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Summary (random tour through the examples)

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

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 Frobenius. an 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 general; but however we there will is do this just in the particular case one where$W(k)$ k W(k) -module is a perfect field of prime interest characteristic which is called the$p$Dieudonné module $M(G)$ of $G$ : . We It define can the be defined in two equivalent ways: 1. as aDieudonné module $M(G)\in W(k).Mod$$W(k)$ -module of an affine unipotent commutative$\mathrm{<span><del\; class="diffmod">\; k</del><ins\; class="diffmod">\; M</ins></span>}$ k M -group scheme equipped to with be two the endomorphisms colimit of$\mathrm{<span><del\; class="diffmod">\; colim</del><ins\; class="diffmod">\; F</ins></span>}{\mathrm{Acu}}_k(G,W_n)$ colim F Acu_k (G, W_n) where and the colimit goes over the codirected system$\underline{W}V:=(W_n{)}_{n\in \mathbb{N}}$ \underline{W}:=(W_n)_{n\in V \mathbb{N}} of satisfying affine the uipotent ‘’Witt-Frobenius commutative identities’‘$k-groups$ with transition maps being the Verschiebung maps $V:W_n\to W_{n+1}$. The module structure is obtained from the operation $W(k)\times\underline{W}\to \underline{W}$ given by