# Contents

## Definition

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 $G:C\mathrm{Ring}\to \mathrm{Set}$ equipped with a transformation $m:G×G\to G$ satisfying the properties spelled out at group object.

The other way is to define it as a functor $\mathrm{Sch}\to \mathrm{Grp}$ from the category of schemes to that of (discrete) groups whose composition with the forgetful functor $\mathrm{Grp}\to \mathrm{Set}$ 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

$G:\left(\mathrm{Sch}/X{\right)}^{\mathrm{op}}\to \mathrm{Grp}$G: (Sch /X)^{op} \to Grp

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 $\mathrm{Grp}$ of groups (instead of with values in Set); an equivalent way to state this is that $f$ needs to satisfy $fm=m\left(f×f\right)$ if $m:G×G\to G$ denotes the group law on $G$.

This construction generalizes to ind-schemes (as for example formal schemes) to that of a formal group scheme.

## Overview

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 ${k}_{\mathrm{sep}}$ of $k$. The importance of étale affine is that the category of them is equivalent to that of Galois modules by $E↦E{\otimes }_{k}{k}_{\mathrm{sep}}={\cup }_{K/k\phantom{\rule{thinmathspace}{0ex}}\mathrm{sep}\phantom{\rule{thinmathspace}{0ex}}\mathrm{fin}}E\left(K\right)$

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 ${\alpha }_{k}:R↦{R}^{+}$ and ${\mu }_{k}:R↦{R}^{×}$ sending a $k$-ring to to its underlying additive- and multiplicative group, respectively. These have the ”function rings” ${O}_{k}\left({\alpha }_{k}\right)=k\left[T\right]$ and ${O}_{\left(}{\mu }_{k}\right)=K\left[T,{T}^{-1}\right]$ and since $\left({O}_{k}⊣{\mathrm{Spec}}_{k}\right):k.\mathrm{Ring}\stackrel{{\mathrm{Spec}}_{k}}{\to }k.\mathrm{Aff}$ 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 $k.\mathrm{Gr}\left({\mu }_{k},{\alpha }_{k}\right)=0$ and call in generalization of this property any group scheme $G$ satisfying $k.\mathrm{Gr}\left(G,{\alpha }_{k}\right)=0$ multiplicative group scheme. (We could have also the idea to call $G$ satisfying $k.\mathrm{Gr}\left({\mu }_{k},G\right)=0$ ”additive” but I didn’t see this.) By some computation of the hom spaces $k.\mathrm{Gr}\left(G,{\mu }_{k}\right)$ involving co- and birings we see that these are again always values of a representable $k$-functor $D\left(G\right)\left(-\right):=\left(-\right).\mathrm{Gr}\left(G{\otimes }_{k}\left(-\right),{\mu }_{\left(-\right)}\right)$; this functor we call the Cartier dual of $G$. If for example $G$ is a finite group scheme $D\left(G\right)$ also is, and moreover $D$ is a contravariant autoequivalence (”duality”) of $k.\mathrm{fin}.\mathrm{comm}.\mathrm{Grp}$; in general it is also a duality in some specific sense. By taking the Cartier dual $D\left({E}_{k}\right)$ of a constant group scheme we obtain the notion of a diagonlizable group scheme. To justify this naming we compute some value $D\left({E}_{k}\right)\left(R\right)={\mathrm{hom}}_{\mathrm{Grp}}\left({E}_{k}{\otimes }_{k}R,{\mu }_{R}\right)\simeq {\mathrm{hom}}_{\mathrm{Grp}}\left({E}_{k},{R}^{×}\right)\simeq {\mathrm{hom}}_{\mathrm{Alg}}\left(k\left[{E}_{k}\right],R\right)$ where $k\left[{E}_{k}\right]$ 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 $G=\mathrm{Spec}\phantom{\rule{thinmathspace}{0ex}}k\left[{E}_{k}\right]$ and recall that a $\zeta \in {E}_{k}\subset k\left[{E}_{k}\right]$ is called a character of $G$ (and one calls a group generated by these ”diagonalizable”). Revisiting the condition $k.\mathrm{Gr}\left(H,{\alpha }_{k}\right)=0$ 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 $\stackrel{^}{X}$ if $X$ is a group scheme) of the Cartier dual of $G$, i.e. $\stackrel{^}{D}\left(G\right)$ is a connected formal group scheme also called local group scheme since a local group scheme $Q={\mathrm{Spec}}_{k}A$ is defined to be the spectrum of a local ring; this requirement in turn is equivalent to $Q\left(K\right)=\mathrm{hom}\left(A,K\right)=\left\{0\right\}$ 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

