From algebraic cobordism to motivic cohomology

Hoyois, Marc

doi:10.1515/crelle-2013-0038

Cited by 88 publications

(140 citation statements)

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“…Proof By Lemma 3.9 it is enough to show that Cell(f q MGL[1/c]) ∈ Σ q T SH(S) eff cell . By [12,Theorem 7.12] and the proof of [34,Theorem 4.7], Using this result we can now prove compatibility of the slice filtration with Cell in the case that E is Landweber exact in the sense of [27]. Note that these spectra are always cellular by [27,Proposition 8.4].…”

Section: The Comparison Theoremmentioning

confidence: 90%

On equivariant and motivic slices

Heard

2019

Algebr. Geom. Topol.

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Let k be a field with a real embedding. We compare the motivic slice filtration of a motivic spectrum over Spec(k) with the C 2 -equivariant slice filtration of its equivariant Betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. In particular, we show that, up to reindexing, the towers agree for all spectra obtained from localized quotients of MGL and MR, and for motivic Landweber exact spectra and their realizations. As a consequence, we deduce that equivariant spectra obtained from localized quotients of MR are even in the sense of Hill-Meier, and give a computation of the slice spectral sequence converging to π * , * BP n /2 for 1 ≤ n ≤ ∞.Published: XX Xxxember 20XX

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Section: The Comparison Theoremmentioning

confidence: 90%

On equivariant and motivic slices

Heard

2019

Algebr. Geom. Topol.

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“…Voevodsky has stated a conjecture [35,Conjecture 9] giving a formula for the slices of S k in terms of the Adams-Novikov spectral sequence for the homotopy groups of S. This conjecture follows from properties of the motivic Thom spectrum MGL [34], together with a result of Hopkins-Morel [17] on the slices of MGL, now available through the paper of M. Hoyois [13]. We give some of the details of the proof of Voevodsky's conjecture, without any claim to originality.…”

Section: Slices Of the Sphere Spectrummentioning

confidence: 99%

“…The conjecture gives a connection of the layer s q S k with the complex of homotopy groups (in degree 2q) arising from the Adams-Novikov spectral sequence. Relying on a result of Hopkins-Morel (now available through the work of Hoyois [13]), we give a sketch of the proof of Voevodsky's conjecture.…”

Section: Introductionmentioning

confidence: 99%

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A comparison of motivic and classical stable homotopy theories

Levine

2013

Journal of Topology

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Let k be an algebraically closed field of characteristic zero. Let c : SH → SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In consequence, c induces an isomorphism c * : πn(E) −→ Πn,0(c(E))(k) for all spectra E and all n ∈ Z.Fix an embedding σ : k → C and let ReB : SH(k) → SH be the associated Betti realization. We show that the slice tower for the motivic sphere spectrum over k, S k , has Betti realization which is strongly convergent. This gives a spectral sequence 'of motivic origin' converging to the homotopy groups of the sphere spectrum S ∈ SH; this spectral sequence at E 2 agrees with the E2 terms in the Adams-Novikov spectral sequence after a reindexing. Finally, we show that, for E a torsion object in SH(k) eff , the Betti realization induces an isomorphism Πn,0(E)(k) → πn(ReBE) for all n, generalizing the Suslin-Voevodsky theorem comparing mod N Suslin homology and mod N singular homology.As a special case, Theorem 1 implies the following corollary:Corollary 2. Let k be an algebraically closed field of characteristic zero. Let S k be the motivic sphere spectrum in SH(k) and S the classical sphere spectrum in SH. Then the constant presheaf functor induces an isomorphismIn fact, the corollary implies the theorem, by a density argument (see Lemma 6.5). Remark.(1) As pointed out by the referee, the functor c is induced by a (left) Quillen functor between model categories (see the proof of Lemma 6.5), so we do achieve a comparison of 'homotopy theories', as stated in the title, rather than just the underlying homotopy categories.(2) The functor c is not full in general. In fact, for a perfect field k, Morel [22, Lemma 3.10, Corollary 6.43] has constructed an isomorphism of Π 0,0 S k (k) with the Grothendieck-Witt group GW(k) of symmetric bilinear forms over k. As long as not every element of k is a square, the augmentation ideal in GW(k) is non-zero, hence c : π 0 (S) → Π 0,0 S k (k) is not surjective. Of course, if k is algebraically closed, then GW(k) = Z by rank, and thus c : π 0 (S) → Π 0,0 S k (k) is an isomorphism. This observation can be viewed as the starting point for our main result.We also have a homotopy analog of the theorem of Suslin-Voevodsky [33, Theorem 8.3] comparing Suslin homology and singular homology with mod N coefficients:Theorem 3. Let k be an algebraically closed field of characteristic zero with an embedding σ : k → C. Then, for all X ∈ Sm/k, all N > 1 and n ∈ Z, the Betti realization associated to σ induces an isomorphism

