aa r X i v : . [ m a t h . AG ] N ov MULTIPLICATIVE PROPERTIES OF THEMULTIPLICATIVE GROUP
BRUNO KAHN
Abstract.
We give a few properties equivalent to the Bloch-Katoconjecture (now the norm residue isomorphism theorem).
Introduction
The Bloch-Kato conjecture, now called the norm residue isomor-phism theorem, was finally proven by Voevodsky in 2011 [19], usingkey inputs from Rost. The proof has many ramifications and involvesa combination of sophisticated motivic techniques, including motivicSteenrod operations, and results of a more combinatorial kind like theexistence of norm varieties.This state of the art gives some interest to the issue of finding amore elementary proof. In this direction, one can consider the earlywork of Thomason on inverting the Bott element in algebraic K -theory[15] as a “stable” version of the conjecture; Levine later gave a motivicversion of Thomason’s theorem in [9]. I wondered how close to thenorm residue isomorphism theorem the latter work takes us; the resultis the following theorem, which was obtained in 2009. Theorem 1.
Let k be an infinite perfect field and let l be a primenumber invertible in k . If l = 2 , assume that k is non-exceptional inthe sense of Harris-Segal: the Galois group of the extension k ( µ ∞ ) /k is torsion-free. Then the following statements are equivalent: (i) The Beilinson-Lichtenbaum conjecture holds modulo l over k . (ii) For all n ≥ and all i > , H i ( G m ⊗ K Mn /l ) = 0 . Here thetensor product is taken in DM eff . (iii) For any n ≥ , any function field K/k , any semi-local K -algebra A and any ideals I, J ⊂ A with I ∩ J = 0 , the map K Mn ( A ) /l → K Mn ( A/I ) /l ⊕ K Mn ( A/J ) /l is injective. Date : November 2, 2017.2010
Mathematics Subject Classification. (iv)
Same as (iii), for A the coordinate ring of ˆ∆ qK,S for all q ≥ and all ∅ 6 = S ⊆ [0 , q ] and I, J defined by sets of vertices.
Here are some explanations on the notation. We assume the readerfamilar with Voevodsky’s category DM eff of effective motivic com-plexes [17, 11, 1]; in (ii) and later, H i is relative to its homotopy t -structure. The Beilinson-Lichtenbaum conjecture is recalled at the endof §
3: it is equivalent to the Bloch-Kato conjecture by [2, 14]. If A is acommutative semi-local ring, we write K M ∗ ( A ) for the Milnor ring of A in the na¨ıve sense, i.e. the quotient of the tensor algebra T ( A ∗ ) by thetwo-sided ideal generated by elements a ⊗ (1 − a ) with a, − a ∈ A ∗ .We shall write K Mn for the associated Nisnevich sheaf on the category Sm of smooth separated k -schemes of finite type.In (iv), we write ˆ∆ ∗ K for the cosimplicial K -scheme whose q -th termˆ∆ qK is the semi-localisation of ∆ qK = Spec K [ t , . . . , t q ] / ( P t i = 1) atits vertices. If i ∈ [0 , q ] (resp. S ⊆ [0 , q ]), we write ˆ∆ qK,i for the i -th faceof ˆ∆ qK and ˆ∆ qK,S = S i ∈ S ˆ∆ qK,i . We shall also write ∂ i for the inclusionˆ∆ qK,i ֒ → ˆ∆ qK ( i -th face map), and ˆ∆ qK, [0 ,q ] =: ∂ ˆ∆ qK .Of course, all statements in Theorem 1 are true since the first oneis. The game we shall play here, however, is to forget about this factand prove the equivalences without using it. Statement (ii) explainsthe title of this note. It is possible that such vanishing holds in moregenerality, which would be one possible direction of attack for a moreelementary proof of [19]. The scant evidence in this direction is a re-markable theorem of Sugiyama [13, Prop. A.1] that the tensor productof Nisnevich sheaves of Q -vector spaces with transfers is exact. Themost appealing leads are of course (iii) and (iv), because of their seem-ingly elementary nature. When I came up with Theorem 1, I tried toprove either of these statements by using the techniques of Guin andNesterenko-Suslin in [3, 12], but was not successful.(Added in November 2017.) When I sent this paper to Voevodskyin June 2017, he answered: I can not say that I knew this particular result, but Ihave encountered some facts of a similar nature andeven tried to prove some of them. Without any suc-cess. . . It is strange that the existing proof is the onlyone known.I am, BTW, partially in connection with my currentinterests, very interested in the elimination of the non-constructive elements from the proof of the BK or, at
ULTIPLICATIVE PROPERTIES OF THE MULTIPLICATIVE GROUP 3 least, from the proof of the Merkurjev-Suslin theoremabout K /l. The main such element is the use of the axiom ofchoice or rather of the existence of well-ordering on anyset quite early in the proof.I am very interested in finding a proof that avoids thispart of the argument.(....)I am sure that I can formalize constructively the state-ment of the BK. I can also formalize constructively mostof my mathematics such as the motivic Steenrod opera-tions.