(1)$0\to {G}^{\circ }\to G\to {\pi }_{0}\left(G\right)\to 0$0\to G^\circ\to G\to \pi_0(G)\to 0

where ${G}^{\circ }$ is connected and ${\pi }_{0}\left(G\right)$ is étale. Such decomposition in exact sequences we obtain in further cases: $0\to {G}^{\mathrm{ex}}\to G\to {G}_{\mathrm{ex}}\to 0$ where

$k$-group${G}^{\mathrm{ex}}$${G}_{\mathrm{ex}}$
formalconnectedétale?p.34
finiteinfinitesimalétalesplits if $k$ is perfectp.35
affinemultiplicativesmooth$G/{G}_{\mathrm{red}}$ is infinitesimalp.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 $G=a×b×c×d$ where $a\in {\mathrm{Fem}}_{k}$ is a formal étale multiplicative $k$ group, $b\in {\mathrm{Feu}}_{k}$ is a formal étale unipotent $k$ group, $c\in {\mathrm{Fim}}_{k}$ is a formal infinitesimal multiplicative $k$ group, and $d\in {\mathrm{Fem}}_{k}$ 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 $G\left(p\right)$ of a group scheme encodes information on the p-torsion of the group scheme $G$. To appreciate the definition of $G\left(p\right)$ we first recall that for any group scheme $G$ we have the relative Frobenius morphism ${F}_{G}:G\to {G}^{\left(p\right)}$ to distinguish it from the absolute Frobenius morphism ${F}_{G}^{\mathrm{abs}}:G\to G$ 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 $G\left[{p}^{n}\right]:=\mathrm{ker}\phantom{\rule{thickmathspace}{0ex}}{F}_{G}^{n}$ where the kernel is taken of the Frobenius iterated $n$-times and the codirected system

$G\left[p\right]\stackrel{p}{\to }G\left[{p}^{2}\right]\stackrel{p}{\to }\dots \stackrel{p}{\to }G\left[{p}^{n}\right]\stackrel{p}{\to }G\left[{p}^{n+1}\right]\stackrel{p}{\to }\dots$G[p]\stackrel{p}{\to}G[p^2]\stackrel{p}{\to}\dots \stackrel{p}{\to}G[p^{n}]\stackrel{p}{\to}G[p^{n+1}]\stackrel{p}{\to}\dots

