scholarly journals Slices of motivic Landweber spectra

2010 ◽  
Vol 9 (1) ◽  
pp. 103-117 ◽  
Author(s):  
Markus Spitzweck
Keyword(s):  
Compact Objects ◽  
Homotopy Category ◽  
Motivic Homotopy

AbstractIn this paper we show that a conjecture of Voevodsky about the slices of the motivic cobordism spectrum implies a statement about the slices of motivic Landweber spectra. Over perfect fields these slices are given by the coefficients of the corresponding topological Landweber spectrum and the motivic Eilenberg MacLane spectrum. We also prove a cohomological version of Landweber exactness which applies to the compact objects of the stable motivic homotopy category.

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 Module categories for group algebras over commutative rings

2013 ◽  
Vol 11 (2) ◽  
pp. 297-329 ◽  
Author(s):  
Dave Benson ◽  
Srikanth B. Iyengar ◽  
Henning Krause
Keyword(s):  
Main Idea ◽  
Commutative Rings ◽  
Compact Objects ◽  
Homotopy Category ◽  
Module Category ◽  
Stable Module Category ◽  
Injective Modules ◽  
Stable Module ◽  
A Finite Group ◽  
Finitely Presented

AbstractWe develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented kG-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits of weakly injective modules. There is also an analogous version of the homotopy category of weakly injective kG-modules and a recollement relating the stable category, the homotopy category, and the derived category of kG-modules.

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 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 On the rational motivic homotopy category

2021 ◽  
Vol 8 ◽  
pp. 533-583
Author(s):  
Frédéric Déglise ◽  
Jean Fasel ◽  
Fangzhou Jin ◽  
Adeel A. Khan
Keyword(s):  
Homotopy Category ◽  
Motivic Homotopy

scholarly journals Spin effects in the effective field theory approach to Post-Minkowskian conservative dynamics

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Zhengwen Liu ◽  
Rafael A. Porto ◽  
Zixin Yang
Keyword(s):  
Field Theory ◽  
Effective Field Theory ◽  
Scattering Problem ◽  
Finite Size ◽  
Effective Field ◽  
Compact Objects ◽  
Scattering Data ◽  
Theory Approach ◽  
Spin Effects ◽  
Radial Action

Abstract Building upon the worldline effective field theory (EFT) formalism for spinning bodies developed for the Post-Newtonian regime, we generalize the EFT approach to Post-Minkowskian (PM) dynamics to include rotational degrees of freedom in a manifestly covariant framework. We introduce a systematic procedure to compute the total change in momentum and spin in the gravitational scattering of compact objects. For the special case of spins aligned with the orbital angular momentum, we show how to construct the radial action for elliptic-like orbits using the Boundary-to-Bound correspondence. As a paradigmatic example, we solve the scattering problem to next-to-leading PM order with linear and bilinear spin effects and arbitrary initial conditions, incorporating for the first time finite-size corrections. We obtain the aligned-spin radial action from the resulting scattering data, and derive the periastron advance and binding energy for circular orbits. We also provide the (square of the) center-of-mass momentum to $$ \mathcal{O}\left({G}^2\right) $$ O G 2 , which may be used to reconstruct a Hamiltonian. Our results are in perfect agreement with the existent literature, while at the same time extend the knowledge of the PM dynamics of compact binaries at quadratic order in spins.

scholarly journals Horizon radiation reaction forces

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Walter D. Goldberger ◽  
Ira Z. Rothstein
Keyword(s):  
Black Hole ◽  
Equations Of Motion ◽  
Finite Size ◽  
Radiation Reaction ◽  
Effective Field ◽  
Compact Objects ◽  
Finite Size Effect ◽  
Binary Black Holes ◽  
Dissipative Effects ◽  
Reaction Forces

Abstract Using Effective Field Theory (EFT) methods, we compute the effects of horizon dissipation on the gravitational interactions of relativistic binary black hole systems. We assume that the dynamics is perturbative, i.e it admits an expansion in powers of Newton’s constant (post-Minkowskian, or PM, approximation). As applications, we compute corrections to the scattering angle in a black hole collision due to dissipative effects to leading PM order, as well as the post-Newtonian (PN) corrections to the equations of motion of binary black holes in non-relativistic orbits, which represents the leading order finite size effect in the equations of motion. The methods developed here are also applicable to the case of more general compact objects, eg. neutron stars, where the magnitude of the dissipative effects depends on non-gravitational physics (e.g, the equation of state for nuclear matter).

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