The quasi-finite fundamental group scheme conjecture for affine space

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Let RR be the base ring, let KK be its fraction field, and let ARn\mathbb{A}^n_R be nn-dimensional affine space with origin 00. Write piqf(ARn,0)pi^{\text{qf}}(\mathbb{A}^n_R,0) for the quasi-finite fundamental group scheme of ARn\mathbb{A}^n_R at the origin, and let pi(AKn,0)pi(\mathbb{A}^n_K,0) denote the fundamental group scheme of the generic fibre at the origin. Quasi-finite fundamental group scheme conjecture. The faithfully flat morphism

π(AKn,0)→πqf(ARn,0)×RK\pi(\mathbb{A}^n_K, 0)\to \pi^{\text{qf}}(\mathbb{A}^n_R, 0)\times_R K

is an isomorphism. The conjecture proposes that the quasi-finite fundamental group scheme over the affine space extends without acquiring additional structure after passage to the generic fibre; the source states that it is not proved, while noting that in characteristic 00 the statement is empty.

References

Primary source

Marco Antei and Jorge A. Esquivel A, “Models of torsors over affine spaces”, arXiv:1809.06236 (2018).

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