-teristic is true if and only if rational semilocal Milnor K-theory groups b K Mn , m ( k ) Q of such fields vanishfor all n > , m > .Proof. Assume the Beilinson conjecture. Then the isomorphism (9) implies H i Zar ( k , Q ( n )) = i = n , where k is a field of prime characteristic. It follows that Zariski cohomology sheaves exceptthe n th cohomology are zero (we use here [22, 4.20]). Now spectral sequence of Theorem 3.4 andCorollary 3.5 imply b K Mn , m ( k ) Q vanish for all n > , m > b K Mn , m ( k ) Q vanish for all n > , m > k of positive characteris-tic. By Theorem 3.4(3) 0 = b K M , − m ( k ) Q = H m ( k , Q ( )) , m . As above, it follows that all Zariski cohomology sheaves of Q ( ) are zero except the second coho-mology sheaf.Assume that the complex Q ( n ) , n >
2, has only one non-zero cohomology sheaf in degree n .Repeating the proof of Theorem 3.4(3) word for word, we obtain that H m Zar ( k , Q ( n + )) = b K Mn + , n − m + ( k ) Q = , m n . By induction, one has that H i Zar ( k , Q ( n )) = i = n . The isomorphism (9) now implies that thenatural homomorphism K Mn ( k ) Q → K n ( k ) Q is an isomorphism, as was to be shown. (cid:3) .2. Remark.
By Corollary 3.7, b K Mn , m ( k ) = b K Mn , m ( k ) ⊗ Z [ / p ] for all n > , m >
0. It follows that b K Mn , m ( k ) Q = b K Mn , m ( k ) ⊗ Z ( p ) . Therefore, b K Mn , m ( k ) Q = b K Mn , m ( k ) ⊗ Z ( p ) = X defined over a finitefield, the higher algebraic K -groups vanish rationally: K i ( X ) Q = , i > . We finish the paper by the following6.3.
Theorem.
Let k be a field of characteristic p > and assume Parshin’s conjecture. Then rationalsemilocal Milnor K-theory groups b K n , m ( k ) Q vanish for all n > , m > .Proof. It follows from [9, p. 203] that H i Zar ( k , Q ( n )) = i = n . The proof of Theorem 6.1 showsthat rational semilocal Milnor K -theory groups b K n , m ( k ) Q vanish for all n > , m >
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EPARTMENT OF M ATHEMATICS , S
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