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\section*{group scheme}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{cartier_duality}{Cartier duality}\dotfill \pageref*{cartier_duality} \linebreak
\noindent\hyperlink{dieudonn_module}{Dieudonn\'e{} module}\dotfill \pageref*{dieudonn_module} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{group scheme} is a [[group object]] in the category of [[schemes]] (or in a category of \emph{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

\begin{displaymath}
G: (Sch /X)^{op} \to Grp
\end{displaymath}
A \textbf{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-object|ind-schemes]] (as for example [[formal scheme|formal schemes]]) to that of a [[formal group scheme]].

\hypertarget{overview}{}\subsection*{{Overview}}\label{overview}

Let $k$ be some base field. We start with the \textbf{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 \textbf{\'e{}tale group scheme}. This is not itself constant but becomes so by scalar extension to the separable closure $k_sep$ of $k$. The importance of \'e{}tale affine is that the category of them is equivalent to that of [[nLab:Galois module|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 \textbf{additive-} and the \textbf{[[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. \textbf{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 $k.Gr(G,\alpha_k)=0$ \textbf{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 \textbf{[[Cartier dual]] of $G$}. If for example $G$ is a \textbf{[[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 \textbf{duality} in some specific sense. By taking the Cartier dual $D(E_k)$ of a constant group scheme we obtain the notion of a \textbf{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 [[nLab: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 \textbf{sub group-schemes} $H$ of $G$ we arrive at the notion of \textbf{unipotent group scheme}. By the structure theorem of \textbf{decomposition of affine groups} we can proof that $G$ is unipotent iff the \textbf{completion of group schemes} (which gives us-by the usual technic of [[nLab:completion]]- a \textbf{formal (group) scheme} $\hat X$ if $X$ is a group scheme) of the Cartier dual of $G$, i.e. $\hat D(G)$ is a \textbf{connected formal group scheme} also called \textbf{local group scheme} since a local group scheme $Q=Spec_k A$ is defined to be the [[nLab: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 \'e{}tale schemes: For any formal group scheme there is an essentially unique [[nLab:exact sequence]]

\begin{equation}
0\to G^\circ\to G\to \pi_0(G)\to 0
\label{connectedetaledecomposition}\end{equation}
where $G^\circ$ is connected and $\pi_0(G)$ is \'e{}tale. Such decomposition in exact sequences we obtain in further cases: $0\to G^{ex}\to G\to G_{ex}\to 0$ where

\newline | finite | [[nLab:infinitesimal object|infinitesimal]] | \'e{}tale | splits if $k$ is perfect |p.35 | | [[nLab:ring spectrum|affine]] | multiplicative | [[unipotent group scheme|unipotent | splits if $k$ is perfect | p.39 | | connected | $G_{red}$ is [[nLab:smooth scheme|smooth]] | $G/G_{red}$ is infinitesimal |p.43 | |

where a \textbf{smooth (group) scheme} is defined to be the spectrum of a finite dimensional (over k) [[nLab:power series]] algebra, a (group) scheme is called \textbf{finite (group) scheme} if we restrict in all necessary definitions to $k$-ring which are finite dimensional $k$-[[nLab:vector space|vector spaces]], and a (group) scheme is called \textbf{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 \'e{}tale multiplicative $k$ group, $b\in Feu_k$ is a formal \'e{}tale 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 \textbf{[[p-divisible group]]} is an extensively studied case because the $p$-divisible group $G(p)$ of a group scheme encodes information on the \textbf{[[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 [[nLab:Frobenius morphism|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

\begin{displaymath}
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
\end{displaymath}
is then called the $p$-divisible group of $G$. As \textbf{cardinality} (in group theory also called \textbf{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 \textbf{[[height of a group scheme|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 \textbf{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 [[nLab:variety]] with a group law (aka. \textbf{algebraic group}). $A(p)$ is called \textbf{[[nLab: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 \textbf{`'exactness'` characterization of $p$-divisible formal groups} by: $G$ is $p$-divisible iff in the connected-\'e{}tale 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 \textbf{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.

(\ldots{})

In cases where $k$ is a field of prime [[nLab: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 \textbf{[[nLab:lambda ring|lambda-rings]]}; the objects $W(R)$ of $\Lambda$ are also called \textbf{[[nLab: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 \textbf{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 \textbf{discrete valuation ring}. The next construction we visit with $W(R)$ is \textbf{Cartier duality of finite Witt groups} (here we forget that $W(R)$ is even a ring): For this note that the \textbf{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 \textbf{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 \textbf{Dieudonn\'e{} 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 \textbf{Dieudonn\'e{} 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

\begin{displaymath}
\sum_{i\gt 0} a_{-i} V^i + a_0 + \sum_{i\gt 0} a_i F^i
\end{displaymath}
(\ldots{})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item For a field $k$ the terminal $k$-scheme $Sp_k k$ is a group scheme in a unique way.


\item An [[affine scheme|affine]] group scheme. Affine group [[variety|varieties]] are called [[linear algebraic group|linear algebraic groups]].


