Regular function


Regular function

:"In complex analysis, see holomorphic function."In mathematics, a regular function in the sense of algebraic geometry is an everywhere-defined, polynomial function on an algebraic variety "V" with values in the field "K" over which "V" is defined.

For example, if "V" is the affine line over "K", the regular functions on "V" make up a commutative ring, under pointwise multiplication of functions, isomorphic with the polynomial ring in one variable over "K". In other words, the regular functions are just polynomials in some natural parameter on the affine line.

More generally, for any affine variety "V", the regular functions make up the coordinate ring of "V", often written "K" ["V"] . This can be expressed in other ways. A regular function is the same as a morphism to the affine line, or in the language of scheme theory a global section of the structure sheaf.

The reason for looking at regular functions becomes more apparent when one allows "V" to be a projective variety. Then regular functions on "V" become rare. For example morphisms from a projective space to the affine line must be constant: regular functions on a projective space are constant functions. The same is true for any connected projective variety (this can be viewed as an algebraic analogue of Liouville's theorem in complex analysis).

In fact taking the function field "K"("V") of an irreducible algebraic curve "V", the functions "F" in the function field may all be realised as morphisms from "V" to the projective line over "K". The image will either be a single point, or the whole projective line (this is a consequence of the completeness of projective varieties). That is, unless "F" is actually constant, we have to attribute to "F" the value ∞ at some points of "V". Now in some sense "F" is no worse behaved at those points than anywhere else: ∞ is just the chosen point at infinity on the projective line, and by using a Möbius transformation we can move it anywhere we wish. But it is in some way inadequate to the needs of geometry to use only the affine line as target for functions, since we shall end up only with constants.

For those reasons, the larger class of rational functions are constantly used in algebraic geometry. For the needs of birational geometry, more generally, morphisms are replaced with morphisms defined on open dense subsets. This brings fresh phenomena in dimension ≥ 1.


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