The Work of Vladimir Voevodsky 101
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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 ×. -
FIELDS MEDAL for Mathematical Efforts R
Recognizing the Real and the Potential: FIELDS MEDAL for Mathematical Efforts R Fields Medal recipients since inception Year Winners 1936 Lars Valerian Ahlfors (Harvard University) (April 18, 1907 – October 11, 1996) Jesse Douglas (Massachusetts Institute of Technology) (July 3, 1897 – September 7, 1965) 1950 Atle Selberg (Institute for Advanced Study, Princeton) (June 14, 1917 – August 6, 2007) 1954 Kunihiko Kodaira (Princeton University) (March 16, 1915 – July 26, 1997) 1962 John Willard Milnor (Princeton University) (born February 20, 1931) The Fields Medal 1966 Paul Joseph Cohen (Stanford University) (April 2, 1934 – March 23, 2007) Stephen Smale (University of California, Berkeley) (born July 15, 1930) is awarded 1970 Heisuke Hironaka (Harvard University) (born April 9, 1931) every four years 1974 David Bryant Mumford (Harvard University) (born June 11, 1937) 1978 Charles Louis Fefferman (Princeton University) (born April 18, 1949) on the occasion of the Daniel G. Quillen (Massachusetts Institute of Technology) (June 22, 1940 – April 30, 2011) International Congress 1982 William P. Thurston (Princeton University) (October 30, 1946 – August 21, 2012) Shing-Tung Yau (Institute for Advanced Study, Princeton) (born April 4, 1949) of Mathematicians 1986 Gerd Faltings (Princeton University) (born July 28, 1954) to recognize Michael Freedman (University of California, San Diego) (born April 21, 1951) 1990 Vaughan Jones (University of California, Berkeley) (born December 31, 1952) outstanding Edward Witten (Institute for Advanced Study, -
Affine Grassmannians in A1-Algebraic Topology
AFFINE GRASSMANNIANS IN A1-ALGEBRAIC TOPOLOGY TOM BACHMANN Abstract. Let k be a field. Denote by Spc(k)∗ the unstable, pointed motivic homotopy category and A1 1 by R ΩGm : Spc(k)∗ →Spc(k)∗ the (A -derived) Gm-loops functor. For a k-group G, denote by GrG the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence A1 A1 R ΩGm G ≃ GrG. We use this to compute the motive M(R ΩGm G) ∈ DM(k, Z[1/e]). 1. Introduction This note deals with the subject of A1-algebraic topology. In other words it deals with with the ∞- category Spc(k) of motivic spaces over a base field k, together with the canonical functor Smk → Spc(k) and, importantly, convenient models for Spc(k). Since our results depend crucially on the seminal papers [1, 2], we shall use their definition of Spc(k) (which is of course equivalent to the other definitions in the literature): start with the category Smk of smooth (separated, finite type) k-schemes, form the universal homotopy theory on Smk (i.e. pass to the ∞-category P(Smk) of space-valued presheaves on k), and A1 then impose the relations of Nisnevich descent and contractibility of the affine line k (i.e. localise P(Smk) at an appropriate family of maps). The ∞-category Spc(k) is presentable, so in particular has finite products, and the functor Smk → Spc(k) preserves finite products. Let ∗ ∈ Spc(k) denote the final object (corresponding to the final k-scheme k); then we can form the pointed unstable motivic homotopy category Spc(k)∗ := Spc(k)/∗. -
Algebraic Geometry of Topological Spaces I 3
ALGEBRAIC GEOMETRY OF TOPOLOGICAL SPACES I GUILLERMO CORTINAS˜ AND ANDREAS THOM Abstract. We use techniques from both real and complex algebraic geometry to study K- theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parametrized version of a theorem of Joseph Gubeladze; we show that if M is a countable, abelian, cancella- tive, torsion-free, seminormal monoid, and X is contractible, then every finitely generated n projective module over C(X)[M] is free. The particular case M = N0 gives a parametrized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case M = Zn. We also give alge- braic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C∗-algebras, and for a homology theory of commutative algebras to vanish on C∗-algebras. These criteria have numerous applications. For example, the vanishing cri- terion applied to nil-K-theory implies that commutative C∗-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild-Kostant-Rosenberg and Loday-Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Be˘ılinson-Soul´eand Farrell-Jones are also given. -
