Field of Definition - Definitions For Affine and Projective Varieties

Definitions For Affine and Projective Varieties

Results and definitions stated below, for affine varieties, can be translated to projective varieties, by replacing An(kalg) with projective space of dimension n − 1 over kalg, and by insisting that all polynomials be homogeneous.

A k-algebraic set is the zero-locus in An(kalg) of a subset of the polynomial ring k. A k-variety is a k-algebraic set that is irreducible, i.e. is not the union of two strictly smaller k-algebraic sets. A k-morphism is a regular function between k-algebraic sets whose defining polynomials' coefficients belong to k.

One reason for considering the zero-locus in An(kalg) and not An(k) is that, for two distinct k-algebraic sets X₁ and X₂, the intersections X₁∩An(k) and X₂∩An(k) can be identical; in fact, the zero-locus in An(k) of any subset of k is the zero-locus of a single element of k if k is not algebraically closed.

A k-variety is called a variety if it is absolutely irreducible, i.e. is not the union of two strictly smaller kalg-algebraic sets. A variety V is defined over k if every polynomial in kalg that vanishes on V is the linear combination (over kalg) of polynomials in k that vanish on V. A k-algebraic set is also an L-algebraic set for infinitely many subfields L of kalg. A field of definition of a variety V is a subfield L of kalg such that V is an L-variety defined over L.

Equivalently, a k-variety V is a variety defined over k if and only if the function field k(V) of V is a regular extension of k, in the sense of Weil. That means every subset of k(V) that is linearly independent over k is also linearly independent over kalg. In other words those extensions of k are linearly disjoint.

André Weil proved that the intersection of all fields of definition of a variety V is itself a field of definition. This justifies saying that any variety possesses a unique, minimal field of definition.