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Definition

Base change

Pulling an object over a base back along a morphism to a new base.

For morphisms X→SX\to S and S′→SS'\to S, the base change of XX along S′→SS'\to S is the morphism

XS′:=X×SS′⟶S′.X_{S'}:=X\times_S S'\longrightarrow S'.

It is formed using the . A morphism f:X→Yf:X\to Y over SS similarly pulls back to

fS′:X×SS′⟶Y×SS′.f_{S'}:X\times_S S'\longrightarrow Y\times_S S'.

On , if R→R′R\to R' changes the base and X=Spec⁡AX=\operatorname{Spec}A, then

XR′≅Spec⁡(A⊗RR′).X_{R'}\cong\operatorname{Spec}(A\otimes_R R').
Remarks

Base change is a categorical pullback, not merely a substitution of coordinates or a restriction of underlying topological spaces. Whether a geometric property is preserved by base change must be checked for that property.