The Zeroth -Stable Homotopy Sheaf of a Motivic Space

Abstract

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.

Torus actions, Morse homology, and the Hilbert scheme of points on affine space

“…Again using that k has characteristic zero, Bachmann showed that the P 1suspension functor Q = Σ ∞ P 1 : H(k) * → SH(k) is conservative on A 1 -simply connected spaces which can be written as homotopy colimits of spaces X + ∧ G m with X ∈ Sm k [3,Theorem 1.3].…”

On Infinite Effectivity of Motivic Spectra and the Vanishing of their Motives

“…Note also that one can compose M k,Λ with the natural functors from the category of unstable motivic spaces and from the S 1 -stable motivic category SH S 1 (k) into SH Λ (k). Now, the recent Theorem 1.3 and Corollary 4.13 of [Bac20] give interesting conservativity statements for these functors (in the case Λ = Z; yet note that Proposition 1.2.5(4) gives an embedding of the corresponding category SH S 1 Λ (k) into SH S 1 (k)); thus one can describe even longer chains of (composable) conservative functors between various motivic categories. 2.…”