Field of Definition - Scheme-theoretic Definitions

Scheme-theoretic Definitions

One advantage of defining varieties over arbitrary fields through the theory of schemes is that such definitions are intrinsic and free of embeddings into ambient affine n-space.

A k-algebraic set is a separated and reduced scheme of finite type over Spec(k). A k-variety is an irreducible k-algebraic set. A k-morphism is a morphism between k-algebraic sets regarded as schemes over Spec(k).

To every algebraic extension L of k, the L-algebraic set associated to a given k-algebraic set V is the fiber product V ×_Spec(k) Spec(L). A k-variety is absolutely irreducible if the associated kalg-algebraic set is an irreducible scheme; in this case, the k-variety is called a variety. An absolutely irreducible k-variety is defined over k if the associated kalg-algebraic set is a reduced scheme. A field of definition of a variety V is a subfield L of kalg such that there exists a k∩L-variety W such that W ×_Spec(k∩L) Spec(k) is isomorphic to V and the final object in the category of reduced schemes over W ×_Spec(k∩L) Spec(L) is an L-variety defined over L.

Analogously to the definitions for affine and projective varieties, a k-variety is a variety defined over k if the stalk of the structure sheaf at the generic point is a regular extension of k; furthermore, every variety has a minimal field of definition.

One disadvantage of the scheme-theoretic definition is that a scheme over k cannot have an L-valued point if L is not an extension of k. For example, the rational point (1,1,1) is a solution to the equation x₁ + ix₂ - (1+i)x₃ but the corresponding Q-variety V has no Spec(Q)-valued point. The two definitions of field of definition are also discrepant, e.g. the (scheme-theoretic) minimal field of definition of V is Q, while in the first definition it would have been Q. The reason for this discrepancy is that the scheme-theoretic definitions only keep track of the polynomial set up to change of basis. In this example, one way to avoid these problems is to use the Q-variety Spec(Q/(x₁2+ x₂2+ 2x₃2- 2x₁x₃ - 2x₂x₃)), whose associated Q-algebraic set is the union of the Q-variety Spec(Q/(x₁ + ix₂ - (1+i)x₃)) and its complex conjugate.