Towards conservativity of 𝔾m–stabilization

Bachmann, Tom; Yakerson, Maria

doi:10.2140/gt.2020.24.1969

Cited by 7 publications

(6 citation statements)

References 24 publications

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“…Remark 5.3.12. The stronger assertion here is obtained by combining the weaker assertion and a conjecture about conservativity of G m -stabilization [BY18]. Conjecture 5.3.11 is reminiscent of an open version of the Cannon-Edwards double suspension theorem [Can79,Edw06].…”

Section: A 1 -Contractibility Of the Koras-russell Threefoldmentioning

confidence: 91%

$${\mathbb A}^1$$ -homotopy Theory and Contractible Varieties: A Survey

Asok

Østvær

2021

Lecture Notes in Mathematics

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Section: A 1 -Contractibility Of the Koras-russell Threefoldmentioning

confidence: 91%

$${\mathbb A}^1$$ -homotopy Theory and Contractible Varieties: A Survey

Asok

Østvær

2021

Lecture Notes in Mathematics

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“…Proof. (1) We have E ∈ SH S 1 (k)(d) ≥0 [2, Lemma 6.2(3)]. Write π d 0 for the homotopy object in the t-structure on SH S 1 (k)(d).…”

Section: The Heart Of Shmentioning

confidence: 99%

“…There is a map such that n elements of are linearly independent if and only if their image under D is nonzero (pick a basis of and let D be the determinant). By adjunction, the composite defines a map , where R denotes the Weil restriction along (see, e.g., [1, §7.6]). By construction, n elements of V have image in linearly independent over if and only if their image under is nonzero.…”

Section: A ‘Formula’ For Closed Pullbackmentioning

confidence: 99%

The Zeroth -Stable Homotopy Sheaf of a Motivic Space

Bachmann

2021

J. Inst. Math. Jussieu

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We establish a kind of ‘degree $0$ Freudenthal ${\mathbb {G}_m}$ -suspension theorem’ in motivic homotopy theory. From this we deduce results about the conservativity of the $\mathbb P^1$ -stabilization functor. In order to establish these results, we show how to compute certain pullbacks in the cohomology of a strictly homotopy-invariant sheaf in terms of the Rost–Schmid complex. This establishes the main conjecture of [2], which easily implies the aforementioned results.

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“…Firstly, recall that there exists a canonical exact monoidal connecting functorfrom SH S 1 (k) into each our motivic category D(k); it send M SH S 1 (X) Remark 5.1.5. Recall also that for each of the categories D ef f (k) there is a "forgetful" t ef f hom −exact functor Ψ D to SH S 1 (k), which is right adjoint to the natural functor SH S 1 (k) → D ef f (easily follows from our description of t ef f hom in terms of homotopy sheaves, see also Lemma 6.2 (2) of [BaY20]).…”

Section: Our Notions and Assumptions In Various Motivic Categoriesmentioning

confidence: 99%

Smooth weight structures and birationality filtrations on motivic categories

Bondarko¹,

Z.²

2021

Preprint

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We study various triangulated motivic categories and introduce a vast family of aisles (these are certain classes of objects) in them. These aisles are defined in terms of the corresponding "motives" (or motivic spectra) of smooth varieties in them; we relate them to the corresponding homotopy t−structures. We describe our aisles in terms of stalks at function fields and prove that they widely generalize the ones corresponding to slice filtrations. Further, the filtrations on the "homotopy hearts" Ht ef f hom of the corresponding effective subcategories that are induced by these aisles can be described in terms of (Nisnevich) sheaf cohomology as well as in terms of the Voevodsky contractions − −1 . Respectively, we express the condition for an object of Ht ef f hom to be weakly birational (i.e., that its n + 1th contraction is trivial or, equivalently, the Nisnevich cohomology vanishes in degrees > n for some n ≥ 0) in terms of these aisles; this statement generalizes well-known results of Kahn and Sujatha. Next, these classes define weight structures w s Smooth (where s = (s j ) are non-decreasing sequences parameterizing our aisles) that vastly generalize the Chow weight structures w Chow defined earlier. Using general abstract nonsense we also construct the corresponding adjacent t−structures t s Smooth and prove that they give the birationality filtrations on Ht ef f hom . Moreover, some of these weight structures induce weight structures on the corresponding n−birational motivic categories (these are the localizations by the levels of the slice filtrations). Our results also yield some new unramified cohomology calculations.

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Towards conservativity of 𝔾m–stabilization

Cited by 7 publications

References 24 publications

$${\mathbb A}^1$$ -homotopy Theory and Contractible Varieties: A Survey

$${\mathbb A}^1$$ -homotopy Theory and Contractible Varieties: A Survey

The Zeroth -Stable Homotopy Sheaf of a Motivic Space

Smooth weight structures and birationality filtrations on motivic categories

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