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\begin{document}

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\section*{category theoretic aspects of the theory of group schemes}

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

\noindent\hyperlink{group_functors_and_affine_group_schemes}{(Group) functors and affine (group) schemes}\dotfill \pageref*{group_functors_and_affine_group_schemes} \linebreak
\noindent\hyperlink{group_schemes}{(Group) schemes}\dotfill \pageref*{group_schemes} \linebreak
\noindent\hyperlink{constant_group_scheme}{Constant (group) scheme}\dotfill \pageref*{constant_group_scheme} \linebreak
\noindent\hyperlink{formal_group_scheme}{Formal (group) scheme}\dotfill \pageref*{formal_group_scheme} \linebreak
\noindent\hyperlink{examples_of_group_schemes}{Examples of (group) schemes}\dotfill \pageref*{examples_of_group_schemes} \linebreak
\noindent\hyperlink{tale_group_scheme}{Étale (group) scheme}\dotfill \pageref*{tale_group_scheme} \linebreak
\noindent\hyperlink{cartier_dual_of_a_finite_flat_commutative_group_scheme}{Cartier dual of a finite flat commutative group scheme}\dotfill \pageref*{cartier_dual_of_a_finite_flat_commutative_group_scheme} \linebreak
\noindent\hyperlink{ptorsion}{p-torsion}\dotfill \pageref*{ptorsion} \linebreak
\noindent\hyperlink{pdivisible_groups}{p-divisible groups}\dotfill \pageref*{pdivisible_groups} \linebreak
\noindent\hyperlink{witt_rings_and_dieudonn_modules}{Witt rings and Dieudonné modules}\dotfill \pageref*{witt_rings_and_dieudonn_modules} \linebreak


\hypertarget{group_functors_and_affine_group_schemes}{}\subsection*{{(Group) functors and affine (group) schemes}}\label{group_functors_and_affine_group_schemes}

Let $k$ be a ring. Let $k.Ring$ denote the category of $k$-rings. Let $k.Fun$ denote the category of (contravariant) functors $X:k:Ring\to Set$. Let $k.Aff$ denote the category of representable $k$-functors; we call this category the \emph{category of affine $k$-schemes} and an object of this category we write as

\begin{displaymath}
Spec_k A:\begin{cases}
k.Ring\to Set
\\
R\mapsto hom(A,R)
\end{cases}
\end{displaymath}
We obtain in this way a functor

\begin{displaymath}
Spec_k:k.Ring\to k.Fun
\end{displaymath}
This functor has a left adjoint

\begin{displaymath}
(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Fun
\end{displaymath}
assigning to a $k$-functor its \emph{ring of functions}. This adjunction restricts to an adjoint equivalence

\begin{displaymath}
(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Aff
\end{displaymath}
and it restricts moreover to an adjoint equivalence

\begin{displaymath}
(O_k\dashv Spec_k):k.Bi Ring\stackrel{Spec_k}{\to} k.Aff.comm.Gr
\end{displaymath}
between the categories of $k$-birings and the category of commutative affine $k$-group schemes. To see this be aware that a $k$-biring is a commutative ring object in ${k.Ring.comm}^{op}\simeq k.Aff.Sch$ (where the latter denotes the category of affine schemes).

\hypertarget{group_schemes}{}\subsection*{{(Group) schemes}}\label{group_schemes}

A $k$-functor is called a \emph{$k$-scheme} if it is a sheaf for the Zariski Grothendieck topology on $k.Ring^{op}$.

We will consider the moral of this op-ing below.

To give more details, recall that the closed sets of the Zariski topology on the [[nLab:spectrum]] $Spec A$ of a $k$-ring $A$ is defined by

\begin{displaymath}
V(I):=\{P\in Spec \, A|I\subseteq P\}
\end{displaymath}
We can characterize the the elements of $V(a)$ also by

\begin{displaymath}
e_a(P)=0\, iff\, P\in V(a)
\end{displaymath}
where

\begin{displaymath}
e_a:\begin{cases}
Spec (A)\to Quot(A/P)
\\
P\mapsto \frac{a\,mod\,P}{1}
\end{cases}
\end{displaymath}
where $Quot(A/P)$ denotes the quotient field (aka. field of fractions) of the [[nLab:integral domain]] $A/P$.

This construction generalizes to $k$-functors by defining an \emph{open subfunctor of a $k$-functor $X$} by

\begin{displaymath}
V(I):R\mapsto\{x\in X(R)| I\subseteq x\}
\end{displaymath}
where $I\subseteq O(X)$. By the above alternative characterization, the assigned set consists precisely of those $x$ for which $f(x)=0$ for all $f\in I$.

\hypertarget{constant_group_scheme}{}\subsection*{{Constant (group) scheme}}\label{constant_group_scheme}

$Sch_k$ is [[nLab:copower|copowered (= tensored)]] over $Set$. We define the \emph{constant $k$-scheme} on a set $E$ by

\begin{displaymath}
E_k:=E\otimes Sp_k k=\coprod_{e\in E}Sp_k k
\end{displaymath}
For a scheme $X$ we compute $M_k(E_k,E)=Set(Sp_k k,X)^E=X(k)^E=Set(E,X(k))$ and see that there is an adjunction

\begin{displaymath}
((-)_k\dashv (-)(k)):Sch_k\to Set
\end{displaymath}
If $E$ is a group $E_k$ is a group scheme.

