Algebraic and Nori Fundamental Gerbes

Abstract

In this paper we extend the generalized algebraic fundamental group constructed in [EH] to general fibered categories using the language of gerbes. As an application we obtain a Tannakian interpretation for the Nori fundamental gerbe defined in [BV] for non smooth non pseudo-proper algebraic stacks.

An embedding problem for finite local torsors over twisted curves

“…Next we introduce two variants of the Nori fundamental gerbe, namely, the étale and local fundamental gerbes (cf. [9, §8], [37, §4, Definition 4.1]). Definition (Cf.…”

“…In §2, we recall the notion of fundamental group schemes and fundamental gerbes , and their tannakian interpretations, following Borne–Vistoli [9] and Tonini–Zhang [37]. In [6], Biswas–Borne studied the fundamental gerbes of proper tame stacks, e.g.…”

“…If is a fibered category of groupoids over a field k of characteristic , we define the absolute Frobenius morphism as the collection of maps for (cf. [37, Notations and conventions]). If is a scheme over k , then this coincides with the one in the previous sense.…”

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

“…For properties of affine gerbes over a field (often improperly called just gerbes) and Tannakian categories used here the reader is referred to [TZ1,Appendix B].…”

“…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 . If X is reduced, quasi-compact and quasi-separated, then X is inflexible if and only if k is algebraically closed in H 0 pO X q [TZ1,Theorem 4.4]. In particular if X is geometrically connected and geometrically reduced, then it is inflexible.…”