Dwyer and friedlander interpreted important arithmetic questions in terms of this galois action on etale ktheory. For further information on rationalization, the reader is refered to section 9 of 18. All schemes will implicitly be assumed to be quasicompact and quasiseparated. Some remarks on units in grothendieckwitt rings journal of algebra the generalized slices of hermitian k theory journal of topology.
In this paper we study mz as a ring in the highly structured models for. Carlo mazza, vladimir voevodsky and charles weibel, lectures in motivic cohomology web pdf as cohomology with coefficients in eilenbergmac lane objects. Etale cohomology etale cohomology of rigid analytic spaces etale cohomology with compact support etale homology etale ktheory etale ktheory of ring spectra etale morava ktheory etale motivic cohomology fcohomology faltings cohomology finite polynomial cohomology finitedimensional motives flat cohomology flat homology. As a consequence of this construction, we also prove a homotopy theoretic generalization of the \etale version of the suslinvoevodsky comparison theorem comparing \etale and motivic cohomology, both with suitable torsion coefficients. Motivic cohomology is an invariant of algebraic varieties and of more general schemes. One can already trace back this fact in the work of fulton and macpherson as one of their example of a bivariant theory, in the etale setting, already uses the six functors formalism. It is one of the important facts in ktheorymotivic cohomology that the gerstentype complexes for quillen ktheory, milnor ktheory or more generally rosts cycle modules are exact for smooth. Among other applications, we prove grothendieckverdier duality in this. Rational homotopy theory is the study of rational homotopy types of spaces and of the properties of spaces and maps that are invariant under rational homotopy equivalence. We prove a topological invariance statement for the morelvoevodsky motivic homotopy category up to inverting exponential characteristics of residue fields. Dimensional homotopy tstructures in motivic homotopy theory.
Equivariant and motivic homotopy theory, group schemes, equivariant. Why is the motivic category defined over the site of. H, the procategory of the homotopy category of cwcomplexes. Skorobogatov the task of these notes is to supply the reader who has little or no experience of simplicial topology with a phrasebook on. In the third part we define the motivic cohomology of a variety in this setting, a setting where it carries cohomology operations. Indeed, it is a consequence of the morelvoevodsky localization theorem that the stable motivic homotopy category sh satisfies nil. Compared to voevodskys theory of motives and his motivic cohomology, the. Whats the relationship between homotopy theory and algebraic. I have also been actively researching the interaction between etale and motivic homotopy theory.
Firstly, we work solely within the framework of motivic homotopy theory and do not appeal to. Etale motivic cohomology and algebraic cycles journal of. The proof we give of the motivic rigidity theorem uses transfer maps in motivic stable homotopy theory and a homological localization theory for motivic symmetric spectra. Modules over motivic cohomology 3 2 homotopy theory of presheaves with transfers let sbe a noetherian and separated scheme of. A 1 homotopy theory is founded on a category called the a 1 homotopy category. Recall that the doldthom theorem asserts that for a c. A topologists introduction to the motivic homotopy theory for. This leads to a theory of motivic spheres s p,q with two indices. Assume that the site is subcanonical, and let shvt be the category of sheaves of sets on this site.
This does not work in our general setting, and it fails already in the concrete setting of monoid schemes. Rational points and zero cycles of degree 1 in a1homotopy theory. Motivic homotopy theory of noncommutative spaces associative dgalgebras is studied in. Sheaf theory, fundamental group and etale homotopy type are earlier witnesses of this. We take a moment to explain why this might be useful for understanding the above motivating questions. Gunnar carlsson, roy joshua, equivariant motivic homotopy theory, arxiv. Perfection in motivic homotopy theory elmanto 2020.
