higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
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 $G:C Ring\to Set$ equipped with a transformation $m:G\times G\to G$ satisfying the properties spelled out at group object.
The other way is to define it as a functor $Sch\to Grp$ from the category of schemes to that of (discrete) groups whose composition with the forgetful functor $Grp\to Set$ 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 $X$ is a functor
A morphism of group schemes $f:G\to H$ 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. $f$ is required to be a natural transformation of functors with values in the category $Grp$ of groups (instead of with values in Set); an equivalent way to state this is that $f$ needs to satisfy $f m=m(f\times f)$ if $m:G\times G\to G$ denotes the group law on $G$.
This construction generalizes to ind-schemes (as for example formal schemes) to that of a formal group scheme.
Let $k$ be some base field. We start with the constant group scheme $E_k$ defined by some classical group $E$ which gives in every component just the group $E$. Next we visit the notion of étale group scheme. This is not itself constant but becomes so by scalar extension to the separable closure $k_sep$ of $k$. The importance of étale affine is that the category of them is equivalent to that of Galois modules by $E\mapsto E \otimes_k k_sep=\cup_{K/k \,sep\,fin} E(K)$
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 $\alpha_k: R\mapsto R^+$ and $\mu_k:R\mapsto R^\times$ sending a $k$-ring to to its underlying additive- and multiplicative group, respectively. These have the ‘’function rings’‘ $O_k(\alpha_k)=k[T]$ and $O_(\mu_k)=K[T,T^{-1}]$ and since $(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Aff$ we note that our basic building blocks $\alpha_k$ and $\mu_k$ are in fact representable $k$-functors aka. affine group schemes. We observe that we have $k.Gr(\mu_k,\alpha_k)=0$ and call in generalization of this property any group scheme $G$ satisfying $k.Gr(G,\alpha_k)=0$ multiplicative group scheme. (We could have also the idea to call $G$ satisfying $k.Gr(\mu_k,G)=0$ ‘’additive’‘ but I didn’t see this.) By some computation of the hom spaces $k.Gr(G,\mu_k)$ involving co- and birings we see that these are again always values of a representable $k$-functor $D(G)(-):=(-).Gr(G\otimes_k (-),\mu_{(-)})$; this functor we call the Cartier dual of $G$. If for example $G$ is a finite group scheme $D(G)$ also is, and moreover $D$ is a contravariant autoequivalence (’‘duality’’) of $k.fin.comm.Grp$; in general it is also a duality in some specific sense. By taking the Cartier dual $D(E_k)$ of a constant group scheme we obtain the notion of a diagonlizable group scheme. To justify this naming we compute some value $D(E_k)(R)=hom_{Grp}(E_k\otimes_k R, \mu_R)\simeq hom_{Grp}(E_k,R^\times)\simeq hom_Alg(k[E_k],R)$ where $k[E_k]$ denotes the group algebra of $E_k$ and the last isomorphism is due to the universal property of group rings; we observe that the last equality tells us that $G=Spec\,k[E_k]$ and recall that a $\zeta\in E_k\subset k[E_k]$ is called a character of $G$ (and one calls a group generated by these ‘’diagonalizable’’). Revisiting the condition $k.Gr(H,\alpha_k)=0$ by which we defined multiplicative group schemes and considering a group scheme $G$ satisfying this condition for all sub group-schemes $H$ of $G$ we arrive at the notion of unipotent group scheme. By the structure theorem of decomposition of affine groups we can proof that $G$ is unipotent iff the completion of group schemes (which gives us-by the usual technic of completion- a formal (group) scheme $\hat X$ if $X$ is a group scheme) of the Cartier dual of $G$, i.e. $\hat D(G)$ is a connected formal group scheme also called local group scheme since a local group scheme $Q=Spec_k A$ is defined to be the spectrum of a local ring; this requirement in turn is equivalent to $Q(K)=hom(A,K)=\{0\}$ 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 $G^\circ$ is connected and $\pi_0(G)$ is étale. Such decomposition in exact sequences we obtain in further cases: $0\to G^{ex}\to G\to G_{ex}\to 0$ where
$k$-group | $G^{ex}$ | $G_{ex}$ | ||
---|---|---|---|---|
formal | connected | étale? | p.34 | |
finite | infinitesimal | étale | splits if $k$ is perfect | p.35 |
affine | multiplicative | smooth | $G/G_{red}$ is infinitesimal | p.43 |
