NORM VARIETIES AND ALGEBRAIC COBORDISM

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1 NORM VARIETIES AND ALGEBRAIC COBORDISM MARKUS ROST Abstract. We outline briefly results and examles related with the bijectivity of the norm residue homomorhism. We define norm varieties and describe some constructions. Further we discuss degree formulas which form a major tool to handle norm varieties. Finally we formulate Hilbert s 90 for symbols which is the hard art of the bijectivity of the norm residue homomorhism, modulo a theorem of Voevodsky. Introduction This text is a brief outline of results and examles related with the bijectivity of the norm residue homomorhism also called Bloch-Kato conjecture and, for the mod 2 case, Milnor conjecture. The starting oint was a result of Voevodsky which he communicated in Voevodsky s theorem basically reduces the Bloch-Kato conjecture to the existence of norm varieties and to what I call Hilbert s 90 for symbols. Unfortunately there is no text available on Voevodsky s theorem. In this exosition is a rime, k is a field with char k and K M n k denotes Milnor s n-th K-grou of k [15], [19]. Elements in K M n k/ of the form u = {a 1,..., a n } mod are called symbols (mod, of weight n). A field extension F of k is called a slitting field of u if u F = 0 in K M n F/. Let be the norm residue homomorhism. h (n,) : K M n k/ H ń et(k, µ n ) {a 1,..., a n } (a 1,..., a n ) 1. Norm Varieties All successful aroaches to the Bloch-Kato conjecture consist of an investigation of aroriate generic slitting varieties of symbols. This goes back to the work of Merkurjev and Suslin on the case n = 2 who studied the K-cohomology of Severi- Brauer varieties [12]. Similarly, for the case = 2 (for n = 3 by Merkurjev, Suslin [14] and the author [18], for all n by Voevodsky [23]) one considers certain quadrics associated with Pfister forms. For a long time it was not clear which sort of varieties one should consider for arbitrary n,. In some cases one knew candidates, but these were non-smooth varieties and desingularizations aeared to be difficult to handle Mathematics Subject Classification. 12G05. Key words and hrases. Milnor s K-ring, Galois cohomology, Cobordism. 1

2 2 MARKUS ROST Finally Voevodsky roosed a surrising characterization of the necessary varieties. It involves characteristic numbers and yields a beautiful relation between symbols and cobordism theory. Definition. Let u = {a 1,..., a n } mod be a symbol. Assume that u 0. A norm variety for u is a smooth roer irreducible variety X over k such that (1) The function field k(x) of X slits u. (2) dim X = d := n 1 1 (3) s d(x) 0 mod Here s d (X) Z denotes the characteristic number of X given by the d-th Newton olynomial in the Chern classes of T X. It is known (by Milnor) that in dimensions d = n 1 the number s d (X) is -divisible for any X. If k C one may rehrase condition (3) by saying that X(C) is indecomosable in the comlex cobordism ring mod. We will observe in section 2 that the conditions for a norm variety are birational invariant. The name norm variety originates from some constructions of norm varieties, see section 3. We conclude this section with the classical examles of norm varieties. Examle. The case n = 2. Assume that k contains a rimitive -th root ζ of unity. For a, b k let A ζ (a, b) be the central simle k-algebra with resentation A ζ (a, b) = u, v u = a, v = b, vu = ζuv The Severi-Brauer variety X(a, b) of A ζ (a, b) is a norm variety for the symbol {a, b} mod. Examle. The case = 2. For a 1,..., a n k one denotes by n a 1,..., a n = 1, a i the associated n-fold Pfister form [9], [21]. The quadratic form ϕ = a 1,..., a n 1 a n is called a Pfister neighbor. The rojective quadric Q(ϕ) defined by ϕ = 0 is a norm variety for the symbol {a 1,..., a n } mod Degree formulas The theme of degree formulas goes back to Voevodsky s first text on the Milnor conjecture (although he never formulated exlicitly a formula ) [22]. In this section we formulate the degree formula for the characteristic numbers s d. It shows the birational invariance of the notion of norm varieties. The first roof of this formula relied on Voevodsky s stable homotoy theory of algebraic varieties. Later we found a rather elementary aroach [11], which is in sirit very close to elementary aroaches to the comlex cobordism ring [16], [4]. For our aroach to Hilbert s 90 for symbols we use also higher degree formulas which again were first settled using Voevodsky s stable homotoy theory [3]. These follow meanwhile also from the general degree formula roved by Morel and Levine [10] in characteristic 0 using factorization theorems for birational mas [1]. 1

