Group Scheme - Examples

Examples

• The multiplicative group G_m has the punctured affine line as its underlying scheme, and as a functor, it sends an S-scheme T to the multiplicative group of invertible global sections of the structure sheaf. It can be described as the diagonalizable group D(Z) associated to the integers. Over an affine base such as Spec A, it is the spectrum of the ring A/(xy − 1), which is also written A. The unit map is given by sending x to one, multiplication is given by sending x to x ⊗ x, and the inverse is given by sending x to x−1. Algebraic tori form an important class of commutative group schemes, defined either by the property of being locally on S a product of copies of G_m, or as groups of multiplicative type associated to finitely generated free abelian groups.

• The general linear group GL_n is an affine algebraic variety that can be viewed as the multiplicative group of the n by n matrix ring variety. As a functor, it sends an S-scheme T to the group of invertible n by n matrices whose entries are global sections of T. Over an affine base, one can construct it as a quotient of a polynomial ring in n2 + 1 variables by an ideal encoding the invertibility of the determinant. Alternatively, it can be constructed using 2n2 variables, with relations describing an ordered pair of mutually inverse matrices.

• For any positive integer n, the group μ_n is the kernel of the nth power map from G_m to itself. As a functor, it sends any S-scheme T to the group of global sections f of T such that fn = 1. Over an affine base such as Spec A, it is the spectrum of A/(xn−1). If n is not invertible in the base, then this scheme is not smooth. In particular, over a field of characteristic p, μ_p is not smooth.

• The additive group G_a has the affine line A1 as its underlying scheme. As a functor, it sends any S-scheme T to the underlying additive group of global sections of the structure sheaf. Over an affine base such as Spec A, it is the spectrum of the polynomial ring A. The unit map is given by sending x to zero, the multiplication is given by sending x to 1 ⊗ x + x ⊗ 1, and the inverse is given by sending x to −x.

• If p = 0 in S for some prime number p, then the taking of pth powers induces an endomorphism of G_a, and the kernel is the group scheme α_p. As a scheme, it is isomorphic to μ_p, but the group structures are different. Over an affine base such as Spec A, it is the spectrum of A/(xp).

• The automorphism group of the affine line is isomorphic to the semidirect product of G_a by G_m, where the additive group acts by translations, and the multiplicative group acts by dilations. The subgroup fixing a chosen basepoint is isomorphic to the multiplicative group, and taking the basepoint to be the identity of an additive group structure identifies G_m with the automorphism group of G_a.

• A smooth genus one curve with a marked point (i.e., an elliptic curve) has a unique group scheme structure with that point as the identity. Unlike the previous positive-dimensional examples, elliptic curves are projective (in particular proper).