MULTIPLICATIVE PROPERTIES OF THE MULTIPLICATIVE GROUP

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1 MULTIPLICATIVE PROPERTIES OF THE MULTIPLICATIVE GROUP BRUNO KAHN Abstract. We give a few properties equivaent to the Boch-Kato conjecture (now the norm residue isomorphism theorem. Introduction The Boch-Kato conjecture, now caed the norm residue isomorphism theorem, was finay proven by Voevodsky in 2011 [19], using key inputs from Rost. The proof has many ramifications and invoves a combination of sophisticated motivic techniques, incuding motivic Steenrod operations, and resuts of a more combinatoria kind ike the existence of norm varieties. This state of the art gives some interest to the issue of finding a more eementary proof. In this direction, one can consider the eary work of Thomason on inverting the Bott eement in agebraic K-theory [15] as a stabe version of the conjecture; Levine ater gave a motivic version of Thomason s theorem in [9]. I wondered how cose to the norm residue isomorphism theorem the atter work takes us; the resut is the foowing theorem, which was obtained in Theorem 1. Let k be an infinite perfect fied and et be a prime number invertibe in k. If = 2, assume that k is non-exceptiona in the sense of Harris-Sega: the Gaois group of the extension k(µ 2 /k is torsion-free. Then the foowing statements are equivaent: (i The Beiinson-Lichtenbaum conjecture hods moduo over k. (ii For a n 1 and a i > 0, H i (G m Kn M / = 0. Here the tensor product is taken in DM eff. (iii For any n 2, any function fied K/k, any semi-oca K- agebra A and any ideas I, J A with I J = 0, the map is injective. K M n (A/ K M n (A/I/ K M n (A/J/ Date: November 2, Mathematics Subject Cassification. 19D45, 14C15 (19E15. 1

2 2 BRUNO KAHN (iv Same as (iii, for A the coordinate ring of ˆ q K,S for a q 2 and a S [0, q] and I, J defined by sets of vertices. Here are some expanations on the notation. We assume the reader famiar with Voevodsky s category DM eff of effective motivic compexes [17, 11, 1]; in (ii and ater, H i is reative to its homotopy t- structure. The Beiinson-Lichtenbaum conjecture is recaed at the end of 3: it is equivaent to the Boch-Kato conjecture by [2, 14]. If A is a commutative semi-oca ring, we write K M (A for the Minor ring of A in the naïve sense, i.e. the quotient of the tensor agebra T (A by the two-sided idea generated by eements a (1 a with a, 1 a A. We sha write Kn M 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 cosimpicia K-scheme whose q-th term ˆ q K is the semi-ocaisation of q K = Spec K[t 0,..., t q ]/( t i = 1 at its vertices. If i [0, q] (resp. S [0, q], we write ˆ q K,i for the i-th face of ˆ q K and ˆ q K,S = ˆ q i S K,i. We sha aso write i for the incusion ˆ q K,i ˆ q K (i-th face map, and ˆ q K,[0,q] =: ˆ q K. Of course, a statements in Theorem 1 are true since the first one is. The game we sha pay here, however, is to forget about this fact and prove the equivaences without using it. Statement (ii expains the tite of this note. It is possibe that such vanishing hods in more generaity, which woud be one possibe direction of attack for a more eementary proof of [19]. The scant evidence in this direction is a remarkabe theorem of Sugiyama [13, Prop. A.1] that the tensor product of Nisnevich sheaves of Q-vector spaces with transfers is exact. The most appeaing eads are of course (iii and (iv, because of their seemingy eementary nature. When I came up with Theorem 1, I tried to prove either of these statements by using the techniques of Guin and Nesterenko-Susin in [3, 12], but was not successfu. (Added in November When I sent this paper to Voevodsky in June 2017, he answered: I can not say that I knew this particuar resut, but I have encountered some facts of a simiar nature and even tried to prove some of them. Without any success... It is strange that the existing proof is the ony one known. I am, BTW, partiay in connection with my current interests, very interested in the eimination of the nonconstructive eements from the proof of the BK or, at

