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

This entry is about a section of the text

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

For a ring $k$ the *category of $k$-rings*, denoted by $M_k:=k/Ring$ 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 $co Psh (M_k)$, is defined to be the category of covariant functors $M_k\to Set$.

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

$Sp_k:\begin{cases}
M_k&\to& co Psh(M_k)
\\
A&\to& M_k(A,-)
\end{cases}$

$Sp_k A$ (and every isomorphic functor) is called an *affine $k$-scheme*. (In most modern texts one uses the notation $Spec$” instead of $Sp$”.) $Sp_k$ restricts to an equivalence between the categories of $k$-rings and the category $Aff Sch_k$ of affine $k$-schemes. We think of this category as of $M_k^{op}$. The functor $Sp_k$ commutes with limits and skalar extension (see below). Consequently $Aff Sch_k$ 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 $f\in O(X):=co Psh (M_k)(X,O_k)$. $O(X)$ is a $k$-ring by component-wise addition and -multiplication.

There is an adjoint equivalence

$(Sp\dashv O):Sch_{aff}\stackrel{O}{\to}Ring_k$

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

The category of $k$-functors has limits.

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

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

Revised on June 1, 2012 16:09:47
by Stephan Alexander Spahn
(178.195.221.252)