We denote the category of $k$-schemes by $k.Sch$.

A \emph{$k$-group scheme} is a [[nLab:group object]] in $k.Sch$. The category of $k$-group schemes we denote by $k.Grp$.

\hypertarget{formal_group_scheme}{}\subsection*{{Formal (group) scheme}}\label{formal_group_scheme}

Let $k$ be a field. A \emph{finite $k$-ring} is defined to be a $k$-algebra which is a finite dimensional $k$-[[nLab:vector space]]. The category of finite $k$-rings we denote by $fin.k.Ring$. A \emph{finite $k$-functor} is defined to be a covariant functor $X:fin.k.Ring\to Set$. The category of finite $k$-functors we denote by $fin.k.Fun$. A \emph{finite $k$-scheme} is defined to be $k$-scheme which is a finite $k$-functor. The category of finite $k$-schemes we denote by $fin.k.Sch$. Analogously we define the category of \emph{finite group schemes}.

A \emph{formal group scheme} is defined to be a codirected colimit of finite $k$-schemes.

Recall that we have a covariant embedding

\begin{displaymath}
Spec_k:k.Ring^{op}\to k.Fun
\end{displaymath}
but we equivalently an embedding

\begin{displaymath}
Spec^*:k.co Ring\to k.Fun
\end{displaymath}
where by $k.co Ring$ we denote the category of $k$-[[nLab:coring|corings]]. A \emph{coring} is a [[nLab:comonoid]] in the category of affine schemes (the latter is the opposite category of $k.Ring$). If we restrict to finite $k$-rings by linear algebra we have a bijection $A^*\otimes R=hom(A,k)\simeq hom(A,R)$ and can write

\begin{displaymath}
Spec_k C: R\mapsto\{u\in A^*\otimes R|\Delta_R u=u\otimes u, \epsilon_R u=1\}
\end{displaymath}
where $C$ is a $k$-coring, $R$ a finite $k$-ring, $\Delta_R$ the skalar-extended comultiplication, $\epsilon_R$ the skalar-extended counit.

\hypertarget{examples_of_group_schemes}{}\subsubsection*{{Examples of (group) schemes}}\label{examples_of_group_schemes}

[[examples of (group) schemes]]

\hypertarget{tale_group_scheme}{}\subsection*{{Étale (group) scheme}}\label{tale_group_scheme}

(see also [[nLab:Grothendieck's Galois theory]])

An étale group scheme over a field $k$ is defined to be a directed colimit

\begin{displaymath}
colim_{(k\hookrightarrow k^\prime)\in T\subseteq Sep} Spec\, k^\prime
\end{displaymath}
where $T$ denotes some set of finite separable field extensions of $k$.

\hypertarget{cartier_dual_of_a_finite_flat_commutative_group_scheme}{}\subsection*{{Cartier dual of a finite flat commutative group scheme}}\label{cartier_dual_of_a_finite_flat_commutative_group_scheme}

Let $G$ be a commutative $k$-group functor (in cases of interest this is a finite flat commutative [[nLab:group scheme]]). Then the \emph{Cartier dual} $D(G)$ of $G$ is defined by

\begin{displaymath}
D(G)(R):=Gr_R(G\otimes_k R,\mu_R)
\end{displaymath}
where $\mu_k$ denotes the group scheme assigning to a ring its multiplicative group $R^\times$ consisting of the invertible elements of $R$.

This definition deserves the name [[nLab:duality]] since we have

\begin{displaymath}
hom(G,D(H))=hom(H,D(G))=hom(G\times H,\mu_k)
\end{displaymath}
\hypertarget{ptorsion}{}\subsection*{{p-torsion}}\label{ptorsion}

Let $s:R\to S$ be a morphism of rings. Then we have an [[nLab:adjunction]]

\begin{displaymath}
(s^*\dashv s_*):S.Mod\stackrel{s_*}{\to} R.Mod
\end{displaymath}
from the category of $S$-[[nLab:module|modules]] to that of $R$-modules where

\begin{displaymath}
s^*:A\mapsto A\otimes_s S
\end{displaymath}
is called \emph{scalar extension} and $s_*$ is called \emph{scalar restriction}.

If $X$ denotes some [[nLab:scheme]] over a $k$-ring for $k$ being a field of characteristic $p$, we define its $p$-torsion component-wise by $X^{(p)}(R):=X(s_* R)$.

\hypertarget{pdivisible_groups}{}\subsubsection*{{p-divisible groups}}\label{pdivisible_groups}

\hypertarget{witt_rings_and_dieudonn_modules}{}\subsection*{{Witt rings and Dieudonné modules}}\label{witt_rings_and_dieudonn_modules}



\end{document}