Regular function

Regular functions

In the particular case that ${\displaystyle Y}$ equals ${\displaystyle \mathbb {A} ^{1}}$ the regular maps ${\displaystyle f:X\rightarrow \mathbb {A} ^{1}}$ are called regular functions, and are algebraic analogs of smooth functions studied in differential geometry. The ring of regular functions (that is the coordinate ring or more abstractly the ring of global sections of the structure sheaf) is a fundamental object in affine algebraic geometry. The only regular function on a projective variety is constant (this can be viewed as an algebraic analogue of Liouville's theorem in complex analysis).