Gabber’s presentation lemma for finite fields
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Amit Hogadi
Abstract
We give a proof of Gabber’s presentation lemma for finite fields. We first prove this lemma in the special case of open subsets of the affine plane using ideas from Poonen’s proof of Bertini’s theorem over finite fields. We then reduce the case of general smooth varieties to this special case.
Acknowledgements
We thank F. Morel for his comments and for answering our questions on the current status of this result. We thank A. Asok, F. Déglise, M. Levine and J. Riou for their comments during the early stage of this project. We thank Anand Sawant and Charanya Ravi for pointing out a mistake in the previous version of the paper. We also thank the referee for numerous suggestions.
References
[1] J. L. Colliot-Thélène, R. Hoobler and B. Kahn, The Bloch–Ogus–Gabber theorem, Algebraic K-theory (Toronto 1996), Fields Inst. Commun. 16, American Mathematical Society, Providence (1997), 31–94. 10.1090/fic/016/02Search in Google Scholar
[2] O. Gabber, Gersten’s conjecture for some complexes of vanishing cycles, Manuscripta Math. 85 (1994), no. 3–4, 323–343. 10.1007/BF02568202Search in Google Scholar
[3] S. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827. 10.2307/2372655Search in Google Scholar
[4]
F. Morel,
[5] D. Mumford, The red book of varieties and schemes. Includes the Michigan lectures (1974) on curves and their Jacobians, Lecture Notes in Math. 1358, Springer, Berlin 1999. 10.1007/978-3-540-46021-3_4Search in Google Scholar
[6] B. Poonen, Bertini theorems over finite fields, Ann. of Math. (2) 160 (2004), no. 3, 1099–1127. 10.4007/annals.2004.160.1099Search in Google Scholar
[7] The Stacks Project Authors, Stack project, http://stacks.math.columbia.edu. Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Kähler–Einstein metrics: From cones to cusps
- Smoothness of definite unitary eigenvarieties at critical points
- Construction and classification of holomorphic vertex operator algebras
- A Frobenius–Nirenberg theorem with parameter
- On the homotopy classification of proper Fredholm maps into a Hilbert space
- Finite quasi-quantum groups of diagonal type
- Curvature estimates for stable free boundary minimal hypersurfaces
- Gabber’s presentation lemma for finite fields
- A new proof of the Tikuisis–White–Winter theorem
Articles in the same Issue
- Frontmatter
- Kähler–Einstein metrics: From cones to cusps
- Smoothness of definite unitary eigenvarieties at critical points
- Construction and classification of holomorphic vertex operator algebras
- A Frobenius–Nirenberg theorem with parameter
- On the homotopy classification of proper Fredholm maps into a Hilbert space
- Finite quasi-quantum groups of diagonal type
- Curvature estimates for stable free boundary minimal hypersurfaces
- Gabber’s presentation lemma for finite fields
- A new proof of the Tikuisis–White–Winter theorem