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“…If S is over a field of characteristic 0, it seems likely that Theorem 0.1 would follow from the work of Spitzweck [16] on Landweber exact spectra. Such an approach would depend heavily on the Hopkins-Morel-Hoyois theorem [7].Conjecture 8 is intertwined with Voevodsky's Conjectures 1, 7 and 10 in [19], that HZ ← 1 → KGL induces isomorphismsand thus that Hom SH(S) (s 0 (1), s 0 (1)) ∼ = H 0 (S, Z). These are known to hold when the base S is smooth over a perfect field by the work of Voevodsky and Levine (see [9, 10.5.1 and 11.3.6; 21]), or singular over a field of characteristic 0 (see [13]).…”

mentioning

confidence: 99%

Slices of co-operations forKGL

Pelaez¹,

Weibel²

2014

Bulletin of the London Mathematical Society

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We verify a conjecture of Voevodsky, concerning the slices of co-operations in motivic K-theory. the notation that E ⊗ A * denotes q (T q ∧ E) ⊗ A q for a motivic spectrum E and a graded ring A * . In this notation, we prove:Theorem 0.1. Assume that S is smooth over a perfect field. Then there is an isomorphism of cosimplicial ring spectraRemark. If S is over a field of characteristic 0, it seems likely that Theorem 0.1 would follow from the work of Spitzweck [16] on Landweber exact spectra. Such an approach would depend heavily on the Hopkins-Morel-Hoyois theorem [7].Conjecture 8 is intertwined with Voevodsky's Conjectures 1, 7 and 10 in [19], that HZ ← 1 → KGL induces isomorphismsand thus that Hom SH(S) (s 0 (1), s 0 (1)) ∼ = H 0 (S, Z). These are known to hold when the base S is smooth over a perfect field by the work of Voevodsky and Levine (see [9, 10.5.1 and 11.3.6; 21]), or singular over a field of characteristic 0 (see [13]).Here is our main result, which evidently implies Theorem 0.1.Theorem 0.2. Let S be a finite-dimensional separated Noetherian scheme. Then we have the following.(a) There are isomorphisms for all n 0:These isomorphisms commute with all of the coface and codegeneracy operators except possibly ∂ 0 and σ 0 .(b) Assume in addition that s 0 (1) → s 0 (KGL) is an isomorphism in SH(S), and that Hom SH(S) (s 0 (1), s 0 (1)) is torsion-free. Then the maps in (a) are the components of an isomorphism of graded cosimplicial motivic ring spectra:The case n = 0 of Theorem 0.2(a), that s 0 (KGL) ⊗ π 2 * KU ∼ = s * KGL, is immediate from the periodicity isomorphism T ∧ KGL ∼ = KGL defining the motivic spectrum KGL and the formula π 2 * KU ∼ = Z[u, u −1 ]. The need to pass to slices is clear at this stage, because π 2n,n KGL ∼ = K n (S) for S smooth over a perfect field.The left side of Theorem 0.2 is algebraic in nature, involving only the cosimplicial ring π 2 * (KU ∧n+1 ) and H = KU 0 (KU ). In fact, π 2 * (KU ∧n+1 ) is the cobar construction C • Γ (R, R) ∼ = KU * ⊗ H ⊗n for the Hopf algebroid (R, Γ) = (KU * , KU * KU ). We devote the first four sections to an analysis of this algebra, focussing on the rings F = KU 0 (CP ∞ ) and H. Much of this material is well known, and due to Frank Adams [1].If E is an oriented motivic spectrum, the projective bundle theorem says that E ∧ P ∞ ∼ = E ⊗ F and hence that E ∧ P ∞∧n ∼ = E ⊗ F ⊗n . Using this, we establish a toy version of Theorem 0.2 in Propositions 5.8 and 6.4 that, as cosimplicial spectra, KGL ∧ P ∞∧•+1 is KGL ⊗ F ⊗•+1 (the cobar construction on F ) and s * (KGL ∧ P ∞∧•+1 ) ∼ = s * (KGL) ⊗ F ⊗•+1 ∼ = s 0 (KGL) ⊗ KU * (CP ∞∧n ). (0.3) Remark 4.5. Lemma 4.4 fails dramatically if u = 0, for example when E = HZ.

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From algebraic cobordism to motivic cohomology

Cited by 88 publications

References 32 publications

On equivariant and motivic slices

On equivariant and motivic slices

A comparison of motivic and classical stable homotopy theories

Slices of co-operations forKGL

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