An Intuitive Introduction to Motivic Homotopy Theory Vladimir Voevodsky
What follows is Vladimir Voevodsky's snapshot of his Fields Medal work on motivic homotopy, plus a little philosophy and \from my point of view the main fun of doing mathematics" Voevodsky (2002). Voevodsky uses extremely simple leading ideas to extend topological methods of homotopy into number theory and abstract algebra. The talk takes the simple ideas right up to the edge of some extremely refined applications. The talk is on-line as streaming video at claymath.org/video/. In a nutshell: classical homotopy studies a topological space M by looking at maps to M from the unit interval on the real line: [0; 1] = fx 2 R j 0 ≤ x ≤ 1 g as well as maps to M from the unit square [0; 1]×[0; 1], the unit cube [0; 1]3, and so on. Algebraic geometry cannot copy these naively, even over the real numbers, since of course the interval [0; 1] on the real line is not defined by a polynomial equation. The only polynomial equal to 0 all over [0; 1] is the constant polynomial 0, thus equals 0 over the whole line. The problem is much worse when you look at algebraic geometry over other fields than the real or complex numbers. Voevodsky shows how category theory allows abstract definitions of the basic apparatus of homotopy, where the category of topological spaces is replaced by any category equipped with objects that share certain purely categorical properties with the unit interval, unit square, and so on. For example, the line in algebraic geometry (over any field) does not have \endpoints" in the usual way but it does come with two naturally distinguished points: 0 and 1. -
Homotopy Theoretic Models of Type Theory
Homotopy Theoretic Models of Type Theory Peter Arndt1 and Krzysztof Kapulkin2 1 University of Oslo, Oslo, Norway [email protected] 2 University of Pittsburgh, Pittsburgh, PA, USA [email protected] Abstract. We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power so that we can soundly inter- pret dependent products and sums in it while also have purely intensional iterpretation of the identity types. On the other hand, those conditions are easy to check and provide a wide class of models that are examined in the paper. 1 Introduction Starting with the Hofmann-Streicher groupoid model [HS98] it has become clear that there exist deep connections between Intensional Martin-L¨ofType Theory ([ML72], [NPS90]) and homotopy theory. These connections have recently been studied very intensively. We start by summarizing this work | a more complete survey can be found in [Awo10]. It is well-known that Identity Types (or Equality Types) play an important role in type theory since they provide a relaxed and coarser notion of equality between terms of a given type. For example, assuming standard rules for type Nat one cannot prove that n: Nat ` n + 0 = n: Nat but there is still an inhabitant n: Nat ` p: IdNat(n + 0; n): Identity types can be axiomatized in a usual way (as inductive types) by form, intro, elim, and comp rules (see eg. [NPS90]). A type theory where we do not impose any further rules on the identity types is called intensional. -
A1-Algebraic Topology
A1-algebraic topology Fabien Morel Abstract. We present some recent results in A1-algebraic topology, which means both in A1-homotopy theory of schemes and its relationship with algebraic geometry. This refers to the classical relationship between homotopy theory and (differential) topology. We explain several examples of “motivic” versions of classical results: the theory of the Brouwer degree, the classification of A1-coverings through the A1-fundamental group, the Hurewicz Theorem and the A1-homotopy of algebraic spheres, and the A1-homotopy classification of vector bundles. We also give some applications and perspectives. Mathematics Subject Classification (2000). 14F05, 19E15, 55P. Keywords. A1-homotopy theory, Milnor K-theory, Witt groups. 1. The Brouwer degree Let n ≥ 1 be an integer and let X be a pointed topological space. We shall denote by πn(X) the n-th homotopy group of X. A basic fact in homotopy theory is: Theorem 1.1. Let n ≥ 1, d ≥ 1 be integers and denote by Sn the n-dimensional sphere. n 1) If d<nthen πd (S ) = 0; n 2) If d = n then πn(S ) = Z. A classical proof uses the Hurewicz Theorem and the computation of the integral singular homology of the sphere. Half of this paper is devoted to explain the analogue of these results in A1-homotopy theory [54], [38]. For our purpose we also recall a more geometric proof of 2) inspired by the definition of Brouwer’s degree. Any continuous map Sn → Sn is homotopic to a C∞-differentiable map f : Sn → Sn. By Sard’s theorem, f has at least one regular value x ∈ Sn, so that f −1(x) is a finite set of points in Sn and for each y ∈ f −1(x), the n n differential dfy : Ty(S ) → Tx(S ) of f at y is an isomorphism. -