This was a few months before his death on September 30th, 2017. Itwill take time for many of us to recover from it.1.
Proof of (i) ⇒ (iii)Recall that the Bloch-Kato conjecture is a special case of the Beilinson-Lichtenbaum conjecture; the statement thus follows from:1.1. Proposition.
We assume the Bloch-Kato conjecture holds modulo l . Let A be a semi-local k -algebra. Let I, J be two ideals of A such that I ∩ J = 0 . If n ≥ , the homomorphism K Mn ( A ) /l → K Mn ( A/I ) /l ⊕ K Mn ( A/J ) /l is injective.Proof. By Kerz [8, Th. 1.2], the norm residue homomorphism K Mn ( A ) /l → H n ´et ( A, µ ⊗ nl )is bijective for n ≥
1. By the usual transfer argument [8, Def. 5.5],we may assume that µ l ⊂ k . Recall that ´etale cohomology with finitecoefficients verifies closed Mayer-Vietoris, as a consequence of properbase change (for closed immersions!). Consider the diagram K Mn − ( A/I ) ⊗ µ l ⊕ K Mn − ( A/J ) ⊗ µ l −−−→ H n − ( A/I, µ ⊗ nl ) ⊕ H n − ( A/J, µ ⊗ nl ) a y y K Mn − ( A/I + J ) ⊗ µ l −−−→ H n − ( A/I + J, µ ⊗ nl ) ∂ y K Mn ( A ) /l −−−→ H n ´et ( A, µ ⊗ nl ) b y y K Mn ( A/I ) /l ⊕ K Mn ( A/J ) /l −−−→ H n ´et ( A/I, µ ⊗ nl ) ⊕ H n ´et ( A/J, µ ⊗ nl ) BRUNO KAHN where the horizontal maps are norm residue isomorphisms and ∂ is theboundary map for the long exact sequence corresponding to the closedcovering Spec A = Spec( A/I ) ∪ Spec(
A/J ). The two squares obviouslycommute, and all horizontal maps are isomorphisms since n ≥
2. But a is surjective, hence ∂ = 0, hence b is injective. (cid:3) Remark.
This proof does not work for n = 1. In fact the conclusionis false: the short exact sequence0 → A ∗ → ( A/I ) ∗ ⊕ ( A/J ) ∗ → ( A/I + J ) ∗ → → l A ∗ → l ( A/I ) ∗ ⊕ l ( A/J ) ∗ ρ −→ l ( A/I + J ) ∗ → A ∗ /l → ( A/I ) ∗ /l ⊕ ( A/J ) ∗ /l → ( A/I + J ) ∗ /l → ρ is finite but may be nontrivial if A/I + J is too disconnected.2. Motivic cohomology and Milnor K -theory For n ≥
0, the n -th motivic complex of Suslin and Voevodsky maybe defined as Z ( n ) = C ∗ ( G ∧ nm )[ − n ]where G ∧ nm denotes the direct summand of L (( A − n ) given by sec-tions trivial at ( A − i × { } × ( A − n − i − (0 ≤ i < n ) and C ∗ isthe Suslin complex [11, Th. 15.2]. We have the following basic results:2.1. Theorem ([14], [8, Th. 1.1]) . We have Z (0) = Z , Z (1) ≃ G m [ − , H i ( Z ( n )) = 0 for i > n and H n ( Z ( n )) = K Mn . Inverting the motivic Bott element, after Thomasonand Levine
Assume that k contains a primitive l -th root of unity: the Nisnevichsheaf µ l is then constant, cyclic of order l . From the exact triangle(3.1) µ l [0] → Z /l (1) → G m /l [ − +1 −→ and the isomorphism Z /l ( n ) ⊗ Z /l (1) ∼ −→ Z /l ( n + 1), we get a map in DM eff :(3.2) Z /l ( n ) ⊗ µ l → Z /l ( n + 1)hence another map Z /l ( n ) → Z /l ( n + 1) ⊗ µ − l which becomes an isomorphism after sheafifying for the ´etale topology.Let i ≤ n : iterating, we get a commutative diagram in HI , the heartof the homotopy t -structure of DM eff : ULTIPLICATIVE PROPERTIES OF THE MULTIPLICATIVE GROUP 5 H i ( Z /l ( n )) → H i ( Z /l ( n + 1) ⊗ µ − l ) → H i ( Z /l ( n + 2) ⊗ µ − l ) → . . . ψ in y ψ in +1 y ψ in +2 y H i ( Rα ∗ α ∗ Z /l ( n )) ∼ −→ H i ( Rα ∗ α ∗ ( Z /l ( n + 1) ⊗ µ − l )) ∼ −→ H i ( Rα ∗ α ∗ ( Z /l ( n + 2) ⊗ µ − l )) ∼ −→ . . . where α is the projection Sm ´et → Sm Nis . We have:3.1.
