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Geometrically Regular Scheme

A geometrically regular scheme over a field k is a scheme X such that the base change X×_(k)k^' is a regular scheme for every finitely generated extension field k^' of k.

If the structure morphism of X is a morphism of finite type, then X is geometrically regular exactly when that structure morphism is a smooth morphism.

See also

Base Change, Extension Field, Morphism of Finite Type, Regular Scheme, Smooth Morphism, Structure Morphism

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References

Grothendieck, A. and Dieudonné, J. "Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, seconde partie." Publ. Math. IHES 24, 5-231, 1965. https://doi.org/10.1007/BF02684322.The Stacks Project Authors. "Geometrically Regular Schemes." §33.12 in The Stacks Project, Tag 038S, 2026. https://stacks.math.columbia.edu/tag/038S.

Cite this as:

Weisstein, Eric W. "Geometrically Regular Scheme." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeometricallyRegularScheme.html

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