category theoretic aspects of the theory of group schemes (Rev #10, changes)

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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~~ latte~~ latter denotes the category of affine 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 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 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 $f\in I$.

$Sch_k$ is 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 group object in $k.Sch$. The category of $k$-group schemes we denote by $k.Grp$.

Let $k$ be a field. A *finite $k$-ring* is defined to be a $k$-algebra which is a finite dimensional $k$-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 $k$-corings. A *coring* is a 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.

(see also 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 $k$.

Let $G$ be a commutative $k$-group functor (in cases of interest this is a finite flat commutative 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 duality since we have

$hom(G,D(H))=hom(H,D(G))=hom(G\times H,\mu_k)$

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

$(s^*\dashv s_*):S.Mod\stackrel{s_*}{\to} R.Mod$

from the category of $S$-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 scheme over a $k$-ring for $k$ being a field of characteristic $p$, we define its $p$-torsion component-wise by $X^{(p)}(R):=X(s_* R)$.