The arithmetic local Nori fundamental group

Romagny, Matthieu; Tonini, Fabio; Zhang, Lei

doi:10.1090/tran/8504

The arithmetic local Nori fundamental group
HTML articles powered by AMS MathViewer

by Matthieu Romagny, Fabio Tonini and Lei Zhang PDF

Trans. Amer. Math. Soc. 374 (2021), 8869-8885 Request permission

Abstract:

In this paper we introduce the local Nori fundamental group scheme of a reduced scheme or algebraic stack over a perfect field $k$. We give particular attention to the case of fields: to any field extension $K/k$ we attach a pro-local group scheme over $k$. We show how this group has many analogies, but also some crucial differences, with the absolute Galois group. We propose two conjectures, analogous to the classical Neukirch-Uchida Theorem and Abhyankar Conjecture, providing some evidence in their favor. Finally we show that the local fundamental group of a normal variety is a quotient of the local fundamental group of an open, of its generic point (as it happens for the étale fundamental group) and even of any smooth neighborhood.

References

Giulia Battiston, A theory of Galois descent for finite inseparable extensions, Proc. Amer. Math. Soc. 146 (2018), no. 1, 69–83. MR 3723121, DOI 10.1090/proc/13713
Niels Borne and Angelo Vistoli, The Nori fundamental gerbe of a fibered category, J. Algebraic Geom. 24 (2015), no. 2, 311–353. MR 3311586, DOI 10.1090/S1056-3911-2014-00638-X
Lukas Brantner and Joe Waldron, Purely inseparable galois theory I: the fundamental theorem, (2020) available at arXiv:2010.15707v2.
Stephen U. Chase, On inseparable Galois theory, Bull. Amer. Math. Soc. 77 (1971), 413–417. MR 277504, DOI 10.1090/S0002-9904-1971-12721-0
Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325
Murray Gerstenhaber, On the Galois theory of inseparable extensions, Bull. Amer. Math. Soc. 70 (1964), 561–566. MR 162794, DOI 10.1090/S0002-9904-1964-11201-5
Murray Gerstenhaber and Avigdor Zaromp, On the Galois theory of purely inseparable field extensions, Bull. Amer. Math. Soc. 76 (1970), 1011–1014. MR 266904, DOI 10.1090/S0002-9904-1970-12535-6
David Harbater, Abhyankar’s conjecture on Galois groups over curves, Invent. Math. 117 (1994), no. 1, 1–25. MR 1269423, DOI 10.1007/BF01232232
Nickolas Heerema, A Galois theory for inseparable field extensions, Trans. Amer. Math. Soc. 154 (1971), 193–200. MR 269632, DOI 10.1090/S0002-9947-1971-0269632-4
N. Jacobson, Galois theory of purely inseparable fields of exponent one, Amer. J. Math. 66 (1944), 645–648. MR 11079, DOI 10.2307/2371772
Shinichi Mochizuki, The local pro-$p$ anabelian geometry of curves, Invent. Math. 138 (1999), no. 2, 319–423. MR 1720187, DOI 10.1007/s002220050381
John N. Mordeson, On a Galois theory for inseparable field extensions, Trans. Amer. Math. Soc. 214 (1975), 337–347. MR 384762, DOI 10.1090/S0002-9947-1975-0384762-8
Shusuke Otabe, On a purely inseparable analogue of the Abhyankar conjecture for affine curves, Compos. Math. 154 (2018), no. 8, 1633–1658. MR 3830548, DOI 10.1112/s0010437x18007194
Shusuke Otabe, An embedding problem for finite local torsors over twisted curves Mathematische Nachrichten, vol. 294, Issue 7, 2021, pp. 1384-1427. DOI 10.1002/mana.201900091.
Shusuke Otabe, Fabio Tonini, and Lei Zhang, A generalized Abhyankar’s conjecture for simple Lie algebras in characteristic $p>5$ (2020), available at arXiv:2003.03240. To appear in Mathematische Annalen.
Florian Pop, On Grothendieck’s conjecture of birational anabelian geometry, Ann. of Math. (2) 139 (1994), no. 1, 145–182. MR 1259367, DOI 10.2307/2946630
Florian Pop, The birational anabelian conjecture revisited, preprint, 2002.
The Stacks Project Authors, Stack Project, http://stacks.math.columbia.edu/.
Fabio Tonini and Lei Zhang, Algebraic and Nori fundamental gerbes, J. Inst. Math. Jussieu 18 (2019), no. 4, 855–897. MR 3963521, DOI 10.1017/s147474801700024x
Kôji Uchida, Isomorphisms of Galois groups of algebraic function fields, Ann. of Math. (2) 106 (1977), no. 3, 589–598. MR 460279, DOI 10.2307/1971069
Lei Zhang, Nori’s fundamental group over a non algebraically closed field, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 4, 1349–1394. MR 3847315