3 NORM VARIETIES AND ALGEBRAIC COBORDISM 3 We fix a rime and a number d of the form d = n 1. For a roer variety X over k let I(X) = deg ( CH 0 (X) ) Z be the image of the degree ma on the grou of 0-cycles. One has I(X) = i(x)z where i(x) is the index of X, i. e., the gcd of the degrees [k(x) : k] of the residue class field extensions of the closed oints x of X. If X has a k-oint (in articular if k is algebraically closed), then I(X) = Z. The grou I(X) is a birational invariant of X. We ut J(X) = I(X) + Z Let X, Y be irreducible smooth roer varieties over k with dim Y = dim X = d and let f : Y X be a morhism. Define deg f as follows: If dim f(y ) < dim X, then deg f = 0. Otherwise deg f N is the degree of the extension k(y )/k(x) of the function fields. Theorem (Degree formula for s d ). Corollary. The class is a birational invariant. s d (Y ) s d (X) = (deg f) s d(x) mod J(X) mod J(X) Z/J(X) Remark. If X has a k-rational oint, then J(X) = Z and the degree formula is emty. The degree formula and the birational invariants s d (X)/ mod J(X) are henomena which are interesting only over non-algebraically closed fields. Over the comlex numbers the only characteristic numbers which are birational invariant are the Todd numbers. We aly the degree formula to norm varieties. Let u be a nontrivial symbol mod and let X be a norm variety for u. Since k(x) slits u, so does any residue class field k(x) for x X. As u is of exonent, it follows that J(X) = Z. Corollary (Voevodsky). Let u be a nontrivial symbol and let X be a norm variety of u. Let further Y be a smooth roer irreducible variety with dim Y = dim X and let f : Y X be a morhism. Then Y is a norm variety for u if and only if deg f is rime to. It follows in articular that the notion of norm variety is birational invariant. Therefore we may call any irreducible variety U (not necessarily smooth or roer) a norm variety of a symbol u if U is birational isomorhic to a smooth and roer norm variety of u. 3. Existence of norm varieties Theorem. Norm varieties exists for every symbol u K M n k/ for every and every n. As we have noted, for the case n = 2 one can take aroriate Severi-Brauer varieties (if k contains the -th roots of unity) and for the case = 2 one can take aroriate quadrics.

4 4 MARKUS ROST In this exosition we describe a roof for the case n = 3 using fix-oint theorems of Conner and Floyd in order to comute the non-triviality of the characteristic numbers. Our first roof for the general case used also Conner-Floyd fix-oint theory. Later we found two further methods which are comaratively simler. However the Conner-Floyd fix-oint theorem is still used in our aroach to Hilbert s 90 for symbols. Let u = {a, b, c} mod with a, b, c k. Assume that k contains a rimitive -th root ζ of unity, let A = A ζ (a, b) and let MS(A, c) = { x A Nrd(x) = c } We call MS(A, c) the Merkurjev-Suslin variety associated with A and c. The symbol u is trivial if and only if MS(A, c) has a rational oint [12]. The variety MS(A, c) is a twisted form of SL(). Theorem. Suose u 0. Then MS(A, c) is a norm variety for u. Let us indicate a roof for a subfield k C (and for > 2). Let U = MS(A, c). It is easy to see that k(u) slits u. Moreover one has dim U = dim A 1 = 2 1. It remains to show that there exists a roer smooth comletion X of U with nontrivial characteristic number. Let Ū = { [x, t] P(A k) Nrd(x) = ct } be the naive comletion of U. We let the grou G = Z/ Z/ act on the algebra A via (r, s) u = ζ r u, (r, s) v = ζ s v This action extends to an action on P(A k) (with the trivial action on k) which induces a G-action on Ū. Let Fix(Ū) be the fixed oint scheme of this action. One finds that Fix(Ū) consists just of the isolated oints [1, ζi c], i = 1,...,, which are all contained in U. The variety U is smooth, but Ū is not. However, by equivariant resolution of singularities [2], there exists a smooth roer G-variety X together with a G- morhism X Ū which is a birational isomorhism and an isomorhism over U. It remains to show that s d (X) 0 mod ( ) For this we may ass to toology and try to comute s d X(C). We note that for odd, the Chern number s d is also a Pontryagin number and deends only on the differentiable structure of the given variety. Note further that X has the same G-fixed oints as Ū since the desingularization took lace only outside U. Consider the variety { [ ] Z = i,j=1 x iju i v j, t P(A k) } i,j=1 x ij = ct This variety is a smooth hyersurface and it is easy to check s d (Z) 0 mod As a G-variety, the variety Z has the same fixed oints as X ( same means that the collections of fix-oints together with the G-structure on the tangent saces are isomorhic). Let M be the differentiable manifold obtained from X(C) and Z(C) by a multi-fold connected sum along corresonding fixed oints. Then M

