scholarly journals Cohomological obstruction theory for Brauer classes and the period-index problem

2010 ◽  
Vol 8 (3) ◽  
pp. 419-435 ◽  
Author(s):  
Benjamin Antieau
Keyword(s):  
Positive Integer ◽  
Homotopy Theory ◽  
Cohomological Dimension ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Stable Homotopy Groups ◽  
Open Subscheme ◽  
Finite Set ◽  
Twisted Sheaves ◽  
Set Of Points

AbstractLet U be a connected noetherian scheme of finite étale cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that α is a class in H2(Uét,ℂm)tors. For each positive integer m, the K-theory of α-twisted sheaves is used to identify obstructions to α being representable by an Azumaya algebra of rank m2. The étale index of α, denoted eti(α), is the least positive integer such that all the obstructions vanish. Let per(α) be the order of α in H2(Uét,ℂm)tors. Methods from stable homotopy theory give an upper bound on the étale index that depends on the period of α and the étale cohomological dimension of U; this bound is expressed in terms of the exponents of the stable homotopy groups of spheres and the exponents of the stable homotopy groups of B(ℤ/(per(α))). As a corollary, if U is the spectrum of a field of finite cohomological dimension d, then , where [] is the integer part of , whenever per(α) is divided neither by the characteristic of k nor by any primes that are small relative to d.

scholarly journals Detection of a new nontrivial family in the stable homotopy of spheres $ \mbf{\pi_{\ast}S} $

2008 ◽  
Vol 39 (1) ◽  
pp. 75-83
Author(s):  
Liu Xiugui ◽  
Jin Yinglong
Keyword(s):  
Spectral Sequence ◽  
Homotopy Theory ◽  
Steenrod Algebra ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Nontrivial Element ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Stable Homotopy Of Spheres ◽  
Gamma S

To determine the stable homotopy groups of spheres is one of the central problems in homotopy theory. Let $ A $ be the mod $ p $ Steenrod algebra and $S$ the sphere spectrum localized at an odd prime $ p $. In this article, it is proved that for $ p\geqslant 7 $, $ n\geqslant 4 $ and $ 3\leqslant s $, $ b_0 h_1 h_n \tilde{\gamma}_{s} \in Ext_A^{s+4,\ast}(\mathbb{Z}_p,\mathbb{Z}_p) $ is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element of order $ p $ in the stable homotopy groups of spheres $ \pi_{p^nq+sp^{2}q+(s+1)pq+(s-2)q-7}S $, where $ q=2(p-1 ) $.

scholarly journals Stable homotopy groups of spheres

2020 ◽  
Vol 117 (40) ◽  
pp. 24757-24763
Author(s):  
Daniel C. Isaksen ◽  
Guozhen Wang ◽  
Zhouli Xu
Keyword(s):  
Homotopy Theory ◽  
Computational Method ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Mathematical Tool ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Current State ◽  
Motivic Homotopy Theory ◽  
Motivic Homotopy

We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence.

Some infinite families in the stable homotopy of spheres

1987 ◽  
Vol 101 (3) ◽  
pp. 477-485 ◽  
Author(s):  
Wen-Hsiung Lin
Keyword(s):  
Spectral Sequence ◽  
Adams Spectral Sequence ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Stable Homotopy Of Spheres

The classical Adams spectral sequence [1] has been an important tool in the computation of the stable homotopy groups of spheres . In this paper we make another contribution to this computation.

Spectra of derived module homomorphisms

1987 ◽  
Vol 101 (2) ◽  
pp. 249-257 ◽  
Author(s):  
Alan Robinson
Keyword(s):  
Spectral Sequence ◽  
Homotopy Theory ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Stable Homotopy Theory ◽  
Ring Spectrum ◽  
Homotopy Invariant ◽  
New Construction

We introduce a new construction in stable homotopy theory. If F and G are module spectra over a ring spectrum E, there is no well-known spectrum of E-module homomorphisms from F to G. Such a construction would not be homotopy invariant, and therefore would not serve much purpose. We show that, provided the rings and modules have A∞ structures, there is a spectrum RHomE(F, G) of derived module homomorphisms which has very pleasant properties. It is homotopy invariant, exact in each variable, and its homotopy groups form the abutment of a hypercohomology-type spectral sequence.

Sign in / Sign up

Export Citation Format

Share Document