is then called the $p$-divisible group of $G$. As cardinality (in group theory also called rank) of this objects we have $\mathrm{card}\left(G\left[{p}^{j}\right]\right)={p}^{j\cdot h}$ for some $h\in ℕ$; this $h$ is called the height of $G$. Moreover we have (p1) the $G\left[{p}^{i}\right]$ are finite group schemes (we assumed this by definition), (p2) the sequences of the form $0\to \mathrm{ker}\phantom{\rule{thinmathspace}{0ex}}{p}^{j}\stackrel{{\iota }_{j}}{↪}\mathrm{ker}{p}^{j+k}\stackrel{{p}^{j}}{\to }\mathrm{ker}{p}^{k}\to 0$ are exact, (p3) $G={\cup }_{j}\mathrm{ker}\phantom{\rule{thinmathspace}{0ex}}{p}^{j}\cdot {\mathrm{id}}_{G}$ and one can show that if we start with any codirected system $\left({G}_{i}{\right)}_{i\in ℕ}$ satisfying (p1)(p2) we have that ${\mathrm{colim}}_{i}{G}_{i}$ satisfies (p3) and $\mathrm{ker}\left({F}_{G}^{n}\right)\simeq {G}_{n}$ - 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 $G\left(p\right)$). Basic examples of $p$-divisible groups are $\left({ℚ}_{p}/{ℤ}_{p}{\right)}_{k}^{h}$ which is (up to isomorphism) the unique example of a constant $p$-divisible group of height $h$ and $A\left(p\right)$ where $A$ is a commutative variety with a group law (aka. algebraic group). $A\left(p\right)$ is called the Barsotti-Tate group of an abelian variety; if the dimension of $A$ is $g$ the height of $A\left(p\right)$ 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 ,($p{1}^{\prime }$), ${\pi }_{0}\left(G\right)\left(\overline{k}\right)\simeq \left({ℚ}_{p}/{ℤ}_{p}{\right)}^{r}$ for some $r\in ℕ$ and ,($p{2}^{\prime }$), ${G}^{\circ }$ 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 $W:k.\mathrm{Ring}.\mathrm{comm}\to \lambda .\mathrm{Ring}↪\mathrm{Set}$ whose image is the category $\Lambda$ of lambda-rings; the objects $W\left(R\right)$ 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 $\left(VdasvW\right)$ forgetting the lambda-structure and the couniversal property? associated to this adjunction states that for a $k$-ring $R$ we have that $W\left(R\right)$ is the couniversal object such that all so called Witt polynomials ${w}_{n}\left({x}_{0},\dots {x}_{n}\right):={x}_{0}^{{p}^{n}}+p\cdot {x}_{1}^{{p}^{n-1}}+{p}^{2}\cdot {x}_{2}^{{p}^{n-2}}+\dots +{p}^{n}\cdot {x}_{n}$ 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 ${V}_{W}\left(R\right):\left({a}_{1},{a}_{2},\dots ,{a}_{n},\dots \right)↦\left(0,{a}_{1},{a}_{2},\dots ,{a}_{n},\dots \right)$ 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 $k$, $W\left(k\right)$ is a discrete valuation ring. The next construction we visit with $W\left(R\right)$ is Cartier duality of finite Witt groups (here we forget that $W\left(R\right)$ is even a ring): For this note that the ring of finite Witt vectors ${W}_{\mathrm{fin}}\left(R\right)$ is an ideal in $W\left(R\right)$ 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 ${W}_{n}\left(R\right)$ denotes the ring of Witt vectors of length $n$. With this notation we find $\mathrm{ker}\left({F}_{{W}_{n}}^{m}\right)\simeq D\left(\mathrm{ker}\left({F}_{{W}_{n}}^{n}\right)$.

Since $W\left(k\right)$ is a ring we can ask of its modules in general; however there is in particular one $W\left(k\right)$-module of interest which is called the Dieudonné module $M\left(G\right)$ of $G$. It can be defined in two equivalent ways: 1. as a $W\left(k\right)$-module $M$ equipped with two endomorphisms of $F$ and $V$ satisfying the ”Witt-Frobenius identities”

(WF1): $\mathrm{FV}=\mathrm{VF}=p$

(WF2): $\mathrm{Fw}={w}^{\left(p\right)}F$

(WF3): $wV=V{w}^{\left(p\right)}$

or 2. as a left module over the Dieudonné ring which is the (noncommutative ring) generated by $W\left(k\right)$ 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

$\sum _{i>0}{a}_{-i}{V}^{i}+{a}_{0}+\sum _{i>0}{a}_{i}{F}^{i}$\sum_{i\gt 0} a_{-i} V^i + a_0 + \sum_{i\gt 0} a_i F^i

(…)

## Examples

• For a field $k$ the terminal $k$-scheme ${\mathrm{Sp}}_{k}k$ 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 $\mu :={𝔾}_{m}$ is a group scheme given by ${𝔾}_{m}\left(S\right)=\Gamma \left(S,{𝒪}_{S}{\right)}^{×}$. 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 ${𝔾}_{m}$ defined by $\left(-{\right)}^{n}:x↦{x}^{n}$ 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 $\alpha :={𝔾}_{a}$ is a group scheme given by ${𝔾}_{a}\left(S\right)=\Gamma \left(S,{𝒪}_{S}\right)$ 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.