3 MULTIPLICATIVE PROPERTIES OF THE MULTIPLICATIVE GROUP 3 east, from the proof of the Merkurjev-Susin theorem about K 2 /. The main such eement is the use of the axiom of choice or rather of the existence of we-ordering on any set quite eary in the proof. I am very interested in finding a proof that avoids this part of the argument. (... I am sure that I can formaize constructivey the statement of the BK. I can aso formaize constructivey most of my mathematics such as the motivic Steenrod operations. This was a few months before his death on September 30th, It wi take time for many of us to recover from it. 1. Proof of (i (iii Reca that the Boch-Kato conjecture is a specia case of the Beiinson- Lichtenbaum conjecture; the statement thus foows from: 1.1. Proposition. We assume the Boch-Kato conjecture hods moduo. Let A be a semi-oca k-agebra. Let I, J be two ideas of A such that I J = 0. If n 2, the homomorphism is injective. K M n (A/ K M n (A/I/ K M n (A/J/ Proof. By Kerz [8, Th. 1.2], the norm residue homomorphism K M n (A/ H ń et(a, µ n is bijective for n 1. By the usua transfer argument [8, Def. 5.5], we may assume that µ k. Reca that étae cohomoogy with finite coefficients verifies cosed Mayer-Vietoris, as a consequence of proper base change (for cosed immersions!. Consider the diagram Kn 1(A/I M µ Kn 1(A/J M µ a H n 1 ét K M n 1(A/I + J µ H n 1 ét Kn M (A/ Het ń b K M n (A/I/ K M n (A/J/ H ń et (A/I, µ n H n 1 ét (A/J, µ n (A/I + J, µ n (A, µ n (A/I, µ n Het ń (A/J, µ n

4 4 BRUNO KAHN where the horizonta maps are norm residue isomorphisms and is the boundary map for the ong exact sequence corresponding to the cosed covering Spec A = Spec(A/I Spec(A/J. The two squares obviousy commute, and a horizonta maps are isomorphisms since n 2. But a is surjective, hence = 0, hence b is injective Remark. This proof does not work for n = 1. In fact the concusion is fase: the short exact sequence 0 A (A/I (A/J (A/I + J 0 yieds a ong exact sequence 0 A (A/I (A/J ρ (A/I + J A / (A/I / (A/J / (A/I + J / 0 so Coker ρ is finite but may be nontrivia if A/I +J is too disconnected. 2. Motivic cohomoogy and Minor K-theory For n 0, the n-th motivic compex of Susin and Voevodsky may be defined as Z(n = C (G n m [ n] where G n m denotes the direct summand of L((A 1 0 n given by sections trivia at (A 1 0 i {1} (A 1 0 n i 1 (0 i < n and C is the Susin compex [11, Th. 15.2]. We have the foowing basic resuts: 2.1. Theorem ([14], [8, Th. 1.1]. We have Z(0 = Z, Z(1 G m [ 1], H i (Z(n = 0 for i > n and H n (Z(n = K M n. 3. Inverting the motivic Bott eement, after Thomason and Levine Assume that k contains a primitive -th root of unity: the Nisnevich sheaf µ is then constant, cycic of order. From the exact triange (3.1 µ [0] Z/(1 G m /[ 1] +1 and the isomorphism Z/(n Z/(1 Z/(n + 1, we get a map in DM eff : (3.2 Z/(n µ Z/(n + 1 hence another map Z/(n Z/(n + 1 µ 1 which becomes an isomorphism after sheafifying for the étae topoogy. Let i n: iterating, we get a commutative diagram in HI, the heart of the homotopy t-structure of DM eff :

5 MULTIPLICATIVE PROPERTIES OF THE MULTIPLICATIVE GROUP 5 H i (Z/(n H i (Z/(n + 1 µ 1 H i (Z/(n + 2 µ 2... ψn i ψn+2 i ψ i n+1 H i (Rα α Z/(n H i (Rα α (Z/(n + 1 µ 1 H i (Rα α (Z/(n + 2 µ 2... where α is the projection Smét Sm Nis. We have: 3.1. Theorem ([9, Th. 1.1]. Assume that k is non exceptiona if = 2. Then the direct imit of the above diagram is a (vertica isomorphism. (For = 2, Levine assumes either char k > 0 or that k contains a square root of 1, but the hypothesis he actuay uses is that k is not exceptiona. The Beiinson-Lichtenbaum conjecture is the statement that ψ i n is an isomorphism for a (i, n such that i n. Hence Theorem 3.1 impies: 3.2. Proposition. Under the assumption of Theorem 3.1, the Beiinson- Lichtenbaum conjecture hods moduo if and ony if the map H i (Z/(n µ H i (Z/(n + 1 is an isomorphism for any (i, n such that i n. 4. Reformuation of Proposition Proposition. a For a n 0, the objects G m Kn M / and Z(n[n] G m / of DM eff are concentrated in cohomoogica degrees 0 (for the homotopy t-structure, and we have isomorphisms H 0 (G m K M n / H 0 (Z(n G m /[n] K M n+1/. b Assume that k is non exceptiona if = 2. Then the foowing statements are equivaent: (i The Beiinson-Lichtenbaum conjecture hods moduo. (ii For a n 1, Z(n G m / Kn+1/[ n] M in DM eff. (iii For a n 1, G m Kn M / Kn+1/[0] M in DM eff. (iv For a n 2, the image of Kn M /[0] under the ocaisation functor ν 0 : DM eff DM o of [6, (4.5] is 0, where DM o is the category of birationa motivic sheaves of [6]. (v For any function fied K/k, any n 2 and any q 0, we have H q (K M n /( ˆ K = 0. Proof. a foows from Theorem 2.1, the isomorphism Z(1 G m [ 1] and the right t-exactness of [6, comment after (5.2]. b We reduce to µ k. Let C n be the cone of (3.2, so that C n Z(n G m /[ 1].