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 $k$-ring which are finite dimensional $k$-vector spaces, and a (group) scheme is called infinitesimal (group) scheme if it is finite and local. If moreover $k$ is a perfect field any finite affine $k$-group $G$ is in a unique way the product of four subgroups $G=a\times b\times c\times d$ where $a\in Fem_k$ is a formal étale multiplicative $k$ group, $b\in Feu_k$ is a formal étale unipotent $k$ group, $c\in Fim_k$ is a formal infinitesimal multiplicative $k$ group, and $d\in Fem_k$ is a infinitesimal unipotent $k$ 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 $p$-divisible group $G(p)$ of a group scheme encodes information on the p-torsion of the group scheme $G$. To appreciate the definition of $G(p)$ we first recall that for any group scheme $G$ we have the relative Frobenius morphism $F_G:G\to G^{(p)}$ to distinguish it from the absolute Frobenius morphism $F^{abs}_G:G\to G$ which is induced by the Frobenius morphism of the underlying ring $k$. 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 $G[p^n]:=ker\; F^n_G$ where the kernel is taken of the Frobenius iterated $n$-times and the codirected system
is then called the $p$-divisible group of $G$. As cardinality (in group theory also called rank) of this objects we have $card(G[p^j])=p^{j\cdot h}$ for some $h\in \mathbb{N}$; this $h$ is called the height of $G$. Moreover we have (p1) the $G[p^i]$ are finite group schemes (we assumed this by definition), (p2) the sequences of the form $0\to ker\, p^j\xhookrightarrow{\iota_j} ker p^{j+k}\stackrel{p^j}{\to}ker p^k\to 0$ are exact, (p3) $G=\cup_j ker\, p^j\cdot id_G$ and one can show that if we start with any codirected system $(G_i)_{i\in \mathbb{N}}$ satisfying (p1)(p2) we have that $colim_i G_i$ satisfies (p3) and $ker( F^n_G)\simeq G_n$ - in other words the properties (p1)(p2) give an equivalent alternative definition of $p$-divisible groups (and (p3) leads some authors to ‘’identify’‘ $G$ and $G(p)$). Basic examples of $p$-divisible groups are $(\mathbb{Q}_p/\mathbb{Z}_p)^h_k$ which is (up to isomorphism) the unique example of a constant $p$-divisible group of height $h$ and $A(p)$ where $A$ is a commutative variety with a group law (aka. algebraic group). $A(p)$ is called the Barsotti-Tate group of an abelian variety; if the dimension of $A$ is $g$ the height of $A(p)$ is $2g$. Now, what about decomposition of $p$-divisible groups? We have even one more equivalent ‘’exactness’‘ characterization of $p$-divisible formal groups by: $G$ is $p$-divisible iff in the connected-étale decomposition given by the exact sequence displayed in (1) we have ,($p1^\prime$), $\pi_0(G)(\overline k)\simeq (\mathbb{Q}_p/\mathbb{Z}_p)^r$ for some $r\in \mathbb{N}$ and ,($p2^\prime$), $G^\circ$ 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 $p$-divisiblity by exact sequences gives rise to propositions on dimensions and subgroups of $p$-divisible groups.
(…)
In cases where $k$ is a field of prime characteristic $p$, there is some special $k$-functor which is a group functor and even a ring functor (a $k$-functor equipped with a ring structure) - namely the functor $W:k.Ring. comm\to \lambda.Ring\hookrightarrow Set$ whose image is the category $\Lambda$ of lambda-rings; the objects $W(R)$ of $\Lambda$ are also called Witt vectors since they are infinite sequences of elements of $R$ (this justifies at least ‘’vectors’’). $W$ possesses a left adjoint $(V\dasv W)$ forgetting the lambda-structure and the couniversal property? associated to this adjunction states that for a $k$-ring $R$ we have that $W(R)$ is the couniversal object such that all so called Witt polynomials $w_n(x_0,\dots x_n):=x_0^{p^n}+p\cdot x_1^{p^{n-1}}+p^2 \cdot x_2^{p^{n-2}}+\dots+p^n\cdot x_n$ are ring homomorphisms. For this special $k$-group $W$ we revisit some construction we have done above for general $k$-groups: we firstly make the eponymous remark that the Verschiebung morphism $V_W(R):(a_1,a_2,\dots,a_n,\dots)\mapsto (0,a_1,a_2,\dots,a_n,\dots)$ 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 $k$, $W(k)$ is a discrete valuation ring. The next construction we visit with $W(R)$ is Cartier duality of finite Witt groups (here we forget that $W(R)$ is even a ring): For this note that the ring of finite Witt vectors $W_fin(R)$ is an ideal in $W(R)$ and we have Frobenius and Verschiebung also in this truncated case; more precisely we have for each $n$ a Frobenius $F_{W_n}:W_n\to W_n$ where $W_n(R)$ denotes the ring of Witt vectors of length $n$. With this notation we find $ker(F^m_{W_n})\simeq D(ker(F^n_{W_n})$.
