Motivic and real étale stable homotopy theory

Bachmann, Tom

doi:10.1112/s0010437x17007710

Cited by 47 publications

(79 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Morel proved a version of Gabber rigidity theorem for a cohomology theory representable in the motivic stable homotopy category by an n-torsion spectrum provided that the base field satisfies a certain assumption on the finiteness of the virtual cohomological 2-dimension. The latest rigidity result was obtained by Bachmann [Ba16,Corollary 40] as a corollary of his study of ρ-inverted stable motivic homotopy category by means of realétale topology. Bachmann showed that a version of Gabber rigidity theorem holds for a cohomology theory representable by a ρ-periodic spectrum with ρ = −[−1] ∈ K MW 1 (k) ∼ = Hom SH(k) (S, S ∧ (A 1 − {0}, 1)).…”

Section: Introductionmentioning

confidence: 99%

Rigidity for linear framed presheaves and generalized motivic cohomology theories

Ananyevskiy

Druzhinin

2018

Advances in Mathematics

View full text Add to dashboard Cite

A rigidity property for the homotopy invariant stable linear framed presheaves is established. As a consequence a variant of Gabber rigidity theorem is obtained for a cohomology theory representable in the motivic stable homotopy category by a φ-torsion spectrum with φ ∈ GW(k) of rank coprime to the (exponential) characteristic of the base field k. It is shown that the values of such cohomology theories at an essentially smooth Henselian ring and its residue field coincide. The result is applicable to cohomology theories representable by n-torsion spectra as well as to the ones representable by η-periodic spectra and spectra related to Witt groups.Research is supported by the Russian Science Foundation grant 14-21-00035. Preliminaries on framed correspondencesAll schemes are supposed to be separated and noetherian.Definition 2.1. Let X be a scheme and Z be a closed subscheme of X. Anétale neighborhood of Z in X is a pair of morphisms (p : U → X, r : Z → U ) where p isétale and p • r = i for the closed

show abstract

Section: Introductionmentioning

confidence: 99%

Rigidity for linear framed presheaves and generalized motivic cohomology theories

Ananyevskiy

Druzhinin

2018

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…We thus need to establish condition 5. We do this in some detail; this style of argument can also be used to make more precise our sketches for conditions [1][2][3][4]…”

Section: Given Morphismsmentioning

confidence: 99%

Some remarks on units in Grothendieck–Witt rings

Bachmann

2018

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…Let ρ denote the map ½ → Σ α ½ induced by taking the nonbasepoint of S 0 to −1 ∈ A 1 0. In [3], Bachmann finds an alternate presentation of the ρinverted stable motivic homotopy category SH A 1 (F )[1/ρ] in terms of the real étale topology. We will not go into the details of the real étale topology, instead sending the reader to [25], especially its first chapter.…”

Section: Bachmann's Theoremmentioning

confidence: 99%

“…We solve this problem using Bachmann's theorem on π ⋆ ½[1/2, 1/η] and the Hu-Kriz-Ormsby comparison of the 2-and (2, η)-complete spheres when F has finite virtual 2-cohomological dimension. 3 In order to state our results, let π top m ½ denote the m-th homotopy group of the topological sphere spectrum. Theorem 1.5.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4.3, we use fracture square methods to prove Theorem 1.5. Finally, in Section 5, we pose several open questions in Section 5 and 3 Recall that vcd 2 (F ) := cd 2 (F ( √ −1)) where cd 2 denotes 2-primary étale cohomological dimension. prove Theorem 5.5 which compares the motivic stable stems and η-periodic motivic stable stems in a range.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Vanishing in Stable Motivic Homotopy sheaves

Ormsby

Röndigs

Østvær

2018

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

We determine systematic regions in which the bigraded homotopy sheaves of the motivic sphere spectrum vanish. amongst these is the sphere spectrum ½ := Σ ∞ P 1 Spec(F ) + . We denote this object by ½ because it is the unit for the symmetric monoidal product on SH A 1 (F ) given by the smash product.Equivalence between P 1 -spectra is detected by the bigraded homotopy sheaves, π m+nα X for m, n ∈ Z, which are defined as the Nisnevich sheafification of the assignmentHere [ , ] SH A 1 (F ) denotes the hom-set in SH A 1 (F ) and Σ m+nα denotes smashing with (S 1 ) ∧m ∧ (A 1 0) ∧n . Since every motivic spectrum is a ½-module, the bigraded sheafplays a fundamental role in stable motivic homotopy theory, analogous to the stable homotopy groups of spheres in topology. We will refer to π m+nα ½ as the (m + nα)-th motivic stable stem, and to the Z-graded sheaf π m+ * α ½ as the m-th Milnor-Witt stem.The motivic stable stems (and their global sections, π m+nα ½ := π m+nα ½(F )) have been objects of intense study since Morel's analysis of the 0-th motivic stable stem in [15]. That paper launched his program [16] to identify the 0-th Milnor-Witt stem with K MW − * , the Milnor-Witt K-theory sheaf, explaining the nomenclature. In further work [19], Morel shows that ½ is connective, meaning that m-th Milnor-Witt stems are 0 for m < 0. Beyond Morel's theorems, little is known about Milnor-Witt stems over a general field. Röndigs-Spitzweck-Østvaer [24] determine the 1-st Milnor-Witt stem as an extension of K M * /24 and a certain sheaf related to Hermitian K-theory; this vastly generalizes work of Ormsby-Østvaer [22] for fields of cohomological dimension less than three. All other computations are limited to specific fields, and are generally only known on global sections 2010 Mathematics Subject Classification. Primary: 14F42.

show abstract

Motivic and real étale stable homotopy theory

Cited by 47 publications

References 37 publications

Rigidity for linear framed presheaves and generalized motivic cohomology theories

Rigidity for linear framed presheaves and generalized motivic cohomology theories

Some remarks on units in Grothendieck–Witt rings

Vanishing in Stable Motivic Homotopy sheaves

Contact Info

Product

Resources

About