As a consequence a variant of gabber rigidity theorem is obtained for a cohomology theory representable in the motivic stable homotopy category by a. We go through the formalism of grothendiecks six operations for these categories. Also, for the module playing the part of z2, we check that the singer construction can be realized as the continous motivic cohomology of an inverse tower of motivic spectra just as it was done in lins work. As background material, we recommend the lectures of dundas dun and levine lev in this volume. Jardine september 23, 1999 introduction let smj k nisbe the smooth nisnevich site for a eld k, and let sshvsmj k nis respectively shvsmj k nis denote the category of simplicial sheaves respectively sheaves on that site.
Summer school on rigidity theorems in a1homotopy theory regensburg, august 15, 2011 organizers. A 1 homotopy theory is founded on a category called the a 1 homotopy category. We prove that under suitable finiteness hypotheses, and assuming that p is invertible on x. Rigidity in motivic homotopy theory, mathematische annalen. In homotopy theory as well as algebraic topology, one typically does not work with an arbitrary topological space to avoid pathologies in pointset topology. My main tools are motivic analogues of the adams and adamsnovikov spectral sequences. Brie y, there is a motivic version of homotopy xed points constructed using geometric versions of eg. A proposal for the establishment of a dfgpriority program. Jardine september 23, 1999 introduction let smj k nisbe the smooth nisnevich site for a eld k, and let sshvsmj k nis respectively shvsmj k nis denote the category of simplicial sheaves respec tively sheaves on that site. A rigidity property for the homotopy invariant stable linear framed presheaves is established. The purpose of this note is to explain the morelvoevodsky result that the.
Relation between motivic homotopy category and the derived. Rigidity in motivic homotopy theory find, read and cite all the research you need on researchgate. The algebraically or combinatorially minded person wishing to study the homotopy theory of spaces may thus content herself with studying simplicial sets. It is one of the important facts in k theory motivic cohomology that the gerstentype complexes for quillen k theory, milnor k theory or more generally rosts cycle modules are exact for smooth. Motivic homotopy theory in problems in homotopy theory. I determine the coefficients of 2complete algebraic cobordism and a type of connective algebraic ktheory in the motivic setting. A proposal for the establishment of a dfgpriority program in. Rigidity for linear framed presheaves and generalized.
Gw k of rank coprime to the exponential characteristic of the base field k. The short answer is that they are very different, but become quite similar if you 1 stabilize, i. It is one of the important facts in ktheorymotivic cohomology that the gerstentype complexes for quillen ktheory, milnor ktheory or more generally rosts. The first stable homotopy groups of motivic spheres annals.
We recall the construction, following the method of morel and voevodsky, of the triangulated category of etale motivic sheaves over a base scheme. Element ar y homo t opy theor y homotop y theory, which is the main part of algebraic topology, studies topological objects up to homotop y equi valence. In mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. 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. Scanned using book scancenter 5033 institute for advanced study. With a similar scope as the summer school it is aimed at graduate students and researchers in algebraic topology and. Rigidity and algebraic models for rational equivariant. Algebraic methods in unstable homotopy theory this is a comprehensive uptodate treatment of unstable homotopy. Open problems in the motivic stable homotopy theory, i contents. Pages 174 from volume 189 2019, issue 1 by oliver rondigs, markus spitzweck, paul arne ostv. Summer school on rigidity theorems in a homotopy theory. Summer course in motivic homotopy theory 3 examples 1.
Peter arndt, abstract motivic homotopy theory, thesis 2017 web, pdf, pdf exposition. Isaksen uniqueness of motivic homotopy theory following schwede. This leads to a theory of motivic spheres s p,q with two indices. Homotop y equi valence is a weak er relation than topological equi valence, i. Motivic cohomology usc dana and david dornsife college. I the homotopy limit problem for karoubis hermitian ktheory 23 was posed by thomason in 1983 43. This research monograph on motivic homotopy theory contains material based on lectures at a summer school at the sophus lie centre in nordfjordeid, norway, in august 2002. Then 0 is the unique proper ideal in f, and is prime since f is an integral domain ab 0, a6 0 implies b 0. 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. The crossfertilization of homotopy theory and algebraic geometry, especially through motivic homotopy theory, derived algebraic geometry and di. Tentative a weil theory is an hqalgebra e such that e1 e. Tensor triangular geometry, stable motivic homotopy theory. In particular, we are able get localization and cancellation for torsion etale motivic complexes, and also to deal with 2torsion, which is missed in the approach of ayoub. Voevodsky, which is based on the theory of motivic complexes of voevodsky.