5 NORM VARIETIES AND ALGEBRAIC COBORDISM 5 is a G-manifold without fixed oints. By the theory of Conner and Floyd [5], [7] alied to (Z/) 2 -manifolds of dimension d = 2 1 one has s d (M) 0 Thus s d (X) s d(z) and the desired non-triviality is established. mod mod The functions Φ n. We conclude this section with examles of norm varieties for the general case. Let a 1, a 2,... be a sequence of elements in k. We define functions Φ n = Φ a1,...,a n in n variables inductively as follows. Φ 0 (t) = t 1 ( Φ n (T 0,..., T 1 ) = Φ n 1 (T 0 ) 1 an Φ n 1 (T i ) ) Here the T i stand for tules of n 1 variables. Let U(a 1,..., a n ) be the variety defined by Φ a1,...,a n 1 (T ) = a n Theorem. Suose that the symbol u = {a 1,..., a n } mod is nontrivial. Then U(a 1,..., a n ) is a norm variety of u. i=1 4. Hilbert s 90 for symbols The bijectivity of the norm residue homomorhisms has always been considered as a sort of higher version of the classical Hilbert s Theorem 90 (which establishes the bijectivity for n = 1). In fact, there are various variants of the Bloch-Kato conjecture which are obvious generalizations of Hilbert s Theorem 90: The Hilbert s Theorem 90 for Kn M of cyclic extensions or the vanishing of the motivic cohomology grou H n+1( k, Z(n) ). In this section we describe a variant which on one hand is very elementary to formulate and on the other hand is the really hard art of the Bloch-Kato conjecture (modulo Voevodsky s theorem). Let u = {a 1,..., a n } Kn M k/ be a symbol. Consider the norm ma N u = F N F/k : F K 1 F K 1 k where F runs through the finite field extensions of k (contained in some algebraic closure of k) which slit u. Hilbert s Theorem 90 for u states that ker N u is generated by the obvious elements. To make this recise, we consider two tyes of basic relations between the norm mas N F/k. Let F 1, F 2 be finite field extensions of k. Then the sequence (1) K 1 (F 1 F 2 ) (N F 1 F 2 /F 1, N F1 F 2 /F 2 ) K 1 F 1 K 1 F 2 N F1 /k+n F2 /k K 1 k is a comlex.

6 6 MARKUS ROST Further, if K/k is of transcendence degree 1, then the sequence (2) K 2 K d K v K 1 κ(v) N K 1 k is a comlex. Here v runs through the valuations of K/k, d K is given by the tame symbols at each v and N is the sum of the norm mas N κ(v)/k. The sum formula N d K = 0 is also known as Weil s formula. We now restrict again to slitting fields of u. The mas in (1) yield a ma R u = (N F1 F 2/F 1, N F1 F 2/F 2 ): K 1 (F 1 F 2 ) K 1 F F 1,F 2 F 1,F 2 F with N u R u = 0. Let C be the cokernel of R u and let N u : C K 1 k be the ma induced by N u. Then the mas in (2) yield a ma S u = K d K : K K 2 K C with N u S u = 0 where K runs through the slitting fields of u of transcendence degree 1 over k (contained in some universal field). Let H 0 (u, K 1 ) be the cokernel of S u and let N u : H 0 (u, K 1 ) K 1 k be the ma induced by N u. Hilbert s 90 for symbols. For every symbol u the norm ma is injective. N u : H 0 (u, K 1 ) K 1 k Examle. If u = 0, then it is easy to see that N u is injective. In fact, it is a trivial exercise to check that N u is injective. Examle. The case n = 1. The slitting fields F of u = {a} mod are exactly the field extensions of k containing a -th root of a. It is an easy exercise to reduce the injectivity of N u (in fact of N u) to the classical Hilbert s Theorem 90, i. e., the exactness of K 1 L 1 σ K 1 L N L/k K 1 k for a cyclic extension L/k of degree with σ a generator of Gal(L/k). Examle. The case n = 2. Assume that k contains a rimitive -th root ζ of unity. The slitting fields F of u = {a, b} mod are exactly the slitting fields of the algebra A ζ (a, b). One can show that H 0 (u, K 1 ) = K 1 A ζ (a, b) with N u corresonding to the reduced norm ma Nrd [13]. Hence in this case Hilbert s 90 for u reduces to the classical fact SK 1 A = 0 for central simle algebras of rime degree [6]. Examle. The case = 2. The slitting fields F of u = {a 1,..., a n } mod 2 are exactly the field extensions of k which slit the Pfister form a 1,..., a n or, equivalently, over which the Pfister neighbor a 1,..., a n 1 a n becomes isotroic. Hilbert s 90 for symbols mod 2 had been first established in [17]. This text considered similar norm mas associated with any quadratic form (which are not injective in general). A treatment of the secial case of Pfister forms is contained in [8].

