Ring of regular functions

Structure sheaf

Given an affine variety X with coordinate ring A, the sheaf of k-algebras ${\displaystyle {\mathcal {O}}_{X}}$ is defined by letting ${\displaystyle {\mathcal {O}}_{X}(U)=\Gamma (U,{\mathcal {O}}_{X})}$ be the ring of regular functions on U. Since X is irreducible, the ring of regular functions can be defined as a subring of K, the field of fractions of the coordinate ring A. An element of ${\displaystyle \varphi \in K}$ is formally a fraction ${\displaystyle b/a}$ with ${\displaystyle a,b\in A,}$ with ${\displaystyle b/a=b'/a'}$ iff ${\displaystyle ab'=a'b.}$ It is regular at${\displaystyle x\in X}$ if it can be written as ${\displaystyle b/a}$ with ${\displaystyle a,b\in A}$ and ${\displaystyle a(x)\neq 0.}$ Moreover, ${\displaystyle \varphi }$ can be regarded as a rational function on the subset of points of X at which it is regular: if ${\displaystyle \varphi =b/a}$ is regular at x, its value there, ${\displaystyle \varphi (x),}$ is defined as ${\displaystyle b(x)/a(x).}$