The eta-inverted sphere over the rationals

Wilson, Glen Matthew

doi:10.2140/agt.2018.18.1857

Cited by 12 publications

(14 citation statements)

References 16 publications

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“…They resolved it up to a conjecture about the classical Adams-Novikov spectral sequence, which was subsequently proved by Andrews-Miller [AM17]. The cases k = R and k = Q (both up to 2-adic completion) were done by Guillou-Isaksen [GI16] and Wilson [Wil18], respectively. All of these authors use the motivic Adams spectral sequence.…”

Section: Main Theoremmentioning

confidence: 99%

$η$-periodic motivic stable homotopy theory over fields

Bachmann¹,

Hopkins²

2020

Preprint

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Over any field of characteristic = 2, we establish a 2-term resolution of the η-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the K(1)-local sphere in classical stable homotopy theory. As applications we determine the η-periodized motivic stable stems and the η-periodized algebraic symplectic and SL-cobordism groups. Along the way we construct Adams operations on the motivic spectrum representing Hermitian K-theory and establish new completeness results for certain motivic spectra over fields of finite virtual 2-cohomological dimension. In an appendix, we supply a new proof of the homotopy fixed point theorem for the Hermitian K-theory of fields. Contents 1. Introduction 1.1. Inverting the motivic Hopf map 1.2. Main results 1.3. Proof sketch 1.4. Organization 1.5. Acknowledgements 1.6. Conventions 1.7. Table of notation 2. Preliminaries and Recollections 2.1. Filtered modules 2.2. Derived completion of abelian groups 2.3. Witt groups 2.4. The homotopy t-structure 2.5. Completion and localization 2.6. Very effective spectra 2.7. Inverting ρ 3. Adams operations for the motivic spectrum KO 3.1. Summary 3.2. Motivic ring spectra KO and KGL 3.3. The Gepner-Snaith and homotopy fixed point theorems 3.4. Action on the Bott element 3.5. The 2-adic fracture square 3.6. Proof of the main theorem 3.7. The motivic orthogonal image of j spectrum 3.8. Geometric operations 4. Cobordism spectra 4.1. Summary 4.2. Real realization 4.3. Homology 4.4. Adams action 5. Some completeness results 5.1. Summary 5.2. Main result 5.3. Remaining proofs 6. The HW-homology of kw 6.1. Summary 6.2. The motivic dual Steenrod algebra 6.3. Spectra employed in the proof Date: June 4, 2020. 1 2 TOM BACHMANN AND MICHAEL J. HOPKINS 6.4. Determination of π * (kw ∧ HW) ∧ 2 41 7. Main theorem 45 8. Applications 48 8.1. Homotopy groups of the η-periodic sphere 48 8.2. Homotopy groups of MSp[η −1 ] 49 8.3. Homotopy groups of MSL[η −1 ] 52 8.4. MSp, MSL and kw 52 8.5. Cellularity results 53 8.6. HW ∧ HW (2) 54 8.7. kw * kw (2) 55 Appendix A. The homotopy fixed point theorem 56 References 58

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Section: Main Theoremmentioning

confidence: 99%

$η$-periodic motivic stable homotopy theory over fields

Bachmann¹,

Hopkins²

2020

Preprint

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“…The global sections of these sheaves are computed for F = C in [1], and, for F = R, the 2-complete global section computation appears in [7]. Calculations for p-adic fields Q p and the rational numbers Q will appear in [33]. Any sheaf computations and computations over a general field are completely open, except that we know we have vanishing under the conditions of Theorem 1.5.…”

Section: Questionsmentioning

confidence: 99%

Vanishing in Stable Motivic Homotopy sheaves

Ormsby

Röndigs

Østvær

2018

Forum of Mathematics, Sigma

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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.

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“…Note that since τ η 4 = 0 in π F * * (S 0,0 ), these classes cannot be seen using Wilson's η-local computations over the rationals [34]. Therefore the theorem provides two new infinite families in the F -motivic stable stems for any field F of characteristic zero.…”

Section: Introductionmentioning

confidence: 98%

“…n+1,n f 0 (KQ) where K M is Milnor K-theory and f 0 (KQ) is the effective cover of Hermitian K-theory. Other infinite computations have been done after inverting the Hopf map η ∈ π F 1,1 (S 0,0 ) by Guillou-Isaksen [16][14], Andrews-Miller [4], Röndigs [31], and Wilson [34]. We refer the reader to [19,Sec.…”

Section: Introductionmentioning

confidence: 99%

Motivic Mahowald invariants over general base fields

Quigley

2019

Preprint

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The motivic Mahowald invariant was introduced in [27] and [28] to study periodicity in the C-and R-motivic stable stems. In this paper, we define the motivic Mahowald invariant over any field F of characteristic not two and use it to study periodicity in the F -motivic stable stems. In particular, we construct lifts of some of Adams' classical v 1 -periodic families [2] and identify them as the motivic Mahowald invariants of powers of 2 + ρη. ContentsAbstract 1 1. Introduction 1 2. Motivic Lin's Theorem and the motivic Mahowald invariant revisited 4 3. v 1 -periodic families over prime fields 6 4. Motivic Mahowald invariant of (2 + ρη) i 11 References 13

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The eta-inverted sphere over the rationals

Cited by 12 publications

References 16 publications

$η$-periodic motivic stable homotopy theory over fields

$η$-periodic motivic stable homotopy theory over fields

Vanishing in Stable Motivic Homotopy sheaves

Motivic Mahowald invariants over general base fields

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