Since $W(k)$ is a ring we can ask of its modules in general; however there is in particular one $W(k)$-module of interest which is called the Dieudonné module $M(G)$ of $G$. It can be defined in two equivalent ways: 1. as a $W(k)$-module $M$ equipped with two endomorphisms of $F$ and $V$ satisfying the ‘’Witt-Frobenius identities’‘
(WF1): $FV=VF=p$
(WF2): $Fw=w^{(p)} F$
(WF3): $w V=V w^{(p)}$
or 2. as a left module over the Dieudonné ring which is the (noncommutative ring) generated by $W(k)$ and two variables $F$ and $V$ satisfying (WF1)(WF2)(WF3) in which case every element of $D_k$ can uniquely be written as a finite sum
(…)
For a field $k$ the terminal $k$-scheme $Sp_k k$ is a group scheme in a unique way.
An affine group scheme. Affine group varieties are called linear algebraic groups.
Complete group varieties are called abelian varieties.
Given any group $G$, one can form the constant group scheme? $G_X$ over $X$.
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 $\mu:=\mathbb{G}_m$ is a group scheme given by $\mathbb{G}_m(S)=\Gamma(S, \mathcal{O}_S)^\times$. 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 $k$-group scheme endomorphisms of $\mathbb{G}_m$ defined by $(-)^n:x\mapsto x^n$ for an integer $n$ are of particular interest. These kernels give the group schemes of the $n$-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.
The additive group scheme assigns to a ring its additive group. Also here the kernels of the powering-by-n map are of interest. These kernels give the group schemes of the $n$-th nilpotent element?.
Group schemes can be constructed by restriction of scalars.
The functor $\alpha:=\mathbb{G}_a$ is a group scheme given by $\mathbb{G}_a(S)=\Gamma(S, \mathcal{O}_S)$ the additive group of the ring of global functions. This group scheme is called the additive group scheme.
connected group scheme? (is synonymous to local group scheme?)
unipotent group scheme (these are Cartier duals of local group schemes)
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 $G$ is a finite flat commutative group scheme (over $X$). The Cartier dual of $G$ is given by the functor $G^D(S)=Hom (G\otimes S, \mathbb{G}_m \otimes S)$. The Hom is taken in the category of group schemes over $S$.
For example, $\alpha_p^D\simeq \alpha_p$.
(main article: Dieudonné module)
There are certain correspondences (Theorem Acu–Theorem Fftc) between certain categories of group schemes and certain categories of Dieudonné modules.
A Dieudonné module is a module over the Dieudonné ring $D_k$ of a field $k$ of prime characteristic $p$.
The Dieudonné ring of $k$ is the ring generated by two objects $F,V$ subject to the relations
where
denotes the endomorphism of the Witt ring $W(k)$ of $k$ given by raising each component of the Witt vectors to the $p$-th power; this means that $\sigma$ is component-wise given by the Frobenius endomorphism of the file $k$.
The Dieudonné ring is a $\mathbb{Z}$-graded ring where the degree $n$-part is the $1$-dimensional free module generated by $V^{-n}$ if $n\lt 0$ and by $F^n$ if $n\gt 0$
(III.5, $Acu_k\simeq Tor_V D_kMod$)
(see also Dieudonné module for more details concerning this theorem)
Let $k$ be a perfect field of prime characteristic $p$. Since $k$ is perfect Frobenius is an automorphism.
On the left we have the category of affine commutative unipotent group schemes. On the right we have the category of all D_k-modules of $V$-torsion. The (contravariant) equivalence is given by
where we recall that how the colimit of the hom space can be multiplied by the generators of the Dieudonné ring.
(III.6, $Feu_k\simeq Tor_V D_kMod$)
(III.6, $Fiu_k\simeq Tor_F D_kMod$)
(III.8, $Torf_p\simeq (fin W(k) Mod,F,V)$)
(III.9, $Fftc\simeq \hat D_k Mod_{fin.len.quot}$)
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)