Theorem ([9, Th. 1.1]) . Assume that k is non exceptional if l = 2 .Then the direct limit of the above diagram is a (vertical) isomorphism. (For l = 2, Levine assumes either char k > k contains asquare root of −
1, but the hypothesis he actually uses is that k is notexceptional.)The Beilinson-Lichtenbaum conjecture is the statement that ψ in is anisomorphism for all ( i, n ) such that i ≤ n . Hence Theorem 3.1 implies:3.2. Proposition.
Under the assumption of Theorem 3.1, the Beilinson-Lichtenbaum conjecture holds modulo l if and only if the map H i ( Z /l ( n )) ⊗ µ l → H i ( Z /l ( n + 1)) is an isomorphism for any ( i, n ) such that i ≤ n . (cid:3) Reformulation of Proposition 3.2
Proposition. a) For all n ≥ , the objects G m ⊗ K Mn /l and Z ( n )[ n ] ⊗ G m /l of DM eff are concentrated in cohomological degrees ≤ (for the homotopy t -structure), and we have isomorphisms H ( G m ⊗ K Mn /l ) ≃ H ( Z ( n ) ⊗ G m /l [ n ]) ≃ K Mn +1 /l. b) Assume that k is non exceptional if l = 2 . Then the followingstatements are equivalent: (i) The Beilinson-Lichtenbaum conjecture holds modulo l . (ii) For all n ≥ , Z ( n ) ⊗ G m /l ∼ −→ K Mn +1 /l [ − n ] in DM eff . (iii) For all n ≥ , G m ⊗ K Mn /l ∼ −→ K Mn +1 /l [0] in DM eff . (iv) For all n ≥ , the image of K Mn /l [0] under the localisationfunctor ν ≤ : DM eff → DM o of [6, (4.5)] is , where DM o isthe category of birational motivic sheaves of [6] . (v) For any function field
K/k , any n ≥ and any q ≥ , we have H q ( K Mn /l ( ˆ∆ ∗ K )) = 0 . Proof. a) follows from Theorem 2.1, the isomorphism Z (1) ≃ G m [ − t -exactness of ⊗ [6, comment after (5.2)]. b) We reduceto µ l ⊂ k . Let C n be the cone of (3.2), so that C n ≃ Z ( n ) ⊗ G m /l [ − BRUNO KAHN
In view of a) and Proposition 3.2, (i) is equivalent to saying that C n isconcentrated in degree n + 1 and that the map K Mn +1 /l ≃ H n +1 ( Z /l ( n + 1)) → H n +1 ( C n )is an isomorphism. This shows that (i) ⇐⇒ (ii).The identity Z ( n ) ⊗ G m /l ≃ G m ⊗ Z ( n − ⊗ ( G m /l )[ − ⇐⇒ (iii) by induction on n (note that (ii) and (iii) areidentical for n = 1).By [6, Prop. 4.2.5], the statement in (iv) is equivalent to K Mn /l be-ing divisible by Z (1) in DM eff , which is implied by (iii). Conversely,if K Mn /l ≃ C (1) for some C ∈ DM eff , Voevodsky’s cancellation theo-rem [18] shows that C ≃ Hom( Z (1) , K Mn /l ) = Hom( G m , K Mn /l )[1] =( K Mn /l ) − [1] = K Mn − /l [1] (compare [7, Prop. 4.3 and Rk. 4.4]).For (iv) ⇐⇒ (v), we use [6, Rk. 4.6.3] (see also [5, Rk. 2.2.6]):let i o : DM o → DM eff be the inclusion. For any C ∈ C ( PST ) whichis A -invariant and satisfies Nisnevich excision, and for any connected Y ∈ Sm ( k ) with function field K , one has a quasi-isomorphism(4.1) ( i o ν ≤ C Nis )( Y ) ≃ R Γ( ˆ∆ ∗ K , C ) . For any
F ∈ HI , one has H q Nis ( X, F ) = 0 for q = 0 for any smoothsemi-local k -scheme X as a consequence of [16, Th. 4.37]. Therefore,the right hand side of (4.1) for C = F [0] is quasi-isomorphic to thecomplex associated to the simplicial abelian group F ( ˆ∆ ∗ K ) , which shows the equivalence of (iv) and (v) by taking F = K Mn /l . Thisconcludes the proof. (cid:3) Remark.
