nLab
k-ring of functions on a k-functor

Definition

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.

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.

Related concepts

ringed space