Lecture Notes on Motivic Cohomology
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What Is the Motivation Behind the Theory of Motives?
What is the motivation behind the Theory of Motives? Barry Mazur How much of the algebraic topology of a connected simplicial complex X is captured by its one-dimensional cohomology? Specifically, how much do you know about X when you know H1(X, Z) alone? For a (nearly tautological) answer put GX := the compact, connected abelian Lie group (i.e., product of circles) which is the Pontrjagin dual of the free abelian group H1(X, Z). Now H1(GX, Z) is canonically isomorphic to H1(X, Z) = Hom(GX, R/Z) and there is a canonical homotopy class of mappings X −→ GX which induces the identity mapping on H1. The answer: we know whatever information can be read off from GX; and are ignorant of anything that gets lost in the projection X → GX. The theory of Eilenberg-Maclane spaces offers us a somewhat analogous analysis of what we know and don’t know about X, when we equip ourselves with n-dimensional cohomology, for any specific n, with specific coefficients. If we repeat our rhetorical question in the context of algebraic geometry, where the structure is somewhat richer, can we hope for a similar discussion? In algebraic topology, the standard cohomology functor is uniquely characterized by the basic Eilenberg-Steenrod axioms in terms of a simple normalization (the value of the functor on a single point). In contrast, in algebraic geometry we have a more intricate set- up to deal with: for one thing, we don’t even have a cohomology theory with coefficients in Z for varieties over a field k unless we provide a homomorphism k → C, so that we can form the topological space of complex points on our variety, and compute the cohomology groups of that topological space. -
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, -
The Simplicial Parallel of Sheaf Theory Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 10, No 4 (1968), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES YUH-CHING CHEN Costacks - The simplicial parallel of sheaf theory Cahiers de topologie et géométrie différentielle catégoriques, tome 10, no 4 (1968), p. 449-473 <http://www.numdam.org/item?id=CTGDC_1968__10_4_449_0> © Andrée C. Ehresmann et les auteurs, 1968, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE COSTACKS - THE SIMPLICIAL PARALLEL OF SHEAF THEORY by YUH-CHING CHEN I NTRODUCTION Parallel to sheaf theory, costack theory is concerned with the study of the homology theory of simplicial sets with general coefficient systems. A coefficient system on a simplicial set K with values in an abe - . lian category 8 is a functor from K to Q (K is a category of simplexes) ; it is called a precostack; it is a simplicial parallel of the notion of a presheaf on a topological space X . A costack on K is a « normalized» precostack, as a set over a it is realized simplicial K/" it is simplicial « espace 8ta18 ». The theory developed here is functorial; it implies that all >> homo- logy theories are derived functors. Although the treatment is completely in- dependent of Topology, it is however almost completely parallel to the usual sheaf theory, and many of the same theorems will be found in it though the proofs are usually quite different. -
Publication List, Dated 2010
References [1] V. A. Voevodski˘ı. Galois group Gal(Q¯ =Q) and Teihmuller modular groups. In Proc. Conf. Constr. Methods and Alg. Number Theory, Minsk, 1989. [2] V. A. Voevodski˘ıand G. B. Shabat. Equilateral triangulations of Riemann surfaces, and curves over algebraic number fields. Dokl. Akad. Nauk SSSR, 304(2):265{268, 1989. [3] G. B. Shabat and V. A. Voevodsky. Drawing curves over number fields. In The Grothendieck Festschrift, Vol. III, volume 88 of Progr. Math., pages 199{227. Birkh¨auser Boston, Boston, MA, 1990. [4] V. A. Voevodski˘ı. Etale´ topologies of schemes over fields of finite type over Q. Izv. Akad. Nauk SSSR Ser. Mat., 54(6):1155{1167, 1990. [5] V. A. Voevodski˘ı. Flags and Grothendieck cartographical group in higher dimensions. CSTARCI Math. Preprints, (05-90), 1990. [6] V. A. Voevodski˘ı.Triangulations of oriented manifolds and ramified coverings of sphere (in Russian). In Proc. Conf. of Young Scientists. Moscow Univ. Press, 1990. [7] V. A. Voevodski˘ı and M. M. Kapranov. 1-groupoids as a model for a homotopy category. Uspekhi Mat. Nauk, 45(5(275)):183{184, 1990. [8] V. A. Voevodski˘ıand M. M. Kapranov. Multidimensional categories (in Russian). In Proc. Conf. of Young Scientists. Moscow Univ. Press, 1990. [9] V. A. Voevodski˘ıand G. B. Shabat. Piece-wise euclidean approximation of Jacobians of algebraic curves. CSTARCI Math. Preprints, (01-90), 1990. [10] M. M. Kapranov and V. A. Voevodsky. Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results). Cahiers Topologie G´eom.Diff´erentielle Cat´eg., 32(1):11{27, 1991. -
Lecture 3. Resolutions and Derived Functors (GL)
