nLab
Demazure, lectures on p-divisible groups, I.1, k-functors

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

Definition

(k-ring, k-functor,affine k-scheme)

For a ring $k$ the category of $k$-rings, denoted by $M_k:=k/\mathrm{Ring}$ is defined to be the category of commutative associative $k$-algebras with unit which are rings. This is equivalently the category of pairs $(R,f:k\to R)$ where $R$ is a Ring and $f$ is a morphism of $k$-algebras.

The category of $k$-functors, denoted by $\mathrm{co}\mathrm{Psh}(M_k)$, is defined to be the category of covariant functors $M_k\to \mathrm{Set}$.

The forgetful functor $O_k:R\to R$ sending a $k$-ring to its underlying set is called affine line.

For the full and faithful contravariant functor

$${\mathrm{Sp}}_k:\{\begin{array}{lll}{M}_k& \to & \mathrm{co}\mathrm{Psh}(M_k)\\ A& \to & M_k(A,-)\end{array}$$Sp_k:\begin{cases} M_k&\to& co Psh(M_k) \\ A&\to& M_k(A,-) \end{cases}

${\mathrm{Sp}}_{k}A$ (and every isomorphic functor) is called an affine $k$-scheme. (In most modern texts one uses the notation ”$\mathrm{Spec}$” instead of ”$\mathrm{Sp}$”.) ${\mathrm{Sp}}_k$ restricts to an equivalence between the categories of $k$-rings and the category $\mathrm{Aff}{\mathrm{Sch}}_k$ of affine $k$-schemes. We think of this category as of $M_k^{\mathrm{op}}$. The functor ${\mathrm{Sp}}_k$ commutes with limits and skalar extension (see below). Consequently $\mathrm{Aff}{\mathrm{Sch}}_k$ is closed under limits and base change.

The affine line $O_k=M_k(k[t],-)$ is an affine $k$-scheme.

A function on a $k$-scheme $X$ is defined to be an object $f\in O(X):=\mathrm{co}\mathrm{Psh}(M_k)(X,O_k)$. $O(X)$ is a $k$-ring by component-wise addition and -multiplication.

Proposition

There is an adjoint equivalence

$$(\mathrm{Sp}⊣O):{\mathrm{Sch}}_{\mathrm{aff}}\stackrel{O}{\to }{\mathrm{Ring}}_k$$(Sp\dashv O):Sch_{aff}\stackrel{O}{\to}Ring_k

of the categories of affine k-schemes and $k$-rings.

Remark

The category of $k$-functors has limits.

The terminal object is $e:R\mapsto \{\varnothing \}$. Products and pullbacks are computed component-wise.

Remark

For $\varphi :k\to k^{\prime }$ the ”base change” functor $(-){\otimes }_k{k}^{\prime }:\mathrm{co}\mathrm{Psh}(M_k)\to \mathrm{co}\mathrm{Psh}(M_{k^{\prime }})$ induced by $(-)\circ \varphi :M_k\to M_{k^{\prime }}$ given by postcompositions with $\varphi$ is called skalar extension.