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Base Change

A base change of a scheme X over a scheme S along a morphism S^'->S is the fiber product

X_(S^')=S^'×_(S)X,

regarded as a scheme over S^'.

If f:X->Y is a morphism of schemes over S, its base change is the morphism f_(S^'):X_(S^')->Y_(S^') induced by f. Many properties of morphisms, including being an open immersion, a closed immersion, a proper morphism, or a smooth morphism, are preserved by base change.

See also

Closed Immersion, Fiber Product, Open Immersion, Proper Morphism, Scheme, Smooth Morphism, Structure Morphism

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References

Grothendieck, A. and Dieudonné, J. "Éléments de géométrie algébrique. I. Le langage des schémas." Publ. Math. IHES 4, 5-228, 1960. https://doi.org/10.1007/BF02684778.The Stacks Project Authors. "Base Change in Algebraic Geometry." §26.18 in The Stacks Project, Tag 01JX, 2026. https://stacks.math.columbia.edu/tag/01JX.

Cite this as:

Weisstein, Eric W. "Base Change." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BaseChange.html

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