This was vastly generalized and studied more thoroughly in. Motivic homotopy theory of group scheme actions illinois. Construction in this section, we go through the construction of the categories da ets. In this thesis, we study applications and connections of voevodskys theory of motives to stable homotopy theory, birational geometry, and arithmetic. The last part, containing the most innovative results, assumes some knowledge of motivic homotopy theory, although precise statements and references are given. This text contains no proofs, for which we refer to the. This is an abstraction of an argument suslin rigidity used by suslin to show determine k. As a cohomology theory, mz represents motivic cohomology groups. Lecture notes on motivic cohomology clay mathematics institute. It is a result which is the direct analogue of the pontryaginthom construction. Workshop \ etale and motivic homotopy theory, titles and abstracts aravind asok.
Motivic homotopy theory or a1homotopy theory is the homotopy theory of smooth. Unstable motivic homotopy theory department of mathematics. For example, for a closed subscheme z of x, there is an. Axiomatic, enriched and motivic homotopy theory springer. For example, the etale version of stable motivic homotopy theory sh ets, the theory of rational motives dms. However, the topological invariance property still fails, at least in positive characteristic see remark 2. Let sm s be the category of nitely presented smooth schemes.
Remarks on motivic homotopy theory over algebraically closed fields po hu, igor kriz and kyle ormsby abstract. Motivic cohomology usc dana and david dornsife college. To compute the homotopy groups of motivic spheres would also yield the classical stable homotopy groups of the spheres, so in this respect a 1 homotopy theory is at least as complicated as classical homotopy theory. For legibility, we shall suppress sin the notation. In the context of motivic homotopy theory, this was first studied by heller and ormsby in ho16, where the authors established a connection between c 2 equivariant homotopy theory and motivic.
Purity theorem in motivic homotopy theory october 24, 2014 in these notes xwill always be a smooth scheme over some noetherian base scheme s. Log geometry, monodromy, and betti realization of varieties defined by. Finally, i have been working in the field of log geometry with sarah scherotzke, nicolo sibilla, and mattia talpo. Etale homotopy theory after artinmazur, friedlander et al. Lately i have been studying these two papers first and second that introduce a new cohomology theory called arakelov motivic cohomology. We consider etale motivic or lichtenbaum cohomology and its relation to algebraic cycles. This implies in particular that sh 1 p of characteristic p 0 schemes is invariant under passing to perfections. The other important technical advantage is the ease with which one. Beginning with an introduction to the homotopy theory of. I will discuss how sensitive di erent categories of homotopy theoretic origin are to existence of rational points or zero cycles of degree 1. If you instead started with all topological spaces, i think you would get something different.
The aim of these notes is to explain this construction. A now classical construction due to kato and nakayama attaches a topological space betti realization to a log scheme over c. Marc hoyois, the six operations in equivariant motivic homotopy theory, arxiv. Here shx denotes the motivic stable homotopy category. If one wishes to study possibly nonsmooth schemes, then different. Whats the relationship between homotopy theory and. We give an geometric interpretation of lichtenbaum cohomology and use it to show that the usual integral cycle maps extend to maps on integral lichtenbaum cohomology. Still following voevodskys path, there is a second possible construction using the a1. This monograph on the homotopy theory of topologized diagrams of spaces and spectra gives an expert account of a subject at the foundation of motivic homotopy theory and the theory of topological modular forms in stable homotopy theory. Galois equivariance and stable motivic homotopy theory. In order to do this, we construct a stable \etale realization functor. Bachmann, tom 2018, motivic and real etale stable homotopy theory, arxiv. The second family of properties relates motivic cohomology to other known invariants of algebraic varieties and rings. Morelvoevodsky motivic homotopy theory in spectral algebraic geometry.