6 6 BRUNO KAHN In view of a and Proposition 3.2, (i is equivaent to saying that C n is concentrated in degree n + 1 and that the map K M n+1/ H n+1 (Z/(n + 1 H n+1 (C n is an isomorphism. This shows that (i (ii. The identity Z(n G m / G m Z(n 1 (G m /[ 1] shows that (ii (iii by induction on n (note that (ii and (iii are identica for n = 1. By [6, Prop ], the statement in (iv is equivaent to K M n / being divisibe by Z(1 in DM eff, which is impied by (iii. Conversey, if K M n / C(1 for some C DM eff, Voevodsky s canceation theorem [18] shows that C Hom(Z(1, K M n / = Hom(G m, K M n /[1] = (K M n / 1 [1] = K M n 1/[1] (compare [7, Prop. 4.3 and Rk. 4.4]. For (iv (v, we use [6, Rk ] (see aso [5, Rk ]: et i o : DM o DM eff be the incusion. For any C C(PST which is A 1 -invariant and satisfies Nisnevich excision, and for any connected Y Sm(k with function fied K, one has a quasi-isomorphism (4.1 (i o ν 0 C Nis (Y RΓ( ˆ K, C. For any F HI, one has H q Nis (X, F = 0 for q 0 for any smooth semi-oca 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 the compex associated to the simpicia abeian group F( ˆ K, which shows the equivaence of (iv and (v by taking F = Kn M /. This concudes the proof Remark. In Proposition 4.1 b, (ii is aso (triviay true for n = 0, but not (iii (see ( Eementary emmas on Minor K-groups Let A be a commutative semi-oca ring, and et I be an idea of A. We write (1 + I = (1 + I A = Ker(A (A/I Lemma. Assume that A/m > 2 for a maxima ideas m of A. Then, with the above notation: (i (ii A (A/I is surjective. Let ā A/I be such that ā, 1 ā (A/I. Then there exists a A such that a ā and a, 1 a A.

7 MULTIPLICATIVE PROPERTIES OF THE MULTIPLICATIVE GROUP 7 (iii Let J be another idea of A, with image J A/I. Then (1 + J (1 + J is surjective. Proof. Let R be the Jacobson radica of A, so that 1+R A. Assume first R = 0: then A is a finite product of fieds and the three statements are obvious (the cardinaity hypothesis is used in (ii. The genera case foows from chasing in the commutative square A A/I A/R A/(R + I Lemma. Keep the assumption of Lemma 5.1. With the above notation, K M (A K M (A/I is surjective with kerne the idea generated by (1 + I. Proof. The first assertion foows from Lemma 5.1 (i. To prove the second one, et us construct a surjective section to the surjection K M (A {(1 + I }K M (A KM (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 1 kis the Steinberg reations: this foows from Lemma 5.1 (ii Proposition. Keep the assumption of Lemma 5.1, and et I, J be two ideas of A. Then the sequence is exact. K M (A K M (A/I K M (A/J K M (A/I + J 0 Proof. Let Ī be the image of I in A/J. diagram Consider the commutative {(1 + I }K M (A K M (A K M (A/I 0 {(1 + Ī }K M (A/J K M (A/J K M (A/I + J 0. By Lemma 5.2, the rows are exact and the midde and right vertica maps are surjective; by Lemma 5.1 (iii, the eft vertica map is aso surjective. The caim now foows from a diagram chase.

8 8 BRUNO KAHN 6. End of proof of Theorem Lemma. Let sloc be the category of semi-oca K-schemes. Let F be a contravariant functor from sloc to abeian groups. Suppose that, for any X sloc and any cosed cover X = Z 1 Z 2, the sequence 0 F (X F (Z 1 F (Z 2 F (Z 1 Z 2 is exact. (Here, Z 1 Z 2 := Z 1 X Z 2 is the scheme-theoretic intersection. Then, for any cosed cover X = Z 1 Z r, the sequence r 0 F (X F (Z j Z k is exact. j=1 F (Z j j<k Proof. Of course this emma is much more genera and the point is to spe out its proof. Let Y = Z 1... Z r 1. By hypothesis, the sequence 0 F (X F (Y F (Z r F (Y Z r is exact and, by induction on r, the map F (Y Z r j<r F (Z j Z r is injective. The concusion foows by chasing in the diagram r 0 F (X F (Z j Z k 0 F (Y F (Z j j=1 j<k r 1 F (Z j j=1 j<k<r F (Z j Z k Lemma (See aso [10, Lemma 2.4]. Let A = (A n n 0 be a simpicia abeian group. Let (A 0 n be the normaised compex of A: A 0 r = i>0 Ker( i : A r A r 1. For q > 0, consider the commutative diagram of compexes (6.1 A 0 q+1 α A q+1 0 A 0 0 q β q a b i=0 A q A 0 q 1 γ 0 j<k q A q 1

