Topological models for stable motivic invariants of regular number rings

Bachmann, Tom; Østvær, Paul Arne

doi:10.48550/arxiv.2102.01618

Cited by 2 publications

(4 citation statements)

References 29 publications

(32 reference statements)

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“…As a consequence the present paper and the second section of [BØ21] present similar results with similar techniques, although [BØ21] has a more direct and a simpler approach. The present paper was written, independently of [BØ21], during the spring of 2020. We wish to thank T. Bachmann and P. A. Østvaer for having allowed us to publish our work despite the overlap with theirs.…”

Section: Introductionsupporting

confidence: 71%

“…Although our results of [Man18] are phrased for the motivic stable homotopy category hSH(K) = SH(K) of a perfect field K, the structure and the arguments of the present paper are essentially the same as those of [Man18]. As a consequence the present paper and the second section of [BØ21] present similar results with similar techniques, although [BØ21] has a more direct and a simpler approach. The present paper was written, independently of [BØ21], during the spring of 2020.…”

Section: Introductionsupporting

confidence: 54%

“…In their recent work [BØ21] the authors have generalized and streamlined the arguments of a previous version of our results which appeared in [Man18]. Although our results of [Man18] are phrased for the motivic stable homotopy category hSH(K) = SH(K) of a perfect field K, the structure and the arguments of the present paper are essentially the same as those of [Man18].…”

Section: Introductionsupporting

confidence: 53%

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Localizations and completions of stable $\infty$-categories

Mantovani¹

2021

Preprint

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We extend some classical results of Bousfield on homology localizations and nilpotent completions to a presentably symmetric monoidal stable ∞-category M admitting a multiplicative left-complete t-structure. If E is a homotopy commutative algebra in M we show that E-nilpotent completion, E-localization, and a suitable formal completion agree on bounded below objects when E satisfies some reasonable conditions. Contents 1. Introduction 1 2. Preliminaries 3 3. Moore objects 9 4. Localization at some homotopy commutative algebras 15 5. Examples and applications 20 6. The E-based Adams-Novikov spectral sequence 23 7. Nilpotent Resolutions 25 Appendix A. Pro-Objects 35 Appendix B. Categorical Recollection 37 References 39

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Section: Introductionsupporting

confidence: 71%

Section: Introductionsupporting

confidence: 54%

Section: Introductionsupporting

confidence: 53%

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Localizations and completions of stable $\infty$-categories

Mantovani¹

2021

Preprint

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“…This is automatic if S is the spectrum of a perfect field.19 The proper intersection assumption gives a canonical isomorphism: NZC ≃ i * (NDX).20 When bad reduction can occur, specializing at a point does not preserve the normal crossing divisor. Nevertheless, for good reduction, one can show that our definition of quadratic intersection degree specializes correctly.21 This is generalized to Z[1/2] and other number rings in[BØ21].22 Recall the trace form ϕ is non-degenerate if and only if Bx is étale over O.…”

mentioning

confidence: 99%

Stable motivic homotopy theory at infinity

Dubouloz¹,

Déglise²,

Østvær³

2021

Preprint

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In this paper, we initiate a study of motivic homotopy theory at infinity. We use the six functor formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under ℓ-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the stable motivic homotopy type at infinity of an algebraic variety recovers the singular complex at infinity of the corresponding analytic space. We coin the notion of homotopically smooth morphisms with respect to a motivic ∞-category and use it to show a generalization to virtual vector bundles of Morel-Voevodsky's purity theorem, which yields an escalated form of Atiyah duality with compact support. Further we study a quadratic refinement of intersection degrees taking values in motivic cohomotopy groups. For relative surfaces, we show the stable motivic homotopy type at infinity witnesses a quadratic version of Mumford's plumbing construction for smooth complex algebraic surfaces. Our main results are also valid for ℓ-adic sheaves, mixed Hodge modules, and more generally motivic ∞-categories.

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Topological models for stable motivic invariants of regular number rings

Cited by 2 publications

References 29 publications

Localizations and completions of stable $\infty$-categories

Localizations and completions of stable $\infty$-categories

Stable motivic homotopy theory at infinity

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