k-ring of functions on a k-functor

Recall that the affine line ${O}_{k}={M}_{k}(k[t],-)$ is an affine $k$-scheme. Let ${M}_{k}$ denote the category of $k$-rings.

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.

There is an adjoint equivalence

$$(\mathrm{Sp}\u22a3O):{\mathrm{Sch}}_{\mathrm{aff}}\stackrel{O}{\to}{\mathrm{Ring}}_{k}$$

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

Revised on June 5, 2012 17:15:43
by Stephan Alexander Spahn
(178.195.231.138)