nLab affine group scheme

Distinguish from the notion of a group of affine transformations (of an affine space) or the (general) affine group.

Definition

An affine group scheme is a group object (hence a group scheme) internal to the category of affine schemes.

Related notions

If $k$ is a commutative ring, the category of affine group $k$-schemes is opposite to the category of commutative Hopf $k$-algebras.

By an affine algebraic group, one additionally assumes that the underlying scheme is over an algebraic variety (in particular reduced) and over a field. The narrower notion then agrees with the notion of a linear algebraic group.

Waterhouse12.1 The Lie algebra of the affine group $k$-scheme $X$ is the Lie algebra of left-invariant $k$-linear derivations $T:\mathcal{O}(X)\to\mathcal{O}(X)$ of the Hopf algebra of regular functions of $X$, which are also morphisms of right comodules: $\Delta\circ T = (id\otimes T)\circ\Delta:\mathcal{O}(X)\to\mathcal{O}(X)\otimes\mathcal{O}(X)$ (one says also that $T$ is left-invariant).

This Lie algebra is also isomorphic to the Lie algebra of $k$-linear $k$-valued derivations of $\mathcal{O}(X)$.

Examples

• The additive group scheme (viewed as a group valued functor on the opposite of the category of affine schemes) assigns to a commutative ring its additive group.

Literature

Comprehensive textbooks include

• James S. Milne, Basic theory of affine group schemes, pdf

• William C. Waterhouse, Introduction to affine group schemes, Graduate texts in mathematics 66, 1979

category: algebra, algebraic geometry