$η$-periodic motivic stable homotopy theory over fields

Abstract

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

Topological models for stable motivic invariants of regular number rings

“…Since f * preserves colimits it commutes with b-periodization by Lemma 3.2. We shall make use of the fact that a map is an equivalence if and only if it is an equivalence after b-periodization and b-completion, see e.g., [BH20a,Lemma 2.16]. Thus to prove fully faithfulness it would also be sufficient, as well as necessary, to prove…”

Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings

“…Coming back to the 2-local behaviour of Grothendieck-Witt theory, we note that sending a symmetric bilinear form to its underlying finitely generated projective module leads to a canonical map GW s cl ( ; ) → K( ; ) hC 2 . The question whether this map is a 2-adic equivalence in positive degrees is known as Thomason's homotopy limit problem [Tho83], which admits a positive solution for many rings in which 2 is invertible, notably by work of Hu, Kriz, and Ormsby [HKO11], Bachmann and Hopkins [BH20] and Berrick, Karoubi, Schlichting and Østvaer [BKSØ15]. In §3.1 we will show: Theorem 2.…”

“…The characteristic 0 case of Theorem 3.1.1 was proven in [HKO11], while the positive odd characteristic case is established in [BKSØ15]. An alternative proof of this theorem is also provided in recent work of Bachmann and Hopkins [BH20]. Special cases of the above theorem were already known before: the case of the field ℂ of complex numbers, for example, can be reduced to the classical equivalence BO ≃ BU hC 2 see, e.g., [BK05, Lemma 7.3].…”

“…This question, first raised by Thomason in [Tho83], is commonly known as the homotopy limit problem. In the case of fields, the following theorem represents the current state of the art; see [HKO11,BKSØ15,BH20]. We recall that the virtual mod 2 cohomological dimension vcd 2 of a field can be defined as the ordinary mod 2 cohomological dimension cd 2 of (the absolute Galois group of) [ √ −1].…”

The stable Adams operations on Hermitian K-theory

“…We expect that these operations will be useful in many situations. For instance, Bachmann and Hopkins recently used them in [BH20] to compute the η-inverted homotopy sheaves of the algebraic symplectic and special linear cobordism spaces. Their construction of Adams operation is quite different in spirit to the one presented here, but we believe that the two constructions coincide.…”