# Contents * Automatic table of contents {: toc} ## (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 *category of affine $k$-schemes* and an object of this category we write as $$Spec_k A:\begin{cases} k.Ring\to Set \\ R\mapsto hom(A,R) \end{cases}$$ We obtain in this way a functor $$Spec_k:k.Ring\to k.Fun$$ This functor has a left adjoint $$(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Fun$$ assigning to a $k$-functor its *ring of functions*. This adjunction restricts to an adjoint equivalence $$(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Aff$$ and it restricts moreover to an adjoint equivalence $$(O_k\dashv Spec_k):k.Bi Ring\stackrel{Spec_k}{\to} k.Aff.comm.Gr$$ 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). ## (Group) schemes A $k$-functor is called a *$k$-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 $$V(I):=\{P\in Spec \, A|I\subseteq P\}$$ We can characterize the the elements of $V(a)$ also by $$e_a(P)=0\, iff\, P\in V(a)$$ where $$e_a:\begin{cases} Spec (A)\to Quot(A/P) \\ P\mapsto \frac{a\,mod\,P}{1} \end{cases}$$ 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 *open subfunctor of a $k$-functor $X$* by $$V(I):R\mapsto\{x\in X(R)| I\subseteq x\}$$ 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 $f\in I$. ## Constant (group) scheme $Sch_k$ is [[nLab:copower|copowered (= tensored)]] over $Set$. We define the _constant $k$-scheme_ on a set $E$ by $$E_k:=E\otimes Sp_k k=\coprod_{e\in E}Sp_k k$$ 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 $$((-)_k\dashv (-)(k)):Sch_k\to Set$$ If $E$ is a group $E_k$ is a group scheme. We denote the category of $k$-schemes by $k.Sch$. A *$k$-group scheme* is a [[nLab:group object]] in $k.Sch$. The category of $k$-group schemes we denote by $k.Grp$. ## Formal (group) scheme Let $k$ be a field. A *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 *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 *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 *finite group schemes*. A *formal group scheme* is defined to be a codirected colimit of finite $k$-schemes. Recall that we have a covariant embedding $$Spec_k:k.Ring^{op}\to k.Fun$$ but we equivalently an embedding $$Spec^*:k.co Ring\to k.Fun$$ where by $k.co Ring$ we denote the category of $k$-[[nLab:coring|corings]]. A *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 $$Spec_k C: R\mapsto\{u\in A^*\otimes R|\Delta_R u=u\otimes u, \epsilon_R u=1\}$$ where $C$ is a $k$-coring, $R$ a finite $k$-ring, $\Delta_R$ the skalar-extended comultiplication, $\epsilon_R$ the skalar-extended counit. ### Examples of (group) schemes [[examples of (group) schemes]] ## Étale (group) scheme (see also [[nLab:Grothendieck's Galois theory]]) An étale group scheme over a field $k$ is defined to be a directed colimit $$colim_{(k\hookrightarrow k^\prime)\in T\subseteq Sep} Spec\, k^\prime$$ where $T$ denotes some set of finite separable field extensions of $k$. ## 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 *Cartier dual* $D(G)$ of $G$ is defined by $$D(G)(R):=Gr_R(G\otimes_k R,\mu_R)$$ 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 $$hom(G,D(H))=hom(H,D(G))=hom(G\times H,\mu_k)$$ ## p-torsion Let $s:R\to S$ be a morphism of rings. Then we have an [[nLab:adjunction]] $$(s^*\dashv s_*):S.Mod\stackrel{s_*}{\to} R.Mod$$ from the category of $S$-[[nLab:module|modules]] to that of $R$-modules where $$s^*:A\mapsto A\otimes_s S$$ is called *scalar extension* and $s_*$ is called *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 $X^{(p)}(R):=X(s_* R)$. ### p-divisible groups ## Witt rings and Dieudonné modules