Use this tag for scheme-theoretic and category-theoretic questions about group schemes, as well as those group schemes that are not algebraic groups. A group scheme G over a scheme S is simply a group object in the category of schemes over S. Finite type group schemes over a field are represented by varieties, and considered algebraic groups; for questions specific to algebraic groups use the [algebraic-groups] tag
A group scheme $G$ over a scheme $S$ is simply a group object in the category of schemes over $S$.
In other words, a group scheme consists of a scheme $G$ over $S$ and $3$ distinguished morphisms of $S$-schemes:
- Multiplication $m \colon G \times G \to G$
- Inverse $i \colon G \to G$
- Identity $e \colon S \to G$
such that for all schemes $T$ over $S$, the morphisms $m$, $i$, and $e$ satisfy the axioms of a group structure on $G(T) = \mathrm{Hom}_S(T, G)$.
By transitivity of the group's action on itself, it must be either smooth or non-reduced everywhere. Thus all reduced group schemes $G$ of finite type over a field $k$ are smooth varieties, making them algebraic groups. Algebraic groups include linear algebraic groups, such as reductive groups and parabolic groups, as well as abelian varieties. Non-algebraic group schemes include non-smooth groups, such as the kernel of the Frobenius $F: \mathbb{G}_a(\mathbb{F}_p) \to \mathbb{G}_a(\mathbb{F}_p)$, as well as infinite-dimensional group schemes such as the positive loop group $L^+G$ of an affine group scheme. The category of affine group schemes is the opposite category of Hopf algebras.