The quasi-finite fundamental group scheme conjecture for affine space

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.

Sources & referencesView supporting material

Primary source

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

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