9 MULTIPLICATIVE PROPERTIES OF THE MULTIPLICATIVE GROUP 9 where α is incusion, β(x = (x, 0,..., 0, γ(x j,k = a = ( i 0 i q and the (i, j, k-component of b is 0 if i j, k k 1 if i = j j if i = k. Then this diagram induces an injection on homoogy. Proof. Obvious. { x si (j, k = (0, 1 0 ese, End of proof of Theorem 1. We saw in 1 that (i (iii; we have (i (ii by the equivaence (i (iii in Proposition 4.1 b. Obviousy, (iii (iv. It remains to show that (iv (i. Suppose that (iv hods in Theorem 1. In view of Proposition 5.3 and (the proof of Lemma 6.1, we get for a q > 0 an exact sequence 0 K M n ( ˆ q K / q j=0 K M n ( ˆ q K,j / j<k K M n ( ˆ q K,j,k /. But Kn M ( ˆ q K KM n ( ˆ q K is surjective, hence the bottom row of (6.1 is exact for A = K M ( ˆ K /. By Lemma 6.2, Condition (v of Proposition 4.1 b hods, and therefore so does its Condition (i. This concudes the proof Remark. For A = K M ( ˆ K /, the homoogy group of the bottom row of (6.1 may be reinterpreted in a more suggestive way: it is ( Coker HZar( 0 i+1 ˆ K, F H0 i+1 oc( ˆ K, F where oc denotes the open-cosed topoogy introduced in [4]. References [1] A. Beiinson, V. Voogodsky A DG guide to Voevodsky s motives, Geom. Funct. Ana. 17 (2008, [2] T. Geisser, M. Levine The Boch-Kato conjecture and a theorem of Susin- Voevodsky, J. Reine Angew. Math. 530 (2001, [3] D. Guin Homoogie du groupe inéaire et K-théorie de Minor des anneaux, J. Ag. 123 (1989, [4] B. Kahn The Geisser-Levine method revisited and agebraic cyces over a finite fied, Math. Ann. 324 (2002, [5] B. Kahn, M. Levine Motives of Azumaya agebras, J. Inst. Math. Jussieu 9 (2010, [6] B. Kahn, R. Sujatha Birationa motives, II: trianguated birationa motives, IMRN 2016, doi: /imrn/rnw184.

10 10 BRUNO KAHN [7] B. Kahn, T. Yamazaki Voevodsky s motives and Wei reciprocity, Duke Math. J. 162 (2013, [8] M. Kerz The Gersten conjecture for Minor K-theory, Invent. Math. 175 (2009, [9] M. Levine Inverting the motivic Bott eement, K-Theory 19 (2000, [10] M. Levine Techniques of ocaization in the theory of agebraic cyces, J. Ag. Geom. 10 (2001, [11] C. Mazza, V. Voevodsky, C. Weibe Lecture notes on motivic cohomoogy, Cay Math. Monographs 2, AMS, Cay Math. Inst., [12] Yu. Nesterenko, A. Susin Homoogy of the genera inear group over a oca ring, and Minor s K-theory (Russian, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989, ; transation in Math. USSR-Izv. 34 (1990, [13] R Sugiyama Motivic homoogy of a semiabeian variety over a perfect fied, Doc. Math. 19 (2014, [14] A. Susin, V. Voevodsky Boch-Kato conjecture and motivic cohomoogy with finite coefficients, in The arithmetic and geometry of agebraic cyces (Banff, AB, 1998, , NATO Sci. Ser. C Math. Phys. Sci., 548, Kuwer, [15] R. Thomason Agebraic K-theory and étae cohomoogy, Ann. Sci. Éc. Norm. Sup. 18 (1985, [16] V. Voevodsky Cohomoogica theory of presheaves with transfers, in Cyces, transfers, and motivic homoogy theories, Ann. of Math. Stud. 143, Princeton Univ. Press, 2000, [17] V. Voevodsky Trianguated categories of motives over a fied, in Cyces, transfers, and motivic homoogy theories, Ann. of Math. Stud. 143, Princeton Univ. Press, 2000, [18] V. Voevodsky Canceation theorem, Doc. Math. 2010, Extra voume: Andrei A. Susin sixtieth birthday, [19] V. Voevodsky On motivic cohomoogy with Z/-coefficients, Annas of Math. 174 (2011, IMJ-PRG, Case 247, 4 pace Jussieu, Paris Cedex 05, France E-mai address: bruno.kahn@imj-prg.fr