In Proposition 4.1 b), (ii) is also (trivially) true for n = 0,but not (iii) (see (3.1)).5. Elementary lemmas on Milnor K -groups Let A be a commutative semi-local ring, and let I be an ideal of A .We write (1 + I ) ∗ = (1 + I ) ∩ A ∗ = Ker( A ∗ → ( A/I ) ∗ ).5.1. Lemma.
Assume that | A/ m | > for all maximal ideals m of A .Then, with the above notation: (i) A ∗ → ( A/I ) ∗ is surjective. (ii) Let ¯ a ∈ A/I be such that ¯ a, − ¯ a ∈ ( A/I ) ∗ . Then there exists a ∈ A such that a ¯ a and a, − a ∈ A ∗ . ULTIPLICATIVE PROPERTIES OF THE MULTIPLICATIVE GROUP 7 (iii)
Let J be another ideal of A , with image ¯ J ⊂ A/I . Then (1 + J ) ∗ → (1 + ¯ J ) ∗ is surjective.Proof. Let R be the Jacobson radical of A , so that 1+ R ⊂ A ∗ . Assumefirst R = 0: then A is a finite product of fields and the three statementsare obvious (the cardinality hypothesis is used in (ii)). The general casefollows from chasing in the commutative square A −−−→ A/I y y
A/R −−−→ A/ ( R + I ) . (cid:3) Lemma.
Keep the assumption of Lemma 5.1. With the abovenotation, K M ∗ ( A ) → K M ∗ ( A/I ) is surjective with kernel the ideal gen-erated by (1 + I ) ∗ .Proof. The first assertion follows from Lemma 5.1 (i). To prove thesecond one, let us construct a surjective section to the surjection K M ∗ ( A ) { (1 + I ) ∗ } K M ∗ ( A ) −→→ K M ∗ ( A/I ) . It suffices to show that the surjective ring homomorphism T (( A/I ) ∗ ) →→ K M ∗ ( A ) / { (1 + I ) ∗ } K M ∗ ( A ) extending the identity map in degree 1kills the Steinberg relations: this follows from Lemma 5.1 (ii). (cid:3) Proposition.
Keep the assumption of Lemma 5.1, and let
I, J betwo ideals of A . Then the sequence K M ∗ ( A ) → K M ∗ ( A/I ) ⊕ K M ∗ ( A/J ) → K M ∗ ( A/I + J ) → is exact.Proof. Let ¯ I be the image of I in A/J . Consider the commutativediagram { (1 + I ) ∗ } K M ∗ ( A ) −−−→ K M ∗ ( A ) −−−→ K M ∗ ( A/I ) −−−→ y y y { (1 + ¯ I ) ∗ } K M ∗ ( A/J ) −−−→ K M ∗ ( A/J ) −−−→ K M ∗ ( A/I + J ) −−−→ . By Lemma 5.2, the rows are exact and the middle and right verticalmaps are surjective; by Lemma 5.1 (iii), the left vertical map is alsosurjective. The claim now follows from a diagram chase. (cid:3)
BRUNO KAHN End of proof of Theorem 1
Lemma.
