basechanging

Base changing, or base change, is a construction in mathematics that replaces the base ring or base space of a given object with another via a morphism to the original base. It is a way to compare objects defined over different bases and to study their behavior in families; the idea appears in algebra, geometry, and number theory in related forms.

In algebra, if φ: R → R' is a ring homomorphism and M is an R-module, the base change of M to R' is the R'-module M ⊗_R R'. For vector spaces, if K → L is a field extension and V is a K-vector space, the base change is V ⊗_K L, a vector space over L. This construction is functorial and preserves many structural properties under suitable hypotheses.

In algebraic geometry, base change refers to forming the fiber product X_{S'} = X ×_S S' of a scheme X over a base S with a morphism S' → S, producing a scheme X_{S'} over S'. The operation also applies to sheaves: a sheaf F on X can be pulled back to X_{S'} to give a base-changed sheaf. Base change is central to the study of families of schemes and their fibers, and its behavior is governed by results such as flat base change, which provides compatibility under flat morphisms.

Base change also appears in other areas, notably Langlands base change in number theory, where automorphic representations are transferred along field extensions. The concept thus provides a unifying framework for changing the underlying base in various mathematical constructions.