On the group of separable quadratic algebras and stacks

TL;DR

This paper investigates the structure of torsors under finite flat group schemes of rank 2 over rings, generalizing quadratic algebras and providing new insights into their classification and properties.

Contribution

It introduces a new framework for understanding the group of torsors of rank 2 group schemes, extending the theory of quadratic algebras and offering novel results.

Findings

Characterization of torsors under finite flat group schemes of rank 2

Generalization of the group of quadratic algebras

New structural results on isomorphism classes

Abstract

The aim of this paper is to study the group of isomorphism classes of torsors of finite flat group schemes of rank 2 over a commutative ring $R$. This, in particular, generalises the group of quadratic algebras (free or projective), which is especially well studied. Our approach however, yields new results even in this case.