Fundamental group scheme

Let ${\displaystyle k}$ be a perfect field and ${\displaystyle X\to {\text{Spec}}(k)}$ a faithfully flat and proper morphism of schemes with ${\displaystyle X}$ a reduced and connected scheme. Assume the existence of a section ${\displaystyle x:{\text{Spec}}(k)\to X}$, then the fundamental group scheme ${\displaystyle \pi _{1}(X,x)}$ of ${\displaystyle X}$ in ${\displaystyle x}$ is defined as the affine group scheme naturally associated to the neutral tannakian category (over ${\displaystyle k}$) of essentially finite vector bundles over ${\displaystyle X}$.

Let ${\displaystyle S}$ be a Dedekind scheme, ${\displaystyle X}$ any connected scheme reduced and ${\displaystyle X\to S}$ a faithfully flat morphism of finite type (not necessarily proper). Assume the existence of a section ${\displaystyle x:S\to X}$. Once we prove that the category of isomorphism classes of torsors over ${\displaystyle X}$ (pointed over ${\displaystyle x}$) under the action of finite and flat ${\displaystyle S}$-group schemes is cofiltered then we define the universal torsor (pointed over ${\displaystyle x}$) as the projective limit of all the torsors of that category. The ${\displaystyle S}$-group scheme acting on it is called the fundamental group scheme and denoted by ${\displaystyle \pi _{1}(X,x)}$ (when ${\displaystyle S}$ is the spectrum of a perfect field the two definitions coincide so that no confusion can arise). The definition has been further generalized to some non reduced schemes.

• Étale fundamental group
• Fundamental group