\item Complete group varieties are called [[abelian variety|abelian varieties]].


\item Given any group $G$, one can form the [[constant group scheme]] $G_X$ over $X$.


\item [[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.


\item 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 \emph{the multiplicative group scheme}. In context of [[p-divisible group|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]].


\item [[diagonalizable group scheme]]. Note that the multiplicative group scheme is diagonalizable.


\item [[multiplicative group scheme]] also called \emph{group scheme of multiplicative type}. Every diagonalizable group scheme is in particular of multiplicative type.


\item The \emph{[[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 object|nilpotent element]].


\item Group schemes can be constructed by [[restriction of scalars]].


\item 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 \emph{the additive group scheme}.


\item [[connected group scheme]] (is synonymous to [[local group scheme]])


\item [[unipotent group scheme]] (these are [[Cartier duality|Cartier duals]] of local group schemes)


\item the kernel of any group scheme morphism is a group scheme.


\item Every [[algebraic group]] is in particular a group scheme.



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{cartier_duality}{}\subsubsection*{{Cartier duality}}\label{cartier_duality}

(main article: [[Cartier duality]])

Suppose now that $G$ is a finite flat commutative group scheme (over $X$). The \textbf{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, $\alpha_p^D\simeq \alpha_p$.

\hypertarget{dieudonn_module}{}\subsubsection*{{Dieudonn\'e{} module}}\label{dieudonn_module}

(main article: [[Dieudonné module]])

There are certain correspondences (\hyperlink{thmAcu}{Theorem Acu}--\hyperlink{thmFftc}{Theorem Fftc}) between certain categories of group schemes and certain categories of Dieudonn\'e{} modules.

\begin{defn}
\label{}\hypertarget{}{}
A \emph{Dieudonn\'e{} module} is a module over the [[Dieudonné ring]] $D_k$ of a field $k$ of prime [[characteristic]] $p$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
The Dieudonn\'e{} ring of $k$ is the ring generated by two objects $F,V$ subject to the relations

\begin{displaymath}
FV=VF=p
\end{displaymath}
\begin{displaymath}
Fw=w^\sigma F
\end{displaymath}
\begin{displaymath}
w V=V w^\sigma
\end{displaymath}
where

\begin{displaymath}
\sigma:\begin{cases}
W(k)\to W(k)
\\
(w_1,w_2,\dots)\mapsto (w_1^p,w_2^p,\dots)
\end{cases}
\end{displaymath}
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 $k$.

\end{defn}
The Dieudonn\'e{} ring is a $\mathbb{Z}$-[[graded ring]] where the degree $n$-part is the $1$-[[dimension|dimensional]] [[free object|free module]] generated by $V^{-n}$ if $n\lt 0$ and by $F^n$ if $n\gt 0$

\begin{theorem}
\label{thmAcu}\hypertarget{thmAcu}{}
(\hyperlink{Demazure}{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 scheme|affine]] commutative [[unipotent group scheme|unipotent group schemes]]. On the right we have the category of all [[Dieudonné ring|D\_k]][[Dieudonné module|-modules]] of $V$-torsion. The (contravariant) equivalence is given by

\begin{displaymath}
M:\begin{cases}
Acu_k&\to& Tor_V D_kMod
\\
G&\mapsto&colim_n Acu_k(G,W_{nk})
\end{cases}
\end{displaymath}
where we recall that how the colimit of the hom space can be multiplied by the generators of the [[Dieudonné ring]].

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
(\hyperlink{Demazure}{III.6}, $Feu_k\simeq Tor_V D_kMod$)

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
(\hyperlink{Demazure}{III.6}, $Fiu_k\simeq Tor_F D_kMod$)

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
(\hyperlink{Demazure}{III.8}, $Torf_p\simeq (fin W(k) Mod,F,V)$)

\end{theorem}
\begin{theorem}
\label{thmFftc}\hypertarget{thmFftc}{}
(\hyperlink{Demazure}{III.9}, $Fftc\simeq \hat D_k Mod_{fin.len.quot}$)

\end{theorem}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[abelian group scheme]]


\item [[dual abelian group scheme]]


\item [[spectral group scheme]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, \emph{Schemas en groupes}, i.e. SGA III-1, III-2, III-3


\item Michel Demazure, [[Pierre Gabriel|P. Gabriel]], \emph{Groupes algebriques}, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970


\item Michel [[Demazure, lectures on p-divisible groups]] \href{http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups}{web}


\item W. Waterhouse, \emph{Introduction to affine group schemes}, GTM 66, Springer 1979.


\item D. Mumford, \emph{Abelian varieties}, 1970, 1985.


\item J. C. Jantzen, \emph{Representations of algebraic groups}, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)



\end{itemize}
[[!redirects group scheme]] [[!redirects group schemes]]

[[!redirects Cartier dual]] [[!redirects Cartier duals]]

[[!redirects group schemes]]

[[!redirects height of a group scheme]]



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