The Chow T-Structure on the -Category of Motivic Spectra
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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. -
An Introduction to Spectra
An introduction to spectra Aaron Mazel-Gee In this talk I’ll introduce spectra and show how to reframe a good deal of classical algebraic topology in their language (homology and cohomology, long exact sequences, the integration pairing, cohomology operations, stable homotopy groups). I’ll continue on to say a bit about extraordinary cohomology theories too. Once the right machinery is in place, constructing all sorts of products in (co)homology you may never have even known existed (cup product, cap product, cross product (?!), slant products (??!?)) is as easy as falling off a log! 0 Introduction n Here is a table of some homotopy groups of spheres πn+k(S ): n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 n πn(S ) Z Z Z Z Z Z Z Z Z Z n πn+1(S ) 0 Z Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 n πn+2(S ) 0 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 n πn+3(S ) 0 Z2 Z12 Z ⊕ Z12 Z24 Z24 Z24 Z24 Z24 Z24 n πn+4(S ) 0 Z12 Z2 Z2 ⊕ Z2 Z2 0 0 0 0 0 n πn+5(S ) 0 Z2 Z2 Z2 ⊕ Z2 Z2 Z 0 0 0 0 n πn+6(S ) 0 Z2 Z3 Z24 ⊕ Z3 Z2 Z2 Z2 Z2 Z2 Z2 n πn+7(S ) 0 Z3 Z15 Z15 Z30 Z60 Z120 Z ⊕ Z120 Z240 Z240 k There are many patterns here. The most important one for us is that the values πn+k(S ) eventually stabilize. -
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. -
The Spectral Sequence Relating Algebraic K-Theory to Motivic Cohomology
Ann. Scient. Éc. Norm. Sup., 4e série, t. 35, 2002, p. 773 à 875. THE SPECTRAL SEQUENCE RELATING ALGEBRAIC K-THEORY TO MOTIVIC COHOMOLOGY BY ERIC M. FRIEDLANDER 1 AND ANDREI SUSLIN 2 ABSTRACT. – Beginning with the Bloch–Lichtenbaum exact couple relating the motivic cohomology of a field F to the algebraic K-theory of F , the authors construct a spectral sequence for any smooth scheme X over F whose E2 term is the motivic cohomology of X and whose abutment is the Quillen K-theory of X. A multiplicative structure is exhibited on this spectral sequence. The spectral sequence is that associated to a tower of spectra determined by consideration of the filtration of coherent sheaves on X by codimension of support. 2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. – Partant du couple exact de Bloch–Lichtenbaum, reliant la cohomologie motivique d’un corps F àsaK-théorie algébrique, on construit, pour tout schéma lisse X sur F , une suite spectrale dont le terme E2 est la cohomologie motivique de X et dont l’aboutissement est la K-théorie de Quillen de X. Cette suite spectrale, qui possède une structure multiplicative, est associée à une tour de spectres déterminée par la considération de la filtration des faisceaux cohérents sur X par la codimension du support. 2002 Éditions scientifiques et médicales Elsevier SAS The purpose of this paper is to establish in Theorem 13.6 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic K-theory of X: p,q p−q Z − −q − − ⇒ (13.6.1) E2 = H X, ( q) = CH (X, p q) K−p−q(X). -
Detecting Motivic Equivalences with Motivic Homology
Detecting motivic equivalences with motivic homology David Hemminger December 7, 2020 Let k be a field, let R be a commutative ring, and assume the exponential characteristic of k is invertible in R. Let DM(k; R) denote Voevodsky’s triangulated category of motives over k with coefficients in R. In a failed analogy with topology, motivic homology groups do not detect iso- morphisms in DM(k; R) (see Section 1). However, it is often possible to work in a context partially agnostic to the base field k. In this note, we prove a detection result for those circumstances. Theorem 1. Let ϕ: M → N be a morphism in DM(k; R). Suppose that either a) For every separable finitely generated field extension F/k, the induced map on motivic homology H∗(MF ,R(∗)) → H∗(NF ,R(∗)) is an isomorphism, or b) Both M and N are compact, and for every separable finitely generated field exten- ∗ ∗ sion F/k, the induced map on motivic cohomology H (NF ,R(∗)) → H (MF ,R(∗)) is an isomorphism. Then ϕ is an isomorphism. Note that if X is a separated scheme of finite type over k, then the motive M(X) of X in DM(k; R) is compact, by [6, Theorem 11.1.13]. The proof is easy: following a suggestion of Bachmann, we reduce to the fact (Proposition 5) that a morphism in SH(k) which induces an isomorphism on homo- topy sheaves evaluated on all separable finitely generated field extensions must be an isomorphism. This follows immediately from Morel’s work on unramified presheaves (see [9]). -
