derived smooth geometry
A group scheme is a group object in the category of schemes (or in a category of some schemes as for instance that of schemes over a fixed base scheme); in particular a group scheme is a group functor. As explained at group object there are two equivalent ways of realizing this:
One way is to define it as a functor equipped with a transformation satisfying the properties spelled out at group object.
The other way is to define it as a functor from the category of schemes to that of (discrete) groups whose composition with the forgetful functor is representable.
Grothendieck emphasized the study of schemes over a fixed base scheme. Following this idea in the functor of points formalism, a group scheme over a scheme is a functor
A morphism of group schemes is a morphism of schemes that is a group homomorphism on any choice of values of points. This is more easily stated by saying that a morphism of group schemes must be a natural transformation between the functor of points; i.e. is required to be a natural transformation of functors with values in the category of groups (instead of with values in Set); an equivalent way to state this is that needs to satisfy if denotes the group law on .
Let be some base field. We start with the constant group scheme defined by some classical group which gives in every component just the group . Next we visit the notion of étale group scheme. This is not itself constant but becomes so by scalar extension to the separable closure of . The importance of étale affine is that the category of them is equivalent to that of Galois modules by
So far these examples ”do nothing” with the (multiplicative and additive) structure of the ring in which we evaluate our group scheme. But the next instances do this: We define the additive- and the multiplicative group scheme by and sending a -ring to to its underlying additive- and multiplicative group, respectively. These have the ”function rings” and and since we note that our basic building blocks and are in fact representable -functors aka. affine group schemes. We observe that we have and call in generalization of this property any group scheme satisfying multiplicative group scheme. (We could have also the idea to call satisfying ”additive” but I didn’t see this.) By some computation of the hom spaces involving co- and birings we see that these are again always values of a representable -functor ; this functor we call the Cartier dual of . If for example is a finite group scheme also is, and moreover is a contravariant autoequivalence (”duality”) of ; in general it is also a duality in some specific sense. By taking the Cartier dual of a constant group scheme we obtain the notion of a diagonlizable group scheme. To justify this naming we compute some value where denotes the group algebra of and the last isomorphism is due to the universal property of group rings; we observe that the last equality tells us that and recall that a is called a character of (and one calls a group generated by these ”diagonalizable”). Revisiting the condition by which we defined multiplicative group schemes and considering a group scheme satisfying this condition for all sub group-schemes of we arrive at the notion of unipotent group scheme. By the structure theorem of decomposition of affine groups we can proof that is unipotent iff the completion of group schemes (which gives us-by the usual technic of completion- a formal (group) scheme if is a group scheme) of the Cartier dual of , i.e. is a connected formal group scheme also called local group scheme since a local group scheme is defined to be the spectrum of a local ring; this requirement in turn is equivalent to hence the first name ”connected”. There is also a connection between connected and étale schemes: For any formal group scheme there is an essentially unique exact sequence
where is connected and is étale. Such decomposition in exact sequences we obtain in further cases: where
where a smooth (group) scheme is defined to be the spectrum of a finite dimensional (over k) power series algebra, a (group) scheme is called finite (group) scheme if we restrict in all necessary definitions to -ring which are finite dimensional -vector spaces, and a (group) scheme is called infinitesimal (group) scheme if it is finite and local. If moreover is a perfect field any finite affine -group is in a unique way the product of four subgroups where is a formal étale multiplicative group, is a formal étale unipotent group, is a formal infinitesimal multiplicative group, and is a infinitesimal unipotent group.