A guide to motivic sheaves 5 i there is a canonical triangulated functor t. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of grothendieck. The asi was the opening workshop of the four month programme new contexts for stable homotopy theory which explored several themes in greater depth. Etale homotopy theory tomer schlank contents 1 introduction and overview 9420 2. Let gl mot be the direct limit of the group schemes gl n. I will discuss how sensitive di erent categories of homotopy theoretic origin are to existence. The power of motivic cohomology as a tool for proving results in algebra and algebraic geometry lies in the interaction of the results. Motivic homotopy theory dexter chua 1 the nisnevich topology 2 2 a1localization 4 3 homotopy sheaves 5 4 thom spaces 6 5 stable motivic homotopy theory 7 6 e ective and very e ective motivic spectra 9 throughout the talk, sis a quasicompact and quasiseparated scheme.
However, it was artin and mazur who realized that we could actually associate a space to our ring actually, a prospace, from which we can extract invariants like higher homotopy groups. Homological algebra for superalgebras of differentiable functions with roytenberg d. From the point of view of the motivic homotopy theory of morel and voevodsky, one would like the motivic cohomology of x x to be representable in the stable motivic homotopy category sh x. A key goal of algebraic geometry is to compute the chow groups of x, because they give strong information about all subvarieties of x. On the conservativity of the functor assigning to a motivic spectrum its motive. Cohomology theories in motivic stable homotopy theory. Etale and motivic homotopy theory, titles and abstracts piotr. We discuss certain calculations in the 2complete motivic stable homotopy category over an algebraically closed. Etale motives compositio mathematica cambridge core. Etale motivic cohomology and algebraic singular homology. I am grateful to the isaac newton institute for providing such an ideal venue, the nato science committee for their funding, and to all the speakers at the conference, whether or not they were. Some remarks on units in grothendieckwitt rings journal of algebra the generalized slices of hermitian ktheory journal of topology on the invertibility of motives of affine.
Etale and motivic homotopy theory, titles and abstracts. The chow groups of x have some of the formal properties of borelmoore homology in topology, but some things are missing. Workshop \etale and motivic homotopy theory, titles and abstracts piotr achinger. A1homotopy invariance in spectral algebraic geometry. A primer for unstable motivic homotopy theory arxiv.
This is the homotopy category for a certain closed model category whose construction requires two steps step 1. It includes the chow ring of algebraic cycles as a special case. Indeed, giving this concrete homotopy requires both an addition and a. Workshop \etale and motivic homotopy theory, titles and abstracts aravind asok. In algebraic geometry and algebraic topology, branches of mathematics, a1 homotopy theory is. Furthermore, we import the ideas from homotopy theory relating cohomology theories and formal groups laws to construct an algebra closely related to the dual of the motivic steenrod. We also show that the homotopy coniveau ltration on algebraic k theory agrees with the gamma ltration, up to small primes. Motivic and real etale stable homotopy theory with alexander vishik motivic equivalence of affine quadrics. The relative picard group and suslins rigidity theorem 47 lecture 8. Prerequisites in algebraic topology the nordfjordeid summer school on motivic homotopy theory.
The first two parts develop a motivic homotopy theory, and are joint work with fabien morel. While most of the applications presented in the papers are. Informally, the theory enriches the category of smooth schemes over a base eld so it also admits simplicial constructions, and then imposes a homotopytheoretic structure in which the a ne line a1 plays the role of the unit. A topologists introduction to the motivic homotopy theory. Over a base eld k, motivic homotopy theory is built from the category of kspaces, denoted spck, which is the category of simplicial presheaves on smk, the category of smooth kschemes. Xy is a universal homeomorphism if it induces a homeomorphism on underlying topological spaces after any base change, or equivalently, if it. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology. Tewhich is the identity on objects in particular tand tehave the same class of objects. Workshop etale and motivic homotopy theory, titles and abstracts. The purpose of this work is to study the notion of bivariant theory introduced by fulton and macpherson in the context of motivic stable homotopy theory, and more generally in the broader. It is designed to be a reference work and could also be useful outside motivic homotopy theory. Specifically, we prove the convergence of motivic analogues of the adams and adams. Let x be a noetherian scheme of finite dimension and denote by rho the additive inverse of the morphism in shx from s to gm corresponding to the unit 1.
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