Quillen's Work in Algebraic K-Theory
J. K-Theory 11 (2013), 527–547 ©2013 ISOPP doi:10.1017/is012011011jkt203 Quillen’s work in algebraic K-theory by DANIEL R. GRAYSON Abstract We survey the genesis and development of higher algebraic K-theory by Daniel Quillen. Key Words: higher algebraic K-theory, Quillen. Mathematics Subject Classification 2010: 19D99. Introduction This paper1 is dedicated to the memory of Daniel Quillen. In it, we examine his brilliant discovery of higher algebraic K-theory, including its roots in and genesis from topological K-theory and ideas connected with the proof of the Adams conjecture, and his development of the field into a complete theory in just a few short years. We provide a few references to further developments, including motivic cohomology. Quillen’s main work on algebraic K-theory appears in the following papers: [65, 59, 62, 60, 61, 63, 55, 57, 64]. There are also the papers [34, 36], which are presentations of Quillen’s results based on hand-written notes of his and on communications with him, with perhaps one simplification and several inaccuracies added by the author. Further details of the plus-construction, presented briefly in [57], appear in [7, pp. 84–88] and in [77]. Quillen’s work on Adams operations in higher algebraic K-theory is exposed by Hiller in [45, sections 1-5]. Useful surveys of K-theory and related topics include [81, 46, 38, 39, 67] and any chapter in Handbook of K-theory [27]. I thank Dale Husemoller, Friedhelm Waldhausen, Chuck Weibel, and Lars Hesselholt for useful background information and advice. 1. -
References Is Provisional, and Will Grow (Or Shrink) As the Course Progresses
Note: This list of references is provisional, and will grow (or shrink) as the course progresses. It will never be comprehensive. See also the references (and the book) in the online version of Weibel's K- Book [40] at http://www.math.rutgers.edu/ weibel/Kbook.html. References [1] Armand Borel and Jean-Pierre Serre. Le th´eor`emede Riemann-Roch. Bull. Soc. Math. France, 86:97{136, 1958. [2] A. K. Bousfield. The localization of spaces with respect to homology. Topology, 14:133{150, 1975. [3] A. K. Bousfield and E. M. Friedlander. Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. In Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, volume 658 of Lecture Notes in Math., pages 80{130. Springer, Berlin, 1978. [4] Kenneth S. Brown and Stephen M. Gersten. Algebraic K-theory as gen- eralized sheaf cohomology. In Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 266{ 292. Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973. [5] William Dwyer, Eric Friedlander, Victor Snaith, and Robert Thoma- son. Algebraic K-theory eventually surjects onto topological K-theory. Invent. Math., 66(3):481{491, 1982. [6] Eric M. Friedlander and Daniel R. Grayson, editors. Handbook of K- theory. Vol. 1, 2. Springer-Verlag, Berlin, 2005. [7] Ofer Gabber. K-theory of Henselian local rings and Henselian pairs. In Algebraic K-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), volume 126 of Contemp. Math., pages 59{70. Amer. Math. Soc., Providence, RI, 1992. [8] Henri Gillet and Daniel R. -
Prize Is Awarded Every Three Years at the Joint Mathematics Meetings