If we now shift our focus to colimits- or more generally to codirected systems of finite group schemes, in particular the notion of p-divisible group is an extensively studied case because the -divisible group of a group scheme encodes information on the p-torsion of the group scheme . To appreciate the definition of we first recall that for any group scheme we have the relative Frobenius morphism to distinguish it from the absolute Frobenius morphism which is induced by the Frobenius morphism of the underlying ring . The passage to the relative Frobenius is necessary since in general it is not true that the absolute Frobenius respects the base scheme. Now we define where the kernel is taken of the Frobenius iterated -times and the codirected system
is then called the -divisible group of . As cardinality (in group theory also called rank) of this objects we have for some ; this is called the height of . Moreover we have (p1) the are finite group schemes (we assumed this by definition), (p2) the sequences of the form are exact, (p3) and one can show that if we start with any codirected system satisfying (p1)(p2) we have that satisfies (p3) and - in other words the properties (p1)(p2) give an equivalent alternative definition of -divisible groups (and (p3) leads some authors to ”identify” and ). Basic examples of -divisible groups are which is (up to isomorphism) the unique example of a constant -divisible group of height and where is a commutative variety with a group law (aka. algebraic group). is called the Barsotti-Tate group of an abelian variety; if the dimension of is the height of is . Now, what about decomposition of -divisible groups? We have even one more equivalent ”exactness” characterization of -divisible formal groups by: is -divisible iff in the connected-étale decomposition given by the exact sequence displayed in (1) we have ,(), for some and ,(), is of finite type (= the spectrum of a Noetherian ring), smooth, and the kernel of its Verschiebung morphism (this is the left adjoint the Frobenius morphism) is finite. Of course this characterization of -divisiblity by exact sequences gives rise to propositions on dimensions and subgroups of -divisible groups.
In cases where is a field of prime characteristic , there is some special -functor which is a group functor and even a ring functor (a -functor equipped with a ring structure) - namely the functor whose image is the category of lambda-rings; the objects of are also called Witt vectors since they are infinite sequences of elements of (this justifies at least ”vectors”). possesses a left adjoint forgetting the lambda-structure and the couniversal property? associated to this adjunction states that for a -ring we have that is the couniversal object such that all so called Witt polynomials are ring homomorphisms. For this special -group we revisit some construction we have done above for general -groups: we firstly make the eponymous remark that the Verschiebung morphism is given by shifting (German: Verschiebung) one component to the right. By abstract nonsense we have also Frobenius. An important proposition concerning the ring of Witt vectors is that for a perfect field , is a discrete valuation ring. The next construction we visit with is Cartier duality of finite Witt groups (here we forget that is even a ring): For this note that the ring of finite Witt vectors is an ideal in and we have Frobenius and Verschiebung also in this truncated case; more precisely we have for each a Frobenius where denotes the ring of Witt vectors of length . With this notation we find .
Since is a ring we can ask of its modules in general; however there is in particular one -module of interest which is called the Dieudonné module of . It can be defined in two equivalent ways: 1. as a -module equipped with two endomorphisms of and satisfying the ”Witt-Frobenius identities”
or 2. as a left module over the Dieudonné ring which is the (noncommutative ring) generated by and two variables and satisfying (WF1)(WF2)(WF3) in which case every element of can uniquely be written as a finite sum
For a field the terminal -scheme is a group scheme in a unique way.
Complete group varieties are called abelian varieties.
Given any group , one can form the constant group scheme? over .
etale group scheme? is the spectrum of a commutative Hopf algebra. In this case the multiplication- resp. inversion- reps. unit map are given by comultiplication? reps. antipodism? resp. counit in the Hopf algebra.
The functor is a group scheme given by . A scheme is sent to the invertible elements of its global functions. This group scheme is called the multiplicative group scheme. In context of p-divisible groups the kernels of the -group scheme endomorphisms of defined by for an integer are of particular interest. These kernels give the group schemes of the -th root of unity.
diagonalizable group scheme. Note that the multiplicative group scheme is diagonalizable.
multiplicative group scheme also called group scheme of multiplicative type. Every diagonalizable group scheme is in particular of multiplicative type.
Group schemes can be constructed by restriction of scalars.
The functor is a group scheme given by the additive group of the ring of global functions. This group scheme is called the additive group scheme.
the kernel of any group scheme morphism is a group scheme.
Every algebraic group is in particular a group scheme.
(main article: Cartier duality)
Suppose now that is a finite flat commutative group scheme (over ). The Cartier dual of is given by the functor . The Hom is taken in the category of group schemes over .
For example, .
(main article: Dieudonné module)
The Dieudonné ring of is the ring generated by two objects subject to the relations
(see also Dieudonné module for more details concerning this theorem)
Let be a perfect field of prime characteristic . Since is perfect Frobenius is an automorphism.
where we recall that how the colimit of the hom space can be multiplied by the generators of the Dieudonné ring.
M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, Schemas en groupes, i.e. SGA III-1, III-2, III-3
Michel Demazure, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970
W. Waterhouse, Introduction to affine group schemes, GTM 66, Springer 1979.
D. Mumford, Abelian varieties, 1970, 1985.
J. C. Jantzen, Representations of algebraic groups, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)