Nice triples and moving lemmas for motivic spaces
- Autores: Panin I.A.1,2
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Afiliações:
- St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
- University of Oslo
- Edição: Volume 83, Nº 4 (2019)
- Páginas: 158-193
- Seção: Articles
- URL: https://ogarev-online.ru/1607-0046/article/view/133797
- DOI: https://doi.org/10.4213/im8819
- ID: 133797
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Resumo
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].
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Sobre autores
Ivan 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
Bibliografia
- I. Panin, Proof of Grothendieck–Serre conjecture on principal $G$-bundles over semi-local regular domains containing a finite field, 2017
- F. Morel, $mathbb A1$-algebraic topology over a field, Lecture Notes in Math., 2052, Springer, Heidelberg, 2012, x+259 pp.
- I. Panin, A. Stavrova, N. Vavilov, “On Grothendieck–Serre's conjecture concerning principal $G$-bundles over reductive group schemes: I”, Compos. Math., 151:3 (2015), 535–567
- V. Voevodsky, “Cohomological theory of presheaves with transfers”, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., 143, Princeton Univ. Press, Princeton, NJ, 2000, 87–137
- I. Panin, “Oriented cohomology theories of algebraic varieties. II (After I. Panin and A. Smirnov)”, Homology Homotopy Appl., 11:1 (2009), 349–405
- F. Morel, V. Voevodsky, “$mathbf A^1$-homotopy theory of schemes”, Inst. Hautes Etudes Sci. Publ. Math., 90 (1999), 45–143
- M. Ojanguren, I. Panin, “Rationally trivial hermitian spaces are locally trivial”, Math. Z., 237:1 (2001), 181–198
- M. Artin, “Comparaison avec la cohomologie classique: cas d'un preschema lisse”, Theorie des topos et cohomologie etale des schemas, Seminaire de geometrie algebrique du Bois-Marie 1963–1964 (SGA 4), v. 3, Lecture Notes in Math., 305, Exp. XI, Springer-Verlag, Berlin–New York, 1973, 64–78
- D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Grad. Texts in Math., 150, Springer-Verlag, New York, 1995, xvi+785 pp.
- M. Ojanguren, I. Panin, “A purity theorem for the Witt group”, Ann. Sci. Ecole Norm. Sup. (4), 32:1 (1999), 71–86
- J.-L. Colliot-Thelène, M. Ojanguren, “Espaces principaux homogènes localement triviaux”, Inst. Hautes Etudes Sci. Publ. Math., 75 (1992), 97–122
- B. Poonen, “Bertini theorems over finite fields”, Ann. of Math. (2), 160:3 (2004), 1099–1127
- F. Charles, B. Poonen, “Bertini irreducibility theorems over finite fields”, J. Amer. Math. Soc., 29:1 (2016), 81–94
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