AMERICAN MATHEMATICAL SOCIETY LEVI L. CONANT PRIZE This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding fi ve years. Levi L. Conant (1857–1916) was a math- ematician who taught at Dakota School of Mines for three years and at Worcester Polytechnic Institute for twenty-fi ve years. His will included a bequest to the AMS effective upon his wife’s death, which occurred sixty years after his own demise. Citation Persi Diaconis The Levi L. Conant Prize for 2012 is awarded to Persi Diaconis for his article, “The Markov chain Monte Carlo revolution” (Bulletin Amer. Math. Soc. 46 (2009), no. 2, 179–205). This wonderful article is a lively and engaging overview of modern methods in probability and statistics, and their applications. It opens with a fascinating real- life example: a prison psychologist turns up at Stanford University with encoded messages written by prisoners, and Marc Coram uses the Metropolis algorithm to decrypt them. From there, the article gets even more compelling! After a highly accessible description of Markov chains from fi rst principles, Diaconis colorfully illustrates many of the applications and venues of these ideas. Along the way, he points to some very interesting mathematics and some fascinating open questions, especially about the running time in concrete situ- ations of the Metropolis algorithm, which is a specifi c Monte Carlo method for constructing Markov chains. The article also highlights the use of spectral methods to deduce estimates for the length of the chain needed to achieve mixing. -
The Unstable Slice Filtration
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 11, November 2014, Pages 5991–6025 S 0002-9947(2014)06116-3 Article electronically published on May 22, 2014 THE UNSTABLE SLICE FILTRATION PABLO PELAEZ Abstract. The main goal of this paper is to construct an analogue of Vo- evodsky’s slice filtration in the motivic unstable homotopy category. The construction is done via birational invariants; this is motivated by the ex- istence of an equivalence of categories between the orthogonal components for Voevodsky’s slice filtration and the birational motivic stable homotopy cate- gories constructed by the author in 2013. Another advantage of this approach is that the slices appear naturally as homotopy fibres (and not as in the sta- ble setting, where they are defined as homotopy cofibres) which behave much better in the unstable setting. 1. Introduction Our main goal is to construct a tower in the motivic unstable homotopy category with similar properties to Voevodsky’s slice tower [26] in the motivic stable homo- topy category. In this section we recall Voevodsky’s definition for the slice filtration and explain why our construction in the unstable setting is a natural analogue. Definitions and notation. In this paper X will denote a Noetherian separated base scheme of finite Krull dimension, SchX the category of schemes of finite type over X and SmX the full subcategory of SchX consisting of smooth schemes over X regarded as a site with the Nisnevich topology. All the maps between schemes will be considered over the base X.GivenY ∈ SchX , all the closed subsets Z of Y will be considered as closed subschemes with the reduced structure. -
Voevodsky-A1-Homotopy-Theory-Icm
Doc. Math. J. DMV 579 1 ÌheÓÖÝ A ¹ÀÓÑÓØÓÔÝ Vladimir Voevodsky Abstract. A1-homotopy theory is the homotopy theory for algebraic varieties and schemes which uses the affine line as a replacement for the unit interval. In the paper I present in detail the basic constructions of the theory following the sequence familiar from standard texbooks on algebraic topology. At the end I define motivic cohomology and algebraic cobordisms and describe algebraic K-theory in terms of this theory. Contents 1 Introduction.................................. 579 2 Spaces..................................... 581 3 Unstablehomotopycategory . 584 4 Spanier-Whiteheadcategory. 587 5 Spectraandthestablehomotopycategory . 591 6 Threecohomologytheories . 595 6.1 Motiviccohomology . 595 6.2 AlgebraicK-theory. 598 6.3 Algebraiccobordism . 601 7 Concludingremarks ............................. 601 1 Introduction In my talk I will outline the foundations of the A1-homotopy theory. This theory is based on the idea that one can define homotopies in the algebro-geometrical context using the affine line A1 instead of the unit interval. My exposition will follow the sequence familiar from the standard textbooks on topological homotopy theory which roughly looks as follows. Let C be a category which we want to study by means of homotopy theory. Usually C itself is not “good enough” and first one has to choose a convenient category of “spaces” Spc which contains C and has good categorical properties (in particular has internal Hom-objects and all small limits and colimits). In topol- ogy C may be the category of CW-complexes and Spc the category of compactly generated spaces ([8, §6.1]). Then one defines the class of weak equivalences on Spc.