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Abstract We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $$X$$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the $${\mathbb P}^{1}$$ suspension of $$X$$; we also analyze a host of variations on this observation. Our approach yields many examples of $${\mathbb A}^{1}$$-$$(n-1)$-connected smooth affine $2n$-folds and strictly quasi-affine $${\mathbb A}^{1}$$-contractible smooth schemes.
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Localization and nilpotent spaces in -homotopy theory
Abstract For a subring $$R$$ of the rational numbers, we study $$R$$ -localization functors in the local homotopy theory of simplicial presheaves on a small site and then in $${\mathbb {A}}^1$$ -homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in $${\mathbb {A}}^1$$ -homotopy theory, paying attention to future applications for vector bundles. We show that $$R$$ -localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $$BGL_n$$ is $${\mathbb {A}}^1$$ -nilpotent when $$n$$ is odd, and analyze the (more complicated) situation where $$n$$ is even as well. We establish analogs of various classical results about rationalization in the context of $${\mathbb {A}}^1$$ -homotopy theory: if $-1$ is a sum of squares in the base field, $${\mathbb {A}}^n \,{\setminus}\, 0$$ is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres.
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- PAR ID:
- 10356074
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 158
- Issue:
- 3
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 654 to 720
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X22007321
