basechange

Base change is a construction in mathematics that describes how an object defined over one base can be transported to a different base via a morphism between bases. The unifying idea is to form a pullback or extension of scalars, producing a new object over the new base in a way compatible with the original structure.

In algebraic geometry, base change is most commonly described for schemes. If X → S is a morphism of schemes and S' → S is another morphism, the base change of X along S' → S is the fiber product X' = X ×_S S', with projection X' → S'. This X' is called the base-changed scheme (often written X_S' or X ×_S S'). The construction is functorial in S' and preserves many geometric properties: if X → S is flat, smooth, or proper, then the base-changed X' → S' retains the corresponding property. It also preserves finite presentation and other finiteness conditions under suitable hypotheses.

In commutative algebra, base change appears through extension of scalars. For a ring homomorphism A → B and an A-module M, the base change of M to B is the B-module M ⊗_A B. This operation underlies changing the base ring for modules and, geometrically, corresponds to pulling back quasi-coherent sheaves along a base change of schemes.

In category theory, base change (or pullback along a morphism) is a general construction in fibered or indexed categories, producing a functor that reindexes objects over a new base object. This abstract viewpoint underpins many instances of base change across mathematics.

Base change is a fundamental tool in descent, cohomology computations, and the construction of families of objects parameterized by a base scheme or base category. See also fiber product, extension of scalars, and pullback.