Group scheme
Encyclopedia
In mathematics
, a group scheme is a type of algebro-geometric
object equipped with a composition law. Group schemes arise naturally as symmetries of scheme
s, and they generalize algebraic group
s, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The category
of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have kernel
s, and there is a well-behaved deformation theory
. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The initial development of the theory of group schemes was due to Alexandre Grothendieck, Michel Raynaud
, and Michel Demazure
in the early 1960s.
in a category of schemes that has fiber products and some final object S. That is, it is an S-scheme G equipped with one of the equivalent sets of data
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 left action of a group scheme G on a scheme X is a morphism G ×S X→ X that induces a left action
of the group G(T) on the set X(T) for any S-scheme T. Right actions are defined similarly. Any group scheme admits natural left and right actions on its underlying scheme by multiplication and conjugation
. Conjugation is an action by automorphisms, i.e., it commutes with the group structure, and this induces linear actions on naturally derived objects, such as its Lie algebra
, and the algebra of left-invariant differential operators.
An S-group scheme G is commutative if the group G(T) is an abelian group for all S-schemes T. There are several other equivalent conditions, such as conjugation inducing a trivial action, or inversion map ι being a group scheme automorphism.
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 OS-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 complete group variety (variety here meaning reduced and geometrically irreducible separated scheme of finite type over a field) is automatically commutative, by an argument involving the action of conjugation on jet spaces of the identity. Complete group varieties are called abelian varieties
. This generalizes to the notion of abelian scheme; a group scheme G over a base S is abelian if the structural morphism from G to S is proper and smooth with geometrically connected fibers They are automatically projective, and they have many applications, e.g., in geometric class field theory
and throughout algebraic geometry. A complete group scheme over a field need not be commutative, however; for example, any finite group scheme is complete.
.
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 Fp, 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 supersingular). If we consider a family of elliptic curves, the p-torsion forms a finite flat group scheme over the parametrizing space, and the supersingular locus is where the fibers are connected. This merging of connected components can be studied in fine detail by passing from a modular scheme to a rigid analytic space
, where supersingular points are replaced by discs of positive radius.
. Given any finite flat commutative group scheme G over S, its Cartier dual is the group of characters, defined as the functor that takes any S-scheme T to the abelian group of group scheme homomorphisms from the base change GT to Gm,T and any map of S-schemes to the canonical map of character groups. This functor is representable by a finite flat S-group scheme, and Cartier duality forms an additive involutive antiequivalence from the category of finite flat commutative S-group schemes to itself. If G is a constant commutative group scheme, then its Cartier dual is the diagonalizable group D(G), and vice versa. If S is affine, then the duality functor is given by the duality of the Hopf algebras of functions.
The definition of Cartier dual extends usefully to much more general situations where the resulting functor on schemes is no longer represented as a group scheme. Common cases include fppf sheaves of commutative groups over S, and complexes thereof. These more general geometric objects can be useful when one wants to work with categories that have good limit behavior. There are cases of intermediate abstraction, such as commutative algebraic groups over a field, where Cartier duality gives an antiequivalence with commutative affine formal group
s, so if G is the additive group Ga, then its Cartier dual is the multiplicative formal group
, and if G is a torus, then its Cartier dual is étale and torsion-free. For loop groups of tori, Cartier duality defines the tame symbol in local geometric class field theory. Laumon introduced a sheaf-theoretic Fourier transform for quasi-coherent modules over 1-motives that specializes to many of these equivalences.
Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Oda's 1967 thesis gave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze p-divisible groups. Galois actions on the group schemes transfer through the equivalences of categories, and the associated deformation theory of Galois representations was used in Wiles
's work on the Shimura-Taniyama conjecture.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a group scheme is a type of algebro-geometric
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
object equipped with a composition law. Group schemes arise naturally as symmetries of scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s, and they generalize algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have kernel
Kernel (category theory)
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra...
s, and there is a well-behaved deformation theory
Deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach...
. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The initial development of the theory of group schemes was due to Alexandre Grothendieck, Michel Raynaud
Michel Raynaud
Michel Raynaud is a French mathematician working in algebraic geometry. Since 1967 he has been a professor at Paris-Sud 11 University.In 1983 he published a proof of the Manin-Mumford conjecture....
, and Michel Demazure
Michel Demazure
Michel Demazure is a French mathematician. He made contributions in the fields of abstract algebra and algebraic geometry, was president of the French Mathematical Society and directed two French science museums.-Biography:...
in the early 1960s.
Definition
A group scheme is a group objectGroup object
In category theory, a branch of mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets...
in a category of schemes that has fiber products and some final object S. That is, it is an S-scheme G equipped with one of the equivalent sets of data
- 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 groupsCategory of groupsIn mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
, such that composition with the forgetful functor to sets is equivalent to the presheaf corresponding to G under the Yoneda embeddingYoneda lemmaIn mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
.
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
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
of functors from schemes to groups (rather than just sets).
A left action of a group scheme G on a scheme X is a morphism G ×S X→ X that induces a left action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of the group G(T) on the set X(T) for any S-scheme T. Right actions are defined similarly. Any group scheme admits natural left and right actions on its underlying scheme by multiplication and conjugation
Inner automorphism
In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
. Conjugation is an action by automorphisms, i.e., it commutes with the group structure, and this induces linear actions on naturally derived objects, such as its Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
, and the algebra of left-invariant differential operators.
An S-group scheme G is commutative if the group G(T) is an abelian group for all S-schemes T. There are several other equivalent conditions, such as conjugation inducing a trivial action, or inversion map ι being a group scheme automorphism.
Constructions
- Given a group G, one can form the constant group scheme GS. 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 transport of structure. As a functor, it takes any S-scheme T to a product of copies of the group G, where the number of copies is equal to the number of connected components of T. GS is affine over S if and only if G is a finite group. However, one can take a 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 schemeFundamental group schemeIn mathematics, the fundamental group scheme is a group scheme canonically associated to a scheme over a Dedekind scheme . It is a generalisation of the étale fundamental group...
, 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 monodromyMonodromyIn mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
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 scalarsRestriction of scalarsIn abstract algebra, restriction of scalars is a procedure of creating a module over a ring R from a module over another ring S, given a homomorphism f : R \to S between them...
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 restrictionWeil restrictionIn mathematics, restriction of scalars is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k...
.
- For any abelian group A, one can form the corresponding diagonalizable groupDiagonalizable groupIn mathematics, an affine group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, the group of diagonal matrices. A diagonalizable group defined over k is said to split over k or k-split if the isomorphism is defined over k. This coincides with the usual notion of split for an...
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 OT 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.
Examples
- The multiplicative group Gm 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 toriAlgebraic torusIn mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory...
form an important class of commutative group schemes, defined either by the property of being locally on S a product of copies of Gm, or as groups of multiplicative type associated to finitely generated free abelian groups.
- The general linear group GLn 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 μn is the kernel of the nth power map from Gm 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, μp is not smooth.
- The additive group Ga 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 Ga, and the kernel is the group scheme αp. As a scheme, it is isomorphic to μp, but the group structures are different. Over an affine base such as Spec A, it is the spectrum of A[x]/(xp).
- The automorphism group of the affine line is isomorphic to the semidirect product of Ga by Gm, 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 Gm with the automorphism group of Ga.
- A smooth genus one curve with a marked point (i.e., an elliptic curveElliptic curveIn mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
) 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).
