Group scheme

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v t e

In

• a triple of
  morphisms
  μ: G ×_S G → G, e: S → G, and ι: G → G, satisfying the usual compatibilities of groups (namely associativity of μ, identity, and inverse axioms)
• a functor from schemes over S to the category of groups, such that composition with the forgetful functor to sets is equivalent to the presheaf corresponding to G under the Yoneda embedding. (See also: group functor.)

A homomorphism of group schemes is a map of schemes that respects multiplication. This can be precisely phrased either by saying that a map f satisfies the equation fμ = μ(f × f), or by saying that f is a natural transformation of functors from schemes to groups (rather than just sets).

A

• Given a group G, one can form the constant group scheme G_S. As a scheme, it is a disjoint union of copies of S, and by choosing an identification of these copies with elements of G, one can define the multiplication, unit, and inverse maps by
  projective limit of finite constant group schemes to get profinite group schemes, which appear in the study of fundamental groups and Galois representations or in the theory of the fundamental group scheme, and these are affine of infinite type. More generally, by taking a locally constant sheaf of groups on S, one obtains a locally constant group scheme, for which monodromy
  on the base can induce non-trivial automorphisms on the fibers.
• The existence of fiber products of schemes allows one to make several constructions. Finite direct products of group schemes have a canonical group scheme structure. Given an action of one group scheme on another by automorphisms, one can form semidirect products by following the usual set-theoretic construction. Kernels of group scheme homomorphisms are group schemes, by taking a fiber product over the unit map from the base. Base change sends group schemes to group schemes.
• Group schemes can be formed from smaller group schemes by taking
  restriction of scalars with respect to some morphism of base schemes, although one needs finiteness conditions to be satisfied to ensure representability of the resulting functor. When this morphism is along a finite extension of fields, it is known as Weil restriction
  .
• For any abelian group A, one can form the corresponding diagonalizable group D(A), defined as a functor by setting D(A)(T) to be the set of abelian group homomorphisms from A to invertible global sections of O_T for each S-scheme T. If S is affine, D(A) can be formed as the spectrum of a group ring. More generally, one can form groups of multiplicative type by letting A be a non-constant sheaf of abelian groups on S.
• For a subgroup scheme H of a group scheme G, the functor that takes an S-scheme T to G(T)/H(T) is in general not a sheaf, and even its sheafification is in general not representable as a scheme. However, if H is finite, flat, and closed in G, then the quotient is representable, and admits a canonical left G-action by translation. If the restriction of this action to H is trivial, then H is said to be normal, and the quotient scheme admits a natural group law. Representability holds in many other cases, such as when H is closed in G and both are affine.^[1]

• 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[x,y]/(xy − 1), which is also written A[x, x⁻¹]. 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[x]/(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[x]. 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. Over an affine base such as Spec A, it is the spectrum of A[x]/(x^p).
• 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).

Suppose that G is a group scheme of finite type over a field k. Let G⁰ be the connected component of the identity, i.e., the maximal connected subgroup scheme. Then G is an extension of a finite étale group scheme by G⁰. G has a unique maximal reduced subscheme G_red, and if k is perfect, then G_red is a smooth group variety that is a subgroup scheme of G. The quotient scheme is the spectrum of a local ring of finite rank.

Any affine group scheme is the spectrum of a commutative Hopf algebra (over a base S, this is given by the relative spectrum of an O_S-algebra). The multiplication, unit, and inverse maps of the group scheme are given by the comultiplication, counit, and antipode structures in the Hopf algebra. The unit and multiplication structures in the Hopf algebra are intrinsic to the underlying scheme. For an arbitrary group scheme G, the ring of global sections also has a commutative Hopf algebra structure, and by taking its spectrum, one obtains the maximal affine quotient group. Affine group varieties are known as linear algebraic groups, since they can be embedded as subgroups of general linear groups.

Complete connected group schemes are in some sense opposite to affine group schemes, since the completeness implies all global sections are exactly those pulled back from the base, and in particular, they have no nontrivial maps to affine schemes. Any

Among the finite flat group schemes, the constants (cf. example above) form a special class, and over an algebraically closed field of characteristic zero, the category of finite groups is equivalent to the category of constant finite group schemes. Over bases with positive characteristic or more arithmetic structure, additional isomorphism types exist. For example, if 2 is invertible over the base, all group schemes of order 2 are constant, but over the 2-adic integers, μ₂ is non-constant, because the special fiber isn't smooth. There exist sequences of highly ramified 2-adic rings over which the number of isomorphism types of group schemes of order 2 grows arbitrarily large. More detailed analysis of commutative finite flat group schemes over p-adic rings can be found in Raynaud's work on prolongations.

Commutative finite flat group schemes often occur in nature as subgroup schemes of abelian and semi-abelian varieties, and in positive or mixed characteristic, they can capture a lot of information about the ambient variety. For example, the p-torsion of an elliptic curve in characteristic zero is locally isomorphic to the constant elementary abelian group scheme of order p², but over F_p, it is a finite flat group scheme of order p² that has either p connected components (if the curve is ordinary) or one connected component (if the curve is

Cartier duality is a scheme-theoretic analogue of Pontryagin duality taking finite commutative group schemes to finite commutative group schemes.

Finite flat commutative group schemes over a perfect field k of positive characteristic p can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the

• Fundamental group scheme
• Geometric invariant theory
• GIT quotient
• Groupoid scheme
• Group-scheme action
• Group-stack
• Invariant theory
• Quotient stack