Let sLoc be the category of semi-local K -schemes. Let F be a contravariant functor from sLoc to abelian groups. Suppose that,for any X ∈ sLoc and any closed cover X = Z ∪ Z , the sequence → F ( X ) → F ( Z ) ⊕ F ( Z ) → F ( Z ∩ Z ) is exact. (Here, Z ∩ Z := Z × X Z is the scheme-theoretic intersec-tion.) Then, for any closed cover X = Z ∪ · · · ∪ Z r , the sequence → F ( X ) → r M j =1 F ( Z j ) → M j End of proof of Theorem 1. We saw in § ⇒ (iii); we have (i) ⇐⇒ (ii) by the equivalence (i) ⇐⇒ (iii) in Proposition 4.1 b).Obviously, (iii) ⇒ (iv). It remains to show that (iv) ⇒ (i).Suppose that (iv) holds in Theorem 1. In view of Proposition 5.3and (the proof of) Lemma 6.1, we get for all q > → K Mn ( ∂ ˆ∆ qK ) /l → q M j =0 K Mn ( ˆ∆ qK,j ) /l → M j For A ∗ = K M ∗ ( ˆ∆ ∗ K ) /l , the homology group of the bottomrow of (6.1) may be reinterpreted in a more suggestive way: it isCoker (cid:16) H ( ˆ∆ i +1 K , F ) → H ( ∂ ˆ∆ i +1 K , F ) (cid:17) where oc denotes the open-closed topology introduced in [4]. References [1] A. Beilinson, V. Vologodsky A DG guide to Voevodsky’s motives , Geom. Funct.Anal. (2008), 1709–1787.[2] T. Geisser, M. Levine The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky , J. Reine Angew. Math. (2001), 55–103.[3] D. Guin Homologie du groupe lin´eaire et K -th´eorie de Milnor des anneaux , J.Alg. (1989), 27–59.[4] B. Kahn The Geisser-Levine method revisited and algebraic cycles over a finitefield , Math. Ann. (2002), 581–617.[5] B. Kahn, M. Levine Motives of Azumaya algebras , J. Inst. Math. Jussieu (2010), 481–599.[6] B. Kahn, R. Sujatha Birational motives, II: triangulated birational motives ,IMRN , doi: 10.1093/imrn/rnw184. [7] B. Kahn, T. Yamazaki Voevodsky’s motives and Weil reciprocity , Duke Math.J. (2013), 2751–2796.[8] M. Kerz The Gersten conjecture for Milnor K -theory , Invent. Math. (2009), 1–33.[9] M. Levine Inverting the motivic Bott element , K -Theory (2000), 1–28.[10] M. Levine Techniques of localization in the theory of algebraic cycles , J. Alg.Geom. (2001), 299–363.[11] C. Mazza, V. Voevodsky, C. Weibel Lecture notes on motivic cohomology, ClayMath. Monographs , AMS, Clay Math. Inst., 2006.[12] Yu. Nesterenko, A. Suslin Homology of the general linear group over a localring, and Milnor’s K -theory (Russian), Izv. Akad. Nauk SSSR Ser. Mat. (1989), 121–146; translation in Math. USSR-Izv. (1990), 121–145.[13] R Sugiyama Motivic homology of a semiabelian variety over a perfect field ,Doc. Math. (2014), 1061–1084.[14] A. Suslin, V. Voevodsky Bloch-Kato conjecture and motivic cohomology withfinite coefficients , in The arithmetic and geometry of algebraic cycles (Banff,AB, 1998), 117–189, NATO Sci. Ser. C Math. Phys. Sci., , Kluwer, 2000.[15] R. Thomason Algebraic K -theory and ´etale cohomology , Ann. Sci. ´Ec. Norm.Sup. (1985), 437–552.[16] V. Voevodsky Cohomological theory of presheaves with transfers , in Cycles,transfers, and motivic homology theories, Ann. of Math. Stud. , PrincetonUniv. Press, 2000, 87–137.[17] V. Voevodsky Triangulated categories of motives over a field , in Cycles, trans-fers, and motivic homology theories, Ann. of Math. Stud. , Princeton Univ.Press, 2000, 188–238.[18] V. Voevodsky Cancellation theorem , Doc. Math. 2010, Extra volume: AndreiA. Suslin sixtieth birthday, 671–685.[19] V. Voevodsky On motivic cohomology with Z /l -coefficients , Annals of Math. (2011), 401–438. IMJ-PRG, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France E-mail address ::