scholarly journals Brown representability in 1-homotopy theory

2011 ◽  
Vol 7 (3) ◽  
pp. 527-539 ◽  
Author(s):  
Niko Naumann ◽  
Markus Spitzweck
Keyword(s):  
Homotopy Theory ◽  
Homotopy Category ◽  
Noetherian Scheme ◽  
Finite Dimensional ◽  
Brown Representability ◽  
Motivic Homotopy

AbstractWe prove the following result announced by V. Voevodsky. If S is a finite dimensional noetherian scheme such that S = ∪αSpec(Rα) for countable rings Rα, then the stable motivic homotopy category over S satisfies Brown representability.

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.

scholarly journals Remarks on motivic homotopy theory over algebraically closed fields

2010 ◽  
Vol 7 (1) ◽  
pp. 55-89 ◽  
Author(s):  
Po Hu ◽  
Igor Kriz ◽  
Kyle Ormsby
Keyword(s):  
Homotopy Theory ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Closed Fields ◽  
Motivic Homotopy Theory ◽  
Stable Homotopy Category ◽  
Motivic Homotopy ◽  
Algebraically Closed Fields

AbstractWe discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.

scholarly journals Motivic and real étale stable homotopy theory

2018 ◽  
Vol 154 (5) ◽  
pp. 883-917 ◽  
Author(s):  
Tom Bachmann
Keyword(s):  
Homotopy Theory ◽  
Finite Dimension ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
The Real ◽  
Rigidity Results ◽  
Noetherian Scheme ◽  
Stable Homotopy Category

Let$S$be a Noetherian scheme of finite dimension and denote by$\unicode[STIX]{x1D70C}\in [\unicode[STIX]{x1D7D9},\mathbb{G}_{m}]_{\mathbf{SH}(S)}$the (additive inverse of the) morphism corresponding to$-1\in {\mathcal{O}}^{\times }(S)$. Here$\mathbf{SH}(S)$denotes the motivic stable homotopy category. We show that the category obtained by inverting$\unicode[STIX]{x1D70C}$in$\mathbf{SH}(S)$is canonically equivalent to the (simplicial) local stable homotopy category of the site$S_{\text{r}\acute{\text{e}}\text{t}}$, by which we mean thesmallreal étale site of$S$, comprised of étale schemes over$S$with the real étale topology. One immediate application is that$\mathbf{SH}(\mathbb{R})[\unicode[STIX]{x1D70C}^{-1}]$is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the$\unicode[STIX]{x1D70C}$-local sphere (over$\mathbb{R}$). As further applications we show that$D_{\mathbb{A}^{1}}(k,\mathbb{Z}[1/2])^{-}\simeq \mathbf{DM}_{W}(k)[1/2]$(improving a result of Ananyevskiy–Levine–Panin), reprove Röndigs’ result that$\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9}[1/\unicode[STIX]{x1D702},1/2])=0$for$i=1,2$and establish some new rigidity results.

scholarly journals Brown representability often fails for homotopy categories of complexes

2011 ◽  
Vol 9 (1) ◽  
pp. 151-160 ◽  
Author(s):  
George Ciprian Modoi ◽  
Jan Šťovíček
Keyword(s):  
Abelian Groups ◽  
Homotopy Category ◽  
Brown Representability ◽  
Localizing Subcategory

AbstractWe show that for the homotopy categoryK(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory ofK(Ab) for which the inclusion intoK(Ab) does not have a right adjoint.

Rigidity in motivic homotopy theory

2008 ◽  
Vol 341 (3) ◽  
pp. 651-675 ◽  
Author(s):  
Oliver Röndigs ◽  
Paul Arne Østvær
Keyword(s):  
Homotopy Theory ◽  
Motivic Homotopy Theory ◽  
Motivic Homotopy

scholarly journals VANISHING OF THE NEGATIVE HOMOTOPY -THEORY OF QUOTIENT SINGULARITIES

2017 ◽  
Vol 18 (3) ◽  
pp. 619-627
Author(s):  
Gonçalo Tabuada
Keyword(s):  
Algebraic Geometry ◽  
Upper Bound ◽  
Homotopy Theory ◽  
Krull Dimension ◽  
Noncommutative Algebraic Geometry ◽  
Noetherian Scheme ◽  
Quotient Singularities ◽  
Cyclic Quotient Singularities

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy$K$-theory groups of a Noetherian scheme$X$of Krull dimension$d$vanish below$-d$. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy$K$-theory groups vanish below$-1$. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy$K$-theory group.

scholarly journals THE TOM DIECK SPLITTING THEOREM IN EQUIVARIANT MOTIVIC HOMOTOPY THEORY

2021 ◽  
pp. 1-70
Author(s):  
DAVID GEPNER ◽  
JEREMIAH HELLER
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Homotopy Theory ◽  
Independent Interest ◽  
Splitting Theorem ◽  
Motivic Homotopy Theory ◽  
Group A ◽  
Motivic Homotopy ◽  
A Finite Group

Abstract We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck’s splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed-point functors and the motivic Adams isomorphism.

Sign in / Sign up

Export Citation Format

Share Document