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Strong Convergence in the Motivic Adams Spectral Sequence

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Strong Convergence in the Motivic Adams Spectral Sequence STRONG CONVERGENCE IN THE MOTIVIC ADAMS SPECTRAL SEQUENCE JONAS IRGENS KYLLING AND GLEN MATTHEW WILSON ABSTRACT. We prove strong convergence results for the motivic Adams spectral sequence of the sphere spectrum over fields with finite virtual cohomological dimension at the prime 2, and over arbitrary fields at odd primes. We show that the motivic Adams spectral sequence is not strongly convergent over number fields. As applications we give bounds on the exponents of the (`; η)-completed motivic stable stems, and calculate the zeroth (`; η)-completed motivic stable stems. CONTENTS 1. Introduction 1 2. Recollections on mod ` motivic cohomology, the motivic Steenrod algebra and the motivic Adams spectral sequence 5 3. A detour to homotopy modules 7 4. Ext at the prime 2 16 5. Ext at odd primes 18 6. Strong convergence for the sphere spectrum 19 7. The `-Bockstein spectral sequence 21 8. Bounds on exponents of motivic stable stems 24 9. The zeroth motivic stable stem 24 References 26 1. INTRODUCTION The analogue of the Adams spectral sequence in motivic homotopy theory has been used successfully to calculate many interesting motivic invariants. However, the convergence properties of the motivic Adams spectral sequence are more subtle than in topology, which has limited its use to fairly special- ized situations. Not even for the motivic sphere spectrum is strong convergence of the motivic Adams spectral sequence over an arbitrary base field guaranteed. The goal of this work is to study the vanishing of the derived E1-term of the motivic Adams spectral sequence of the sphere spectrum over a general field. This vanishing is the crucial ingredient to obtain strong convergence from conditional convergence, following Boardman [Boa99]. For formal reasons the abutment of the motivic Adams spectral sequence is the H-completion of the target spectrum, where H arXiv:1901.03399v1 [math.AT] 10 Jan 2019 is the mod ` motivic cohomology spectrum. The H-completion was identified to be the (`; η)-completion of the target spectrum by Hu, Kriz and Ormsby [HKO11] under some strict bounded cellularity assump- tions on the target spectrum. Recently a different proof was given by Mantovani [Man18] which drops the cellularity assumption. Combined with vanishing of the derived E1-term this implies strong con- vergence to the homotopy groups of the (`; η)-completed spectrum. In topology it is usually sufficient for the target spectrum to satisfy some finiteness assumptions on its cohomology for the Adams spectral sequence to be strongly convergent. Working at the prime 2, this assumption is usually sufficient in motivic homotopy theory over fields with finitely many square 2010 Mathematics Subject Classification. 14F42 (primary), 55S10, 55T15 (secondary). Key words and phrases. Motivic homotopy theory, motivic Adams spectral sequence, motivic cohomology, motivic stable ho- motopy group. The authors were partially supported by the RCN Frontier Research Group Project no. 250399. 1 2 JONAS IRGENS KYLLING AND GLEN MATTHEW WILSON classes as well. But there are many interesting fields with infinitely many square classes, for instance the rationals, over which we would like to compute motivic invariants. One approach could be to consider the E2-page not as an F2-module, but as a module over mod M 2 Milnor K-theory K∗ (F )=2, and then hope that the successive Er-pages remain finitely generated M M K∗ (F )=2-modules. However, if K∗ (F )=2 is not Noetherian (equivalently, there are infinitely many M square classes), then kernels of finitely generated K∗ (F )=2-modules need not be finitely generated. And indeed this issue occurs in practice when computing the E2-page for the motivic sphere spectrum over the rationals with the ρ-Bockstein spectral sequence. Over Q the E1-page is a finitely generated M K∗ (Q)=2-module. However, on some groups the d1-differential is given by multiplication by ρ, hence M M the kernel contains a copy of the ρ-torsion in K∗ (Q)=2, which is an infinitely generated K∗ (Q)=2- module. What saves the day are vanishing regions in the E2-page analogous to the vanishing lines we have in topology. For a good motivic spectrum X the E2-page of the motivic Adams spectral sequence is Ext F (H (F ; =2);H (X; =2)) Ext A∗;∗ ∗;∗ Z ∗;∗ Z . These -groups are rather computable, and most of this paper is concerned with their properties. For the motivic sphere spectrum over the real numbers at the prime 2, s;(t;w) Guillou and Isaksen have shown that there are vanishing regions for Ext (H∗;∗( ; =2);H∗;∗( ; =2)) A∗R;∗ R Z R Z when t − s > 0 and t − s − w 6= 0. However when t − s = 0 and t − s − w ≥ 0 is even there is an infinite h0-tower in the Ext-group. Over a general field this infinite h0-tower is tensored with mod 2 Milnor K-theory, and is spread out in the cone w ≤ t − s ≤ 0 on the E2-page. Inside of this cone there can be infinitely many differentials exiting a particular tridegree of the motivic Adams spectral sequence, and it is not clear if the derived E1-terms vanish here. With the help of the extension s s 0 ! Ext (HR ;HR (S)) ⊗ k∗ ! Ext (H∗;∗;H∗;∗(S)) A∗R;∗ ∗;∗ ∗;∗ Z=2[ρ] A∗;∗ Z=2[ρ] s+1 R R ! Tor (Ext (H∗;∗;H∗;∗(S)); k∗) ! 