7 NORM VARIETIES AND ALGEBRAIC COBORDISM 7 Remark. One can show that the grou H 0 (u, K 1 ) as defined above is also the quotient of F K 1 F by the R-trivial elements in ker N u. This is quite analogous to the descrition of K 1 A of a central simle algebra A: The grou K 1 A is the quotient of A by the subgrou of R-trivial elements in the kernel of Nrd: A F. Similarly for the case = 2: In this case the injectivity of N u is related with the fact that for Pfister neighbors ϕ the kernel of the sinor norm SO(ϕ) k /(k ) 2 is R-trivial. In our aroach to Hilbert s 90 for symbols one needs a arameterization of the slitting fields of symbols. Definition. Let u = {a 1,..., a n } mod be a symbol. A -generic slitting variety for u is a smooth variety X over k such that for every slitting field F of u there exists a finite extension F /F of degree rime to and a morhism Sec F X. Theorem. Suose char k = 0. Let m 3 and suose for n m and every symbol u = {a 1,..., a n } mod over all fields over k there exists a -generic slitting variety for u of dimension n 1 1. Then Hilbert s 90 holds for such symbols. The roof of this theorem is outlined in [20]. For n = 2 one can take here the Severi-Brauer varieties and for n = 3 the Merkurjev-Suslin varieties. Hence we have: Corollary. Suose char k = 0. Then Hilbert s 90 holds for symbols of weight 3. References [1] D. Abramovich, K. Karu, K. Matsuki, and J. W lodarczyk, Torification and factorization of birational mas, J. Amer. Math. Soc. 15 (2002), no. 3, (electronic). [2] E. Bierstone and P. D. Milman, Canonical desingularization in characteristic zero by blowing u the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, [3] S. Borghesi, Algebraic Morava K-theories and the higher degree formula, thesis, 2000, Evanston, htt:// [4] S. Buoncristiano and D. Hacon, The geometry of Chern numbers, Ann. of Math. (2) 118 (1983), no. 1, 1 7. [5] P. E. Conner and E. E. Floyd, Differentiable eriodic mas, Academic Press Inc., Publishers, New York, 1964, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 33. [6] P. K. Draxl, Skew fields, London Mathematical Society Lecture Note Series, vol. 81, Cambridge University Press, Cambridge, [7] E. E. Floyd, Actions of (Z ) k without stationary oints, Toology 10 (1971), [8] B. Kahn, La conjecture de Milnor (d arès V. Voevodsky), Astérisque (1997), no. 245, Ex. No. 834, 5, , Séminaire Bourbaki, Vol. 1996/97. [9] T. Y. Lam, The algebraic theory of quadratic forms, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1980, Revised second rinting, Mathematics Lecture Note Series. [10] M. Levine and F. Morel, Algebraic cobordism I, rerint, 2002, htt:// [11] A. S. Merkurjev, Degree formula, notes, May 2000, htt:// rost/chain-lemma.html. [12] A. S. Merkurjev and A. A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorhism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, , (Russian), [Math. USSR Izv. 21 (1983), ]. [13], The grou of K 1 -zero-cycles on Severi-Brauer varieties, Nova J. Algebra Geom. 1 (1992), no. 3, [14], Norm residue homomorhism of degree three, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 2, (Russian), [Math. USSR Izv. 36 (1991), no. 2, ], also: LOMIrerint (1986).

8 8 MARKUS ROST [15] J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970), [16] D. Quillen, Elementary roofs of some results of cobordism theory using Steenrod oerations, Advances in Math. 7 (1971), (1971). [17] M. Rost, On the sinor norm and A 0 (X, K 1 ) for quadrics, rerint, 1988, htt:// rost/sinor.html. [18], Hilbert 90 for K 3 for degree-two extensions, Prerint, [19], Chow grous with coefficients, Doc. Math. 1 (1996), No. 16, (electronic). [20], Chain lemma for slitting fields of symbols, rerint, 1998, htt:// rost/chain-lemma.html. [21] W. Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischen Wissenschaften, vol. 270, Sringer-Verlag, Berlin, [22] V. Voevodsky, The Milnor conjecture, rerint, 1996, Max-Planck-Institute for Mathematics, Bonn, htt:// [23], On 2-torsion in motivic cohomology, rerint, 2001, htt:// Deartment of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA address: rost@math.ohio-state.edu URL: htt://