A regular scheme is a locally Noetherian scheme 
whose local ring 
is a regular local
ring for every point 