Let U be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar Conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck'sétale fundamental group πé t 1 (U ). In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori's profinite fundamental group scheme π N (U ), and give a partial answer to it.
In this paper, we study a certain extension of Nori's fundamental group in the case where a base field is of characteristic 0 and give structure theorems about it. As a result for a smooth projective curve with genus g > 1, we prove that Nori's fundamental group acts faithfully on the category of unipotent bundles on the universal covering. In the case when g = 1, we give a more finer result.
In his previous paper, the author proposed as a problem a purely inseparable analogue of the Abhyankar conjecture for affine curves in positive characteristic and gave a partial answer to it, which includes a complete answer for finite local nilpotent group schemes. In the present paper, motivated by the Abhyankar conjectures with restricted ramifications due to Harbater and Pop, we study a refined version of the analogous problem, based on a recent work on tamely ramified torsors due to Biswas–Borne, which is formulated in terms of root stacks. We study an embedding problem to conclude that the refined analogue is true in the solvable case.
Let X be a smooth proper variety over a field k and suppose that the degree map
${\mathrm {CH}}_0(X \otimes _k K) \to \mathbb {Z}$
is isomorphic for any field extension
$K/k$
. We show that
$G(\operatorname {Spec} k) \to G(X)$
is an isomorphism for any
$\mathbb {P}^1$
-invariant Nisnevich sheaf with transfers G. This generalises a result of Binda, Rülling and Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge–Witt cohomology is a
$\mathbb {P}^1$
-invariant Nisnevich sheaf with transfers.
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