The Nori fundamental gerbe of a fibered category

Abstract

We extend Nori's theory of the fundamental group scheme to a theory of the fundamental gerbe, which applies to schemes, algebraic stacks, and more general fibered categories, even in absence of a rational point. We give a tannakian interpretation of the fundamental gerbe in terms of essentially finite bundles, extending Nori's correspondence for complete varieties with a rational point. We also show how our formalism allows a natural formulation of Grothendieck's section conjecture in arbitrary characteristic. † Supported in part by the Labex CEMPI (ANR-11-LABX-0007-01) and Anr ARIVAF (ANR-10-JCJC 0107). ‡ Supported in part by the PRIN project "Geometria delle varietà algebriche e dei loro spazi di moduli" from MIUR.

Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors

“…Remark 1.2. By [BV,p. 13,Theorem 5.7] X admits a Nori fundamental gerbe if and only if it is inflexible; in this case, the Nori étale fundamental gerbe of X is the maximal proétale quotient of the Nori fundamental gerbe of X .…”

“…Uniqueness of the extension follows from the fact that VectpΓq ÝÑ VectpX q is fully faithful. As u is Nori reduced we have u˚O X » O Γ (see Lemma 1.22 or [BV,Lemma 7.11,pp. 22]).…”

“…Indeed, let f : Y ÝÑ X be a torsor under a finite group scheme G over k corresponding to u : X ÝÑ B G. Then u˚pkrGsq » f˚O Y , where krGs is the regular representation. Applying [BV,Lemma 7.15] to Spec k ÝÑ B G we see that krGs is finite in VectpB Gq " Rep G and thus f˚O Y is a finite vector bundle.…”

“…Let X be a pseudo-proper algebraic stack of finite type over k. If X is inflexible then H 0 pO X q " k (see [BV,Lemma 7.4]), while the converse holds if X is reduced (see Remark 1.2). Theorem 1.10 ( [BV,p. 22,Theorem 7.9,Corollary 7.10]).…”

“…This is a direct consequence of Theorem 1.10 and the universal property of Π N X . Moreover φ˚O X » O Γ (see [BV,Lemma 7.11]). One of the key ingredients in the paper is the following result.…”