Motivic Cohomology, K-Theory and Topological Cyclic Homology
Motivic cohomology, K-theory and topological cyclic homology Thomas Geisser? University of Southern California, [email protected] 1 Introduction We give a survey on motivic cohomology, higher algebraic K-theory, and topo- logical cyclic homology. We concentrate on results which are relevant for ap- plications in arithmetic algebraic geometry (in particular, we do not discuss non-commutative rings), and our main focus lies on sheaf theoretic results for smooth schemes, which then lead to global results using local-to-global methods. In the first half of the paper we discuss properties of motivic cohomology for smooth varieties over a field or Dedekind ring. The construction of mo- tivic cohomology by Suslin and Voevodsky has several technical advantages over Bloch's construction, in particular it gives the correct theory for singu- lar schemes. But because it is only well understood for varieties over fields, and does not give well-behaved ´etalemotivic cohomology groups, we discuss Bloch's higher Chow groups. We give a list of basic properties together with the identification of the motivic cohomology sheaves with finite coefficients. In the second half of the paper, we discuss algebraic K-theory, ´etale K- theory and topological cyclic homology. We sketch the definition, and give a list of basic properties of algebraic K-theory, sketch Thomason's hyper- cohomology construction of ´etale K-theory, and the construction of topolog- ical cyclic homology. We then give a short overview of the sheaf theoretic properties, and relationships between the three theories (in many situations, ´etale K-theory with p-adic coefficients and topological cyclic homology agree). -
Motivic Homotopy Theory Dexter Chua
Motivic Homotopy Theory Dexter Chua 1 The Nisnevich topology 2 2 A1-localization 4 3 Homotopy sheaves 5 4 Thom spaces 6 5 Stable motivic homotopy theory 7 6 Effective and very effective motivic spectra 9 Throughout the talk, S is a quasi-compact and quasi-separated scheme. Definition 0.1. Let SmS be the category of finitely presented smooth schemes over S. Since we impose the finitely presented condition, this is an essentially small (1-)category. The starting point of motivic homotopy theory is the 1-category P(SmS) of presheaves on SmS. This is a symmetric monoidal category under the Cartesian product. Eventually, we will need the pointed version P(SmS)∗, which can be defined either as the category of pointed objects in P(SmS), or the category of presheaves with values in pointed spaces. This is symmetric monoidal 1-category under the (pointwise) smash product. In the first two chapters, we will construct the unstable motivic category, which fits in the bottom-right corner of the following diagram: LNis P(SmS) LNisP(SmS) ≡ SpcS L 1 Lg1 A A 1 LgNis 1 1 A LA P(SmS) LA ^NisP(SmS) ≡ SpcS ≡ H(S) The first chapter will discuss the horizontal arrows (i.e. Nisnevich localization), and the second will discuss the vertical ones (i.e. A1-localization). 1 Afterwards, we will start \doing homotopy theory". In chapter 3, we will discuss the motivic version of homotopy groups, and in chapter 4, we will discuss Thom spaces. In chapter 5, we will introduce the stable motivic category, and finally, in chapter 6, we will introduce effective and very effective spectra. -
Spectra and Stable Homotopy Theory
