Cohomology Operations

2019 ◽  
pp. 213-231
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Homotopy Category ◽  
Perfect Field ◽  
Motivic Cohomology ◽  
Steenrod Operations ◽  
Axiomatic Treatment ◽  
Cohomology Operations

This chapter concerns cohomology operations. Although motivic cohomology was originally defined for smooth varieties over the perfect field 𝑘, it is more useful to view it as a functor defined on the pointed 𝔸1-homotopy category Ho ·, discussed previously in chapter 12. After defining cohomology operations and giving a few examples, the chapter turns to an axiomatic treatment of the motivic Steenrod operations. The motivic Milnor operations are then presented. Thereafter, this chapter demonstrates that the sequence of Milnor operations 𝑄𝑖 is exact on the reduced cohomology of the suspension Σ‎𝔛 attached to a Rost variety 𝑋, using the degree map 𝑡𝒩. It concludes with Voevodsky's motivic degree theorem.

An Overview of the Proof

2019 ◽  
pp. 3-22
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Motivic Cohomology ◽  
Norm Residue ◽  
Cohomology Operations ◽  
Definition Of ◽  
Special Case

This chapter provides the main steps in the proof of Theorems A and B regarding the norm residue homomorphism. It also proves several equivalent (but more technical) assertions in order to prove the theorems in question. This chapter also supplements its approach by defining the Beilinson–Lichtenbaum condition. It thus begins with the first reductions, the first of which is a special case of the transfer argument. From there, the chapter presents the proof that the norm residue is an isomorphism. The definition of norm varieties and Rost varieties are also given some attention. The chapter also constructs a simplicial scheme and introduces some features of its cohomology. To conclude, the chapter discusses another fundamental tool—motivic cohomology operations—as well as some historical notes.

scholarly journals Tensor Structure on Smooth Motives

2011 ◽  
Vol 9 (1) ◽  
pp. 57-101 ◽  
Author(s):  
Anandam Banerjee
Keyword(s):  
Characteristic Zero ◽  
Full Subcategory ◽  
Resolution Of Singularities ◽  
Homotopy Category ◽  
Tensor Structure ◽  
Perfect Field ◽  
Base Scheme

AbstractRecently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme S generated by the motives of smooth projective S-schemes, assuming that S is itself smooth over a perfect field. In both constructions, the tensor structure requires ℚ-coefficients. In this article, it is shown how to provide a tensor structure on the homotopy category mentioned above, when S is semi-local and essentially smooth over a field of characteristic zero. This is done by defining a pseudo-tensor structure on the DG category of motives constructed by Levine, which induces a tensor structure on its homotopy category.

scholarly journals Motivic Steenrod operations in characteristic p

2020 ◽  
Vol 8 ◽  
Author(s):  
Eric Primozic
Keyword(s):  
Quadratic Forms ◽  
Chow Groups ◽  
Base Field ◽  
Characteristic P ◽  
Motivic Cohomology ◽  
Steenrod Operations ◽  
Adem Relations

Abstract For a prime p and a field k of characteristic $p,$ we define Steenrod operations $P^{n}_{k}$ on motivic cohomology with $\mathbb {F}_{p}$ -coefficients of smooth varieties defined over the base field $k.$ We show that $P^{n}_{k}$ is the pth power on $H^{2n,n}(-,\mathbb {F}_{p}) \cong CH^{n}(-)/p$ and prove an instability result for the operations. Restricted to mod p Chow groups, we show that the operations satisfy the expected Adem relations and Cartan formula. Using these new operations, we remove previous restrictions on the characteristic of the base field for Rost’s degree formula. Over a base field of characteristic $2,$ we obtain new results on quadratic forms.

scholarly journals Norms in motivic homotopy theory

Astérisque ◽  
2021 ◽  
Vol 425 ◽  
Author(s):  
Tom BACHMANN ◽  
Marc HOYOIS
Keyword(s):  
Homotopy Theory ◽  
Galois Theory ◽  
Homotopy Category ◽  
Motivic Cohomology ◽  
Power Operations ◽  
Motivic Homotopy Theory ◽  
Slice Filtration ◽  
Motivic Homotopy ◽  
Algebraic Cobordism ◽  
Spectrum Structure

If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes : \mathcal{S}\mathcal{H}(S') \to \mathcal{S}\mathcal{H}(S)$, where $\mathcal{S}\mathcal{H}(S)$ is the $\mathbb{P}^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a  normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $H\mathbb{Z}$, the homotopy $K$-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb{Z}$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.

The Norm Residue Theorem in Motivic Cohomology

2019 ◽  
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Large Scale ◽  
Residue Theorem ◽  
Geometric Constructions ◽  
Complete Proof ◽  
Motivic Cohomology ◽  
Symmetric Powers ◽  
Norm Residue ◽  
Cohomology Operations ◽  
Many Sources ◽  
First Time

This book presents the complete proof of the Bloch–Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups. Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The book draws on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduces the key figures behind its development. It proceeds to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. It then addresses symmetric powers of motives and motivic cohomology operations. The book unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

scholarly journals On the Functoriality of the Slice Filtration

2013 ◽  
Vol 11 (1) ◽  
pp. 55-71 ◽  
Author(s):  
Pablo Pelaez
Keyword(s):  
Finite Type ◽  
Characteristic Zero ◽  
Resolution Of Singularities ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Canonical Structure ◽  
Motivic Cohomology ◽  
Ring Spectrum ◽  
Stable Homotopy Category ◽  
Slice Filtration

AbstractLet k be a field with resolution of singularities, and X a separated k-scheme of finite type with structure map g. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along g. Restricting the field further to the case of characteristic zero, we are able to compute the slices of Weibel's homotopy invariant K-theory [24] extending the result of Levine [10], and also the zero slice of the sphere spectrum extending the result of Levine [10] and Voevodsky [23]. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum HZXsf which is stable under pullback and that all the slices have a canonical structure of strict modules over HZXsf. If we consider rational coefficients and assume that X is geometrically unibranch then relying on the work of Cisinski and Déglise [4], we deduce that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum HZX ⊗ ℚ and that the slices have transfers. This proves several conjectures of Voevodsky [22, conjectures 1, 7, 10, 11] in characteristic zero.

scholarly journals Motivic cohomology spectral sequence and Steenrod operations

2016 ◽  
Vol 152 (10) ◽  
pp. 2113-2133 ◽  
Author(s):  
Serge Yagunov
Keyword(s):  
Spectral Sequence ◽  
Prime Number ◽  
Explicit Formula ◽  
Spectral Sequences ◽  
Motivic Cohomology ◽  
Steenrod Operations ◽  
Local Coefficients ◽  
Power Operations

For a prime number$p$, we show that differentials$d_{n}$in the motivic cohomology spectral sequence with$p$-local coefficients vanish unless$p-1$divides$n-1$. We obtain an explicit formula for the first non-trivial differential$d_{p}$, expressing it in terms of motivic Steenrod$p$-power operations and Bockstein maps. To this end, we compute the algebra of operations of weight$p-1$with$p$-local coefficients. Finally, we construct examples of varieties having non-trivial differentials$d_{p}$in their motivic cohomology spectral sequences.

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