Basic properties
When working over a field, one often can analyze a group scheme by treating it as an extension of group schemes with distinguished properties. Any group scheme G of finite type is an extension of the connected component of the identity (i.e., the maximal connected subgroup scheme) by a constant group scheme. If G is connected, then it has a unique maximal reduced subscheme Gred, which is a smooth group variety that is a normal subgroup of G. The quotient group Ginf is the infinitesimal quotient, and is the spectrum of a local Hopf algebra of finite rank.Any affine group scheme is the spectrum
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
of a commutative Hopf algebra
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
(over a base S, this is given by the relative spectrum of an OS-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 complete group variety (variety here meaning reduced and geometrically irreducible separated scheme of finite type over a field) is automatically commutative, by an argument involving the action of conjugation on jet spaces of the identity. Complete group varieties are called abelian varieties
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
. This generalizes to the notion of abelian scheme; a group scheme G over a base S is abelian if the structural morphism from G to S is proper and smooth with geometrically connected fibers They are automatically projective, and they have many applications, e.g., in geometric class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
and throughout algebraic geometry. A complete group scheme over a field need not be commutative, however; for example, any finite group scheme is complete.
Finite flat group schemes
A group scheme G over a noetherian scheme S is finite and flat if and only if OG is a locally free OS-module of finite rank. The rank is a locally constant function on S, and is called the order of G. The order of a constant group scheme is equal to the order of the corresponding group, and in general, order behaves well with respect to base change and finite flat restriction of scalarsRestriction of scalars
In abstract algebra, restriction of scalars is a procedure of creating a module over a ring R from a module over another ring S, given a homomorphism f : R \to S between them...
.
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 Fp, 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 supersingular). If we consider a family of elliptic curves, the p-torsion forms a finite flat group scheme over the parametrizing space, and the supersingular locus is where the fibers are connected. This merging of connected components can be studied in fine detail by passing from a modular scheme to a rigid analytic space
Rigid analytic space
In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. They were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing p-adic elliptic curves with bad reduction using the multiplicative group...
, where supersingular points are replaced by discs of positive radius.
Cartier duality
Cartier duality is a scheme-theoretic analogue of Pontryagin dualityPontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...
. Given any finite flat commutative group scheme G over S, its Cartier dual is the group of characters, defined as the functor that takes any S-scheme T to the abelian group of group scheme homomorphisms from the base change GT to Gm,T and any map of S-schemes to the canonical map of character groups. This functor is representable by a finite flat S-group scheme, and Cartier duality forms an additive involutive antiequivalence from the category of finite flat commutative S-group schemes to itself. If G is a constant commutative group scheme, then its Cartier dual is the diagonalizable group D(G), and vice versa. If S is affine, then the duality functor is given by the duality of the Hopf algebras of functions.
The definition of Cartier dual extends usefully to much more general situations where the resulting functor on schemes is no longer represented as a group scheme. Common cases include fppf sheaves of commutative groups over S, and complexes thereof. These more general geometric objects can be useful when one wants to work with categories that have good limit behavior. There are cases of intermediate abstraction, such as commutative algebraic groups over a field, where Cartier duality gives an antiequivalence with commutative affine formal group
Formal group
In mathematics, a formal group law is a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between...
s, so if G is the additive group Ga, then its Cartier dual is the multiplicative formal group
, and if G is a torus, then its Cartier dual is étale and torsion-free. For loop groups of tori, Cartier duality defines the tame symbol in local geometric class field theory. Laumon introduced a sheaf-theoretic Fourier transform for quasi-coherent modules over 1-motives that specializes to many of these equivalences.Dieudonné modules
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 Dieudonné ring D = W(k){F,V}/(FV − p), which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vectors of k. F and V are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors. Dieudonne and Cartier constructed an antiequivalence of categories between finite commutative group schemes over k of order a power of "p" and modules over D with finite W(k)-length. The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf CW of Witt co-vectors. This sheaf is more or less dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by taking a direct limit of finite length Witt vectors under successive Verschiebung maps V: Wn → Wn+1, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected p-group schemes correspond to D-modules for which F is nilpotent, and étale group schemes correspond to modules for which F is an isomorphism.Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Oda's 1967 thesis gave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze p-divisible groups. Galois actions on the group schemes transfer through the equivalences of categories, and the associated deformation theory of Galois representations was used in Wiles
Andrew Wiles
Sir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory...
's work on the Shimura-Taniyama conjecture.