Lecture 3. Resolutions and derived functors (GL) This lecture is intended to be a whirlwind introduction to, or review of, reso- lutions and derived functors { with tunnel vision. That is, we'll give unabashed preference to topics relevant to local cohomology, and do our best to draw a straight line between the topics we cover and our ¯nal goals. At a few points along the way, we'll be able to point generally in the direction of other topics of interest, but other than that we will do our best to be single-minded. Appendix A contains some preparatory material on injective modules and Matlis theory. In this lecture, we will cover roughly the same ground on the projective/flat side of the fence, followed by basics on projective and injective resolutions, and de¯nitions and basic properties of derived functors. Throughout this lecture, let us work over an unspeci¯ed commutative ring R with identity. Nearly everything said will apply equally well to noncommutative rings (and some statements need even less!). In terms of module theory, ¯elds are the simple objects in commutative algebra, for all their modules are free. The point of resolving a module is to measure its complexity against this standard. De¯nition 3.1. A module F over a ring R is free if it has a basis, that is, a subset B ⊆ F such that B generates F as an R-module and is linearly independent over R. It is easy to prove that a module is free if and only if it is isomorphic to a direct sum of copies of the ring. -
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)/∗. -
Arithmetic Duality Theorems
Arithmetic Duality Theorems Second Edition J.S. Milne Copyright c 2004, 2006 J.S. Milne. The electronic version of this work is licensed under a Creative Commons Li- cense: http://creativecommons.org/licenses/by-nc-nd/2.5/ Briefly, you are free to copy the electronic version of the work for noncommercial purposes under certain conditions (see the link for a precise statement). Single paper copies for noncommercial personal use may be made without ex- plicit permission from the copyright holder. All other rights reserved. First edition published by Academic Press 1986. A paperback version of this work is available from booksellers worldwide and from the publisher: BookSurge, LLC, www.booksurge.com, 1-866-308-6235, [email protected] BibTeX information @book{milne2006, author={J.S. Milne}, title={Arithmetic Duality Theorems}, year={2006}, publisher={BookSurge, LLC}, edition={Second}, pages={viii+339}, isbn={1-4196-4274-X} } QA247 .M554 Contents Contents iii I Galois Cohomology 1 0 Preliminaries............................ 2 1 Duality relative to a class formation . ............. 17 2 Localfields............................. 26 3 Abelianvarietiesoverlocalfields.................. 40 4 Globalfields............................. 48 5 Global Euler-Poincar´echaracteristics................ 66 6 Abelianvarietiesoverglobalfields................. 72 7 An application to the conjecture of Birch and Swinnerton-Dyer . 93 8 Abelianclassfieldtheory......................101 9 Otherapplications..........................116 AppendixA:Classfieldtheoryforfunctionfields............126 -
A Algebraic Varieties
A Algebraic Varieties A.1 Basic definitions Let k[X1,X2,...,Xn] be a polynomial algebra over an algebraically closed field k with n indeterminates X1,...,Xn. We sometimes abbreviate it as k[X]= k[X1,X2,...,Xn]. Let us associate to each polynomial f(X)∈ k[X] its zero set n V(f):= {x = (x1,x2,...,xn) ∈ k | f(x)= f(x1,x2,...,xn) = 0} n 2in the n-fold product set k of k. For any subset S ⊂ k[X] we also set V(S) = f ∈S V(f). Then we have the following properties: n (i) 2V(1) =∅,V(0) =k . = (ii) i∈I V(Si) V( i∈I Si). (iii) V(S1) ∪ V(S2) = V(S1S2), where S1S2 := {fg | f ∈ S1,g∈ S2}. The inclusion ⊂ of (iii) is clear. We will prove only the inclusion ⊃. For x ∈ V(S1S2) \ V(S2) there is an element g ∈ S2 such that g(x) = 0. On the other hand, it ∀ ∀ follows from x ∈ V(S1S2) that f(x)g(x) = 0( f ∈ S1). Hence f(x)= 0( f ∈ S1) and x ∈ V(S1). So the part ⊃ was also proved. By (i), (ii), (iii) the set kn is endowed with the structure of a topological space by taking {V(S) | S ⊂ k[X]} to be its closed subsets. We call this topology of kn the Zariski topology. The closed subsets V(S) of kn with respect to it are called algebraic sets in kn. Note that V(S)= V(S), where S denotes the ideal of k[X] generated by S. -
Derived Functors of /-Adic Completion and Local Homology