## Properties

### Cartier duality

(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 ${G}^{D}\left(S\right)=\mathrm{Hom}\left(G\otimes S,{𝔾}_{m}\otimes S\right)$. The Hom is taken in the category of group schemes over $S$.

For example, ${\alpha }_{p}^{D}\simeq {\alpha }_{p}$.

### Dieudonné module

(main article: Dieudonné module)

There are certain correspondences (Theorem AcuTheorem Fftc) between certain categories of group schemes and certain categories of Dieudonné modules.

###### Definition

A Dieudonné module is a module over the Dieudonné ring ${D}_{k}$ of a field $k$ of prime characteristic $p$.

###### Definition

The Dieudonné ring of $k$ is the ring generated by two objects $F,V$ subject to the relations

$\mathrm{FV}=\mathrm{VF}=p$FV=VF=p
$\mathrm{Fw}={w}^{\sigma }F$Fw=w^\sigma F
$wV=V{w}^{\sigma }$w V=V w^\sigma

where

$\sigma :\left\{\begin{array}{l}W\left(k\right)\to W\left(k\right)\\ \left({w}_{1},{w}_{2},\dots \right)↦\left({w}_{1}^{p},{w}_{2}^{p},\dots \right)\end{array}$\sigma:\begin{cases} W(k)\to W(k) \\ (w_1,w_2,\dots)\mapsto (w_1^p,w_2^p,\dots) \end{cases}

denotes the endomorphism of the Witt ring $W\left(k\right)$ 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 $ℤ$-graded ring where the degree $n$-part is the $1$-dimensional free module generated by ${V}^{-n}$ if $n<0$ and by ${F}^{n}$ if $n>0$

###### Theorem

(III.5, ${\mathrm{Acu}}_{k}\simeq {\mathrm{Tor}}_{V}{D}_{\mathrm{kMod}}$)

(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

$M:\left\{\begin{array}{lll}{\mathrm{Acu}}_{k}& \to & {\mathrm{Tor}}_{V}{D}_{\mathrm{kMod}}\\ G& ↦& {\mathrm{colim}}_{n}{\mathrm{Acu}}_{k}\left(G,{W}_{\mathrm{nk}}\right)\end{array}$M:\begin{cases} Acu_k&\to& Tor_V D_kMod \\ G&\mapsto&colim_n Acu_k(G,W_{nk}) \end{cases}

where we recall that how the colimit of the hom space can be multiplied by the generators of the Dieudonné ring.

###### Theorem

(III.6, ${\mathrm{Feu}}_{k}\simeq {\mathrm{Tor}}_{V}{D}_{\mathrm{kMod}}$)

###### Theorem

(III.6, ${\mathrm{Fiu}}_{k}\simeq {\mathrm{Tor}}_{F}{D}_{\mathrm{kMod}}$)

###### Theorem

(III.8, ${\mathrm{Torf}}_{p}\simeq \left(\mathrm{fin}W\left(k\right)\mathrm{Mod},F,V\right)$)

###### Theorem

(III.9, $\mathrm{Fftc}\simeq {\stackrel{^}{D}}_{k}{\mathrm{Mod}}_{\mathrm{fin}.\mathrm{len}.\mathrm{quot}}$)

## References

• 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)

Revised on November 11, 2013 23:09:39 by Urs Schreiber (145.116.131.252)