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Regular Scheme

A regular scheme is a locally Noetherian scheme X whose local ring O_(X,x) is a regular local ring for every point x in X.

Every open subscheme of a regular scheme is regular. For an algebraic variety over a perfect field, regularity is equivalent to being a smooth variety.

See also

Algebraic Variety, Local Ring, Open Subscheme, Perfect Field, Regular Local Ring, Scheme, Smooth Variety

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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. "Regular Schemes." §28.9 in The Stacks Project, Tag 02IR, 2026. https://stacks.math.columbia.edu/tag/02IR.

Cite this as:

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

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