0 1 A∗R;∗ we extend the vanishing lines of Guillou and Isaksen to general fields. This is where the assumption on the finiteness of the virtual cohomological dimension of the base field enters, since if not the outer terms in the extension can be nonzero for fixed stem and weight and any s. At odd primes we use a Bockstein spectral sequence to extend the topological vanishing lines of Adams to the motivic Ext-groups. At odd primes in high Adams filtration the Er-pages of the motivic Adams spectral sequences for the sphere spectrum and the motivic cohomology spectrum are the same. Thus we would expect there to be infinitely many differentials leaving a particular tridegree over special base fields (of course, this is only necessary for the derived E1-term to be nonzero). Hence it seems reasonable to doubt that the motivic Adams spectral sequence is strongly convergent in general. In Corollary 7.8 we show that the motivic Adams spectral sequence is not strongly convergent over number fields. However, with the help of vanishing regions in the Ext-groups we can show strong convergence for the motivic Adams spectral sequence for the sphere spectrum in positive stems. For S=`n the outlook is not as grim, and we prove strong convergence for S=`n over any field of characteristic not ` with finite virtual cohomological dimension if ` = 2 in Corollary 6.5. This suggests a general strategy for adapting classical proofs with the Adams spectral sequence to the motivic setting. Prove the required properties for S=`n and then pass to the limit over n, assuming that the properties are well behaved. Motivic cohomology is often considered a generalization of singular cohomology to smooth schemes. However, other generalizations are possible; a notable one is the generalized motivic cohomology spec- trum He Z of Bachmann [Bac16]. This spectrum is a closer approximation to the motivic sphere spectrum. In particular they have the same zeroth motivic homotopy group. For us the likeness manifests itself in their Ext-groups. It is simpler to describe the vanishing region as an isomorphism of Ext-groups in high filtration between the motivic sphere spectrum and He Z, and then compute the Ext-group of He Z. As part of this we compute the motivic cohomology of several motivic spectra and homotopy modules related to He Z. This seems to be an interesting computation in itself. For instance, motivic cohomology ∗;∗ W 1;−1 ∗;∗ ∗;∗ 2 1 of the Witt K-theory sheaf is H (K∗ ) = Σ A =A (τ; Sq +ρ Sq ), which as far as we know has no classical analogue (its complex realization is zero). STRONG CONVERGENCE IN THE MOTIVIC ADAMS SPECTRAL SEQUENCE 3 As a simple corollary of our computations we give a bound on the exponent of the torsion in the ^ completed motivic stable stems. For instance, in positive stems the exponent of the 2-torsion in πt;w(S2,η) is bounded by 2maxfd(t−w+1)=2e;t+vcd(F )g. This is a partial answer to a question in [ALP17]. As a second application we compute the (2; η)-completed zero-line of the motivic stable stems in Section 9. This is ^ essentially the same computation as done for π0;0(S2,η) by Morel in [Mor99]. The computation makes Morel’s pull-back square [Mor04b, Theoreme 5.3] appear less mysterious, since the pull-back square is suggested by the computation of the Ext-group. We hope this work will be useful to applications of the motivic Adams spectral sequence over general base fields. Up to now, most work with the motivic Adams spectral sequence has been at the prime 2 over C, R, or other fields with finitely many square classes. Previous work. One of the first applications of the motivic Adams spectral sequence was Morel’s proof ^ of Milnor’s conjecture on quadratic forms in [Mor99]. The proof consists of a computation of π0;0(S2,η). A more systematic study of the motivic Adams spectral sequence was carried out by Dugger and Isak- sen [DI10]. They calculate the C-motivic stable stems in a large range. Later Dugger and Isaksen [DI17a] computed the first four Milnor-Witt stems over the real numbers by using the ρ-Bockstein spectral se- Ext (H ( ; =2);H ( ; =2)) quence to calculate A∗R;∗ ∗;∗ R Z ∗;∗ R Z in a range and observing that the motivic Adams spectral sequence collapses in the first four Milnor-Witt stems.
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