Tentative an ordinary theory is a theory whose cohomology groups are functorial with respect to correspondences. We show that extensions of algebraically closed fields induce full and faithful functors between the respective motivic stable homotopy. The notation tht 1 2 is very similar to a notation for homotopy. Isaksen compute motivic stable homotopy groups at odd primes. Rigidity in motivic homotopy theory article pdf available in mathematische annalen 34. For an oriented theory, this should be equivalent to the. Rational homotopy theory 3 it is clear that for all r, sn r is a strong deformation retract of xr, which implies that hkxr 0 if k 6 0,n. Minicourse on motivic homotopy theory, by matthias wendt as the title suggests, the goal of the lectures is to provide a tour through some of the basic constructions in motivic homotopy, slightly geared towards the recent applications in algebraic classi cation problems. In generality, homotopy theory is the study of mathematical contexts in which functions or rather homomorphisms are equipped with a concept of homotopy between them, hence with a concept of equivalent deformations of morphisms, and then iteratively with homotopies of homotopies between those, and so forth.
This converts the usual homotopy limit problem into a comparison of. Rigidity in etale motivic stable homotopy theory arxiv. Krishna tata jinhyun park department of mathematical sciences kaist, daejeon, korea february 21, 20 at yeosu symposium jinhyun park semitopologization in motivic homotopy theory. These notes contain a brief introduction to rational homotopy theory. Cohomology theories in motivic stable homotopy theory. Rigidity in equivariant stable homotopy theory irakli patchkoria for any nite group g, we show that the 2local gequivariant stable homotopy category, indexed on a complete guni. It originated as a topic in algebraic topology but nowadays it is studied as an independent discipline. A guide to etale motivic sheaves joseph ayoub abstract. Recall that the doldthom theorem asserts that for a c.
Pdf on jul 1, 2011, oliver rondigs and others published erratum to. It is a result which is the direct analogue of the pontryaginthom construction. Rigidity in equivariant stable homotopy theory irakli patchkoria for any nite group g, we show that the 2local gequivariant stable homotopy category, indexed on a complete guniverse, has a unique equivariant model in the sense of quillen model categories. Besides algebraic topology, the theory has also been in used in other areas of mathematics such as algebraic geometry e. Isaksen find an algebraic model for rational motivic homotopy theory. Within this algebra, i discover a motivic analogue of the alpha family and determine its behavior within the motivic adamsnovikov spectral sequence.
Instead, one assumes a space is a reasonable space. An introductory reference to motivic homotopy theory is voevodskys icm address voe98. Informally, the theory enriches the category of smooth schemes over a base eld so it also admits simplicial constructions, and then imposes a homotopy theoretic structure in which the a ne line a1 plays the role of the unit. Axiomatic, enriched and motivic homotopy theory, nato sci. Motivic tambara functors with marc hoyois norms in motivic homotopy theory with jean fasel on the effectivity of spectra representing motivic cohomology theories. The first stable homotopy groups of motivic spheres. Feb 08, 2008 rigidity in motivic homotopy theory rigidity in motivic homotopy theory rondigs, oliver. The notation catht 1,t 2 or t ht 1 2 denotes the homotopy theory of functors from the. Assume that the site is subcanonical, and let shvt be the category of sheaves of sets on this site. Cisinski, to the generalization of the rigidity theorem of suslin and voevodsky, which is based on the theory of motivic complexes of voevodsky. Course on homotopy theory first semester 201220 this is a course jointly taught by moritz groth and ieke moerdijk, and it is part of the mastermath program. Rigidity for linear framed presheaves and generalized motivic cohomology theories.
The relative picard group and suslins rigidity theorem 47 lecture 8. If the homotopy category of cand the homotopy category of q tequivariant spectra are equivalent as triangulated categories, then there exists a quillen equivalence between cand the model category of q t equivariant spectra. Salch the ravenel conjectures in nonclassical settingsequivariant, motivic, andor. Motivic homotopy theory, 147220, universitext series in mathematics, springerverlag, 2007. Let gl mot be the direct limit of the group schemes gl n. Furthermore, the homomorphism induced in reduced homology by the inclusion xr. We show that kqcan be obtained as the motivic homotopy xed points of the c 2action on kgl. This is the homotopy category for a certain closed model category whose construction requires two steps step 1. In particular, we are able get localization and cancellation for torsion etale motivic complexes, and also to deal with 2. There is a canonical map from algebraic hermitian ktheory to the z 2homotopy. This thesis is concerned with the application of certain computational methods from stable algebraic topology in motivic homotopy theory over padic fields. Introduction this overview of rational homotopy theory consists of an extended version of. The proof we give of the motivic rigidity theorem uses transfer maps in motivic stable homotopy theory and a homological localization theory for. The relative picard group and suslins rigidity theorem.
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