Spectra and stable homotopy theory Lectures delivered by Michael Hopkins Notes by Akhil Mathew Fall 2012, Harvard Contents Lecture 1 9/5 x1 Administrative announcements 5 x2 Introduction 5 x3 The EHP sequence 7 Lecture 2 9/7 x1 Suspension and loops 9 x2 Homotopy fibers 10 x3 Shifting the sequence 11 x4 The James construction 11 x5 Relation with the loopspace on a suspension 13 x6 Moore loops 13 Lecture 3 9/12 x1 Recap of the James construction 15 x2 The homology on ΩΣX 16 x3 To be fixed later 20 Lecture 4 9/14 x1 Recap 21 x2 James-Hopf maps 21 x3 The induced map in homology 22 x4 Coalgebras 23 Lecture 5 9/17 x1 Recap 26 x2 Goals 27 Lecture 6 9/19 x1 The EHPss 31 x2 The spectral sequence for a double complex 32 x3 Back to the EHPss 33 Lecture 7 9/21 x1 A fix 35 x2 The EHP sequence 36 Lecture 8 9/24 1 Lecture 9 9/26 x1 Hilton-Milnor again 44 x2 Hopf invariant one problem 46 x3 The K-theoretic proof (after Atiyah-Adams) 47 Lecture 10 9/28 x1 Splitting principle 50 x2 The Chern character 52 x3 The Adams operations 53 x4 Chern character and the Hopf invariant 53 Lecture 11 8/1 x1 The e-invariant 54 x2 Ext's in the category of groups with Adams operations 56 Lecture 12 10/3 x1 Hopf invariant one 58 Lecture 13 10/5 x1 Suspension 63 x2 The J-homomorphism 65 Lecture 14 10/10 x1 Vector fields problem 66 x2 Constructing vector fields 70 Lecture 15 10/12 x1 Clifford algebras 71 x2 Z=2-graded algebras 73 x3 Working out Clifford algebras 74 Lecture 16 10/15 x1 Radon-Hurwitz numbers 77 x2 Algebraic topology of the vector field problem 79 x3 The homology of -
Introduction to Motives
Introduction to motives Sujatha Ramdorai and Jorge Plazas With an appendix by Matilde Marcolli Abstract. This article is based on the lectures of the same tittle given by the first author during the instructional workshop of the program \number theory and physics" at ESI Vienna during March 2009. An account of the topics treated during the lectures can be found in [24] where the categorical aspects of the theory are stressed. Although naturally overlapping, these two independent articles serve as complements to each other. In the present article we focus on the construction of the category of pure motives starting from the category of smooth projective varieties. The necessary preliminary material is discussed. Early accounts of the theory were given in Manin [21] and Kleiman [19], the material presented here reflects to some extent their treatment of the main aspects of the theory. We also survey the theory of endomotives developed in [5], this provides a link between the theory of motives and tools from quantum statistical mechanics which play an important role in results connecting number theory and noncommutative geometry. An extended appendix (by Matilde Marcolli) further elaborates these ideas and reviews the role of motives in noncommutative geometry. Introduction Various cohomology theories play a central role in algebraic geometry, these co- homology theories share common properties and can in some cases be related by specific comparison morphisms. A cohomology theory with coefficients in a ring R is given by a contra-variant functor H from the category of algebraic varieties over a field k to the category of graded R-algebras (or more generally to a R-linear tensor category). -
Stable Motivic Invariants Are Eventually Étale Local
STABLE MOTIVIC INVARIANTS ARE EVENTUALLY ETALE´ LOCAL TOM BACHMANN, ELDEN ELMANTO, AND PAUL ARNE ØSTVÆR Abstract. We prove a Thomason-style descent theorem for the motivic sphere spectrum, and deduce an ´etale descent result applicable to all motivic spectra. To this end, we prove a new convergence result for the slice spectral sequence, following work by Levine and Voevodsky. This generalizes and extends previous ´etale descent results for special examples of motivic cohomology theories. Combined with ´etale rigidity results, we obtain a complete structural description of the ´etale motivic stable category. Contents 1. Introduction 2 1.1. What is done in this paper? 