A1-Algebraic Topology
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Slices of Hermitian K-Theory and Milnor's Conjecture on Quadratic Forms
Slices of hermitian K-theory and Milnor’s conjecture on quadratic forms Oliver R¨ondigs Paul Arne Østvær October 15, 2018 Abstract We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor’s conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology. Contents 1 Introduction 2 2 Slices and the slice spectral sequence 4 3 Algebraic and hermitian K-theory 8 4 Slices 11 4.1 Algebraic K-theory..................................... 11 4.2 Homotopy orbit K-theory ................................. 11 4.3 Hermitian K-theory .................................... 14 4.4 Themysterioussummandistrivial . ......... 17 4.5 HigherWitt-theory............................... ...... 21 5 Differentials 21 5.1 Mod-2 algebraic K-theory................................. 21 arXiv:1311.5833v2 [math.KT] 30 May 2017 5.2 Hermitian K-theory,I ................................... 23 5.3 HigherWitt-theory............................... ...... 25 5.4 Hermitian K-theory,II................................... 27 6 Milnor’s conjecture on quadratic forms 28 7 Hermitian K-groups 36 A Maps between motivic Eilenberg-MacLane spectra 40 1 1 Introduction Suppose that F is a field of characteristic char(F ) = 2. In [33] the Milnor K-theory of F is defined 6 in terms of generators and relations by KM (F )= T ∗F ×/(a (1 a)); a = 0, 1. ∗ ⊗ − 6 Here T ∗F × is the tensor algebra of the multiplicative group of units F ×. -
MILNOR K-THEORY and MOTIVIC COHOMOLOGY 1. Introduction A
MILNOR K-THEORY AND MOTIVIC COHOMOLOGY MORITZ KERZ Abstract. These are the notes of a talk given at the Oberwolfach Workshop K-Theory 2006. We sketch a proof of Beilinson’s conjecture relating Milnor K-theory and motivic cohomology. For detailed proofs see [4]. 1. Introduction A — semi-local commutative ring with infinite residue fields k — field Z(n) — Voevodsky’s motivic complex [8] Definition 1.1. M M × ⊗n × K∗ (A) = (A ) /(a ⊗ (1 − a)) a, 1 − a ∈ A n Beilinson conjectured [1]: Theorem 1.2. A/k essentially smooth, |k| = ∞. Then: M n η : Kn (A) −→ Hzar(A, Z(n)) is an isomorphism for n > 0. The formerly known cases are: Remark 1.3. — A=k a field (Nesterenko-Suslin [6], Totaro [10]) — surjectivity of η (Gabber [3], Elbaz-Vincent/M¨uller-Stach [2], Kerz/M¨uller-Stach [5]) — η ⊗ Q is isomorphic (Suslin) — injectivity for A a DVR, n = 3 (Suslin-Yarosh [9]) 2. General idea of proof X = Spec A We have a morphism of Gersten complexes which we know to be exact except possibly M at Kn (A): Date: 7/20/06. The author is supported by Studienstiftung des deutschen Volkes. 1 2 MORITZ KERZ (1) M M M 0 / Kn (A) / ⊕x∈X(0) Kn (x) / ⊕x∈X(1) Kn−1(x) n n n−1 0 / Hzar(A), Z(n)) / ⊕x∈X(0) Hzar(x, Z(n)) / ⊕x∈X(1) Hzar (x, Z(n − 1)) So it suffices to prove: Theorem 2.1 (Main Result). A/k regular, connected, infinite residue fields, F = Q(A). -
Theoretic Timeline Converging on Motivic Cohomology, Then Briefly Discuss Algebraic 퐾-Theory and Its More Concrete Cousin Milnor 퐾-Theory
AN OVERVIEW OF MOTIVIC COHOMOLOGY PETER J. HAINE Abstract. In this talk we give an overview of some of the motivations behind and applica- tions of motivic cohomology. We first present a 퐾-theoretic timeline converging on motivic cohomology, then briefly discuss algebraic 퐾-theory and its more concrete cousin Milnor 퐾-theory. We then present a geometric definition of motivic cohomology via Bloch’s higher Chow groups and outline the main features of motivic cohomology, namely, its relation to Milnor 퐾-theory. In the last part of the talk we discuss the role of motivic cohomology in Voevodsky’s proof of the Bloch–Kato conjecture. The statement of the Bloch–Kato conjec- ture is elementary and predates motivic cohomolgy, but its proof heavily relies on motivic cohomology and motivic homotopy theory. Contents 1. A Timeline Converging to Motivic Cohomology 1 2. A Taste of Algebraic 퐾-Theory 3 3. Milnor 퐾-theory 4 4. Bloch’s Higher Chow Groups 4 5. Properties of Motivic Cohomology 6 6. The Bloch–Kato Conjecture 7 6.1. The Kummer Sequence 7 References 10 1. A Timeline Converging to Motivic Cohomology In this section we give a timeline of events converging on motivic cohomology. We first say a few words about what motivic cohomology is. 푝 op • Motivic cohomology is a bigraded cohomology theory H (−; 퐙(푞))∶ Sm/푘 → Ab. • Voevodsky won the fields medal because “he defined and developed motivic coho- mology and the 퐀1-homotopy theory of algebraic varieties; he proved the Milnor conjectures on the 퐾-theory of fields.” Moreover, Voevodsky’s proof of the Milnor conjecture makes extensive use of motivic cohomology. -