nLab affine group scheme

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


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

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

By an affine algebraic group, one additionally assumes that the underlying scheme is over an algebraic variety (in particular reducible) 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 kk-scheme XX is the Lie algebra of left-invariant kk-linear derivations T:𝒪(X)𝒪(X)T:\mathcal{O}(X)\to\mathcal{O}(X) of the Hopf algebra of regular functions of XX, which are also morphisms of right comodules: ΔT=(idT)Δ:𝒪(X)𝒪(X)𝒪(X)\Delta\circ T = (id\otimes T)\circ\Delta:\mathcal{O}(X)\to\mathcal{O}(X)\otimes\mathcal{O}(X) (one says also that TT is left-invariant).

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


  • 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.


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

Last revised on August 22, 2023 at 17:09:14. See the history of this page for a list of all contributions to it.