2 1.2. Bott elements and multiplicativity of the Moore spectrum 4 1.3. Some applications 4 1.4. Overview of ´etale motivic cohomology theories 4 1.5. Terminology and notation 5 1.6. Acknowledgements 6 2. Preliminaries 6 2.1. Endomorphisms of the motivic sphere 6 2.2. ρ-completion 7 2.3. Virtual ´etale cohomological dimension 8 3. Bott-inverted spheres 8 3.1. τ-self maps 8 3.2. Cohomological Bott elements 9 3.3. Spherical Bott elements 10 4. Construction of spherical Bott elements 10 4.1. Construction via roots of unity 10 4.2. Lifting to higher ℓ-powers 10 4.3. Construction by descent 11 4.4. Construction over special fields 12 4.5. Summary of τ-self maps 13 5. Slice convergence 13 6. Spheres over fields 18 7. Main result 19 Appendix A. Multiplicative structures on Moore objects 23 A.1. Definitions and setup 23 A.2. Review of Oka’s results 23 arXiv:2003.04006v2 [math.KT] 20 Mar 2020 A.3. -
Chromatic Structures in Stable Homotopy Theory
CHROMATIC STRUCTURES IN STABLE HOMOTOPY THEORY TOBIAS BARTHEL AND AGNES` BEAUDRY Abstract. In this survey, we review how the global structure of the stable homotopy category gives rise to the chromatic filtration. We then discuss computational tools used in the study of local chromatic homo- topy theory, leading up to recent developments in the field. Along the way, we illustrate the key methods and results with explicit examples. Contents 1. Introduction1 2. A panoramic view of the chromatic landscape4 3. Local chromatic homotopy theory 14 4. K(1)-local homotopy theory 29 5. Finite resolutions and their spectral sequences 35 6. Chromatic splitting, duality, and algebraicity 46 References 57 1. Introduction At its core, chromatic homotopy theory provides a natural approach to 0 the computation of the stable homotopy groups of spheres π∗S . Histori- cally, the first few of these groups were computed geometrically through the classification of stably framed manifolds, using the Pontryagin{Thom iso- 0 fr morphism π∗S ∼= Ω∗ . However, beginning with the work of Serre, it soon arXiv:1901.09004v2 [math.AT] 29 Apr 2019 turned out that algebraic tools were more effective, both for the computation of specific low-degree values as well as for establishing structural results. In 0 particular, Serre proved that π∗S is a degreewise finitely generated abelian 0 group with π0S ∼= Z and that all higher groups are torsion. Serre's method was essentially inductive: starting with the knowledge of 0 0 0 the first n groups π0S ; : : : ; πn−1S , one can in principle compute πnS . Said 0 differently, Serre worked with the Postnikov filtration of π∗S , in which the 0 (n + 1)st filtration quotient is given by πnS . -
Duality in Algebra and Topology
DUALITY IN ALGEBRA AND TOPOLOGY W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR Dedicated to Clarence W. Wilkerson, on the occasion of his sixtieth birthday Contents 1. Introduction 1 2. Spectra, S-algebras, and commutative S-algebras 7 3. Some basic constructions with modules 13 4. Smallness 20 5. Examples of smallness 27 6. Matlis lifts 29 7. Examples of Matlis lifting 33 8. Gorenstein S-algebras 36 9. A local cohomology theorem 41 10. Gorenstein examples 44 References 47 1. Introduction In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can be extended to the more general rings that come up in homotopy theory. Amongst the rings we work with are the di®erential graded ring of cochains on a space X, the dif- ferential graded ring of chains on the loop space X, and various ring spectra, e.g., the Spanier-Whitehead duals of ¯nite spectra or chro- matic localizations of the sphere spectrum. Maybe the most important contribution of this paper is the concep- tual framework, which allows us to view all of the following dualities ² Poincar¶e duality for manifolds ² Gorenstein duality for commutative rings ² Benson-Carlson duality for cohomology rings of ¯nite groups Date: October 3, 2005. 1 2 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR ² Poincar¶e duality for groups ² Gross-Hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon.