Nice triples and moving lemmas for motivic spaces
- Authors: Panin I.A.1,2
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Affiliations:
- St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
- University of Oslo
- Issue: Vol 83, No 4 (2019)
- Pages: 158-193
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/133797
- DOI: https://doi.org/10.4213/im8819
- ID: 133797
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Abstract
This paper contains geometric tools developed to solve the finite-fieldcase of the Grothendieck–Serre conjecture in [1]. It turns out thatthe same machinery can be applied to solve some cohomological questions.In particular, for any presheaf of $S^1$-spectra $E$ on the category of$k$-smooth schemes, all its Nisnevich sheaves of $\mathbf{A}^1$-stablehomotopy groups are strictly homotopy invariant. This shows that $E$ is$\mathbf{A}^1$-local if and only if all its Nisnevich sheaves of ordinarystable homotopy groups are strictly homotopy invariant. The latter resultwas obtained by Morel [2] in the case when the field $k$ is infinite.However, when $k$ is finite, Morel's proof does not work since it usesGabber's presentation lemma and there is no published proof of that lemma.We do not use Gabber's presentation lemma. Instead, we develop the machineryof nice triples invented in [3]. This machinery is inspired byVoevodsky's technique of standard triples [4].
Keywords
About the authors
Ivan Alexandrovich Panin
St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences; University of Oslo
Email: paniniv@gmail.com
Doctor of physico-mathematical sciences
References
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