Notes On Fundamental Groupoid Schemes
Pavan Adroja, Sanjay Amrutiya
TL;DR
The work develops a groupoid-augmented Tannakian approach to fundamental group schemes, generalizing Nori-type constructions to groupoid schemes and computing explicit objects for real forms such as anisotropic conics, Klein bottles, and abelian varieties. It leverages Deligne's Tannakian duality to realize categories of bundles as representations of affine (groupoid) schemes, enabling concrete identifications like $\Pi^{\rm EN}(A) \simeq \Pi^{\rm uni}(A)\times_k \Pi^{\rm N}(A)$ for abelian varieties and explicit basepoint analyses yielding $\Pi^{\rm S}(C,\tau) \simeq C\times_{\mathbb{R}} C$ in the anisotropic conic case. The paper then clarifies how morphisms of groupoids are reflected in their representation categories, showing that faithful flatness corresponds to representations factoring through the morphism while closed immersions are witnessed by the presence of all objects as subquotients of pullback representations. Overall, the results provide a unified, representation-theoretic framework for fundamental groupoid schemes in contexts lacking $k$-rational points, with explicit computations on classical real forms and a clear decomposition phenomenon for abelian varieties.
Abstract
In this article, we study the various fundamental groupoid schemes corresponding to Tannakian categories of certain types of vector bundles. We compute fundamental groupoid scheme of anisotropic conic, Klein bottle and abelian varieties. Additionally, we study the relation among various fundamental groupoid schemes by considering their representations.