JOURNAL OF ALGEBRA 149, 438453 (1992) Derived Functors of /-adic Completion and Local Homology J. P. C. GREENLEES AND J. P. MAY Department q/ Mathematics, Unil;ersi/y of Chicago, Chicago, Illinois 60637 Communicated by Richard G. Swan Received June 1. 1990 In recent topological work [2], we were forced to consider the left derived functors of the I-adic completion functor, where I is a finitely generated ideal in a commutative ring A. While our concern in [2] was with a particular class of rings, namely the Burnside rings A(G) of compact Lie groups G, much of the foundational work we needed was not restricted to this special case. The essential point is that the modules we consider in [2] need not be finitely generated and, unless G is finite, say, the ring ,4(G) is not Noetherian. There seems to be remarkably little information in the literature about the behavior of I-adic completion in this generality. We presume that interesting non-Noetherian commutative rings and interesting non-finitely generated modules arise in subjects other than topology. We have therefore chosen to present our algebraic work separately, in the hope that it may be of value to mathematicians working in other fields. One consequence of our study, explained in Section 1, is that I-adic com- pletion is exact on a much larger class of modules than might be expected from the key role played by the Artin-Rees lemma and that the deviations from exactness can be computed in terms of torsion products. However, the most interesting consequence, discussed in Section 2, is that the left derived functors of I-adic completion usually can be computed in terms of certain local homology groups, which are defined in a fashion dual to the definition of the classical local cohomology groups of Grothendieck. -
Algebraic K-Theory (University of Washington, Seattle, 1997) 66 Robert S
http://dx.doi.org/10.1090/pspum/067 Selected Titles in This Series 67 Wayne Raskind and Charles Weibel, Editors, Algebraic K-theory (University of Washington, Seattle, 1997) 66 Robert S. Doran, Ze-Li Dou, and George T. Gilbert, Editors, Automorphic forms, automorphic representations, and arithmetic (Texas Christian University, Fort Worth, 1996) 65 M. Giaquinta, J. Shatah, and S. R. S. Varadhan, Editors, Differential equations: La Pietra 1996 (Villa La Pietra, Florence, Italy, 1996) 64 G. Ferreyra, R. Gardner, H. Hermes, and H. Sussmann, Editors, Differential geometry and control (University of Colorado, Boulder, 1997) 63 Alejandro Adem, Jon Carlson, Stewart Priddy, and Peter Webb, Editors, Group representations: Cohomology, group actions and topology (University of Washington, Seattle, 1996) 62 Janos Kollar, Robert Lazarsfeld, and David R. Morrison, Editors, Algebraic geometry—Santa Cruz 1995 (University of California, Santa Cruz, July 1995) 61 T. N. Bailey and A. W. Knapp, Editors, Representation theory and automorphic forms (International Centre for Mathematical Sciences, Edinburgh, Scotland, March 1996) 60 David Jerison, I. M. Singer, and Daniel W. Stroock, Editors, The legacy of Norbert Wiener: A centennial symposium (Massachusetts Institute of Technology, Cambridge, October 1994) 59 William Arveson, Thomas Branson, and Irving Segal, Editors, Quantization, nonlinear partial differential equations, and operator algebra (Massachusetts Institute of Technology, Cambridge, June 1994) 58 Bill Jacob and Alex Rosenberg, Editors, K-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara, July 1992) 57 Michael C. Cranston and Mark A. Pinsky, Editors, Stochastic analysis (Cornell University, Ithaca, July 1993) 56 William J. Haboush and Brian J. -
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. -
CATERADS and MOTIVIC COHOMOLOGY Contents Introduction 1 1. Weighted Multiplicative Presheaves 3 2. the Standard Motivic Cochain
CATERADS AND MOTIVIC COHOMOLOGY B. GUILLOU AND J.P. MAY Abstract. We define “weighted multiplicative presheaves” and observe that there are several weighted multiplicative presheaves that give rise to motivic cohomology. By neglect of structure, weighted multiplicative presheaves give symmetric monoids of presheaves. We conjecture that a suitable stabilization of one of the symmetric monoids of motivic cochain presheaves has an action of a caterad of presheaves of acyclic cochain complexes, and we give some fragmentary evidence. This is a snapshot of work in progress. Contents Introduction 1 1. Weighted multiplicative presheaves 3 2. The standard motivic cochain complex 5 3. A variant of the standard motivic complex 7 4. The Suslin-Friedlander motivic cochain complex 8 5. Towards caterad actions 10 6. The stable linear inclusions caterad 11 7. Steenrod operations? 13 References 14 Introduction The first four sections set the stage for discussion of a conjecture about the multiplicative structure of motivic cochain complexes. In §1, we define the notion of a “weighted multiplicative presheaf”, abbreviated WMP. This notion codifies formal structures that appear naturally in motivic cohomology and presumably elsewhere in algebraic geometry. We observe that WMP’s determine symmetric monoids of presheaves, as defined in [11, 1.3], by neglect of structure. In §§2–4, we give an outline summary of some of the constructions of motivic co- homology developed in [13, 17] and state the basic comparison theorems that relate them to each other and to Bloch’s higher Chow complexes. There is a labyrinth of definitions and comparisons in this area, and we shall just give a brief summary overview with emphasis on product structures.