# nLab k-ring of functions on a k-functor

###### Definition

Recall that the affine line ${O}_{k}={M}_{k}\left(k\left[t\right],-\right)$ 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 $f\in O\left(X\right):=\mathrm{co}\mathrm{Psh}\left({M}_{k}\right)\left(X,{O}_{k}\right)$. $O\left(X\right)$ is a $k$-ring by component-wise addition and -multiplication.

###### Proposition

$\left(\mathrm{Sp}⊣O\right):{\mathrm{Sch}}_{\mathrm{aff}}\stackrel{O}{\to }{\mathrm{Ring}}_{k}$(Sp\dashv O):Sch_{aff}\stackrel{O}{\to}Ring_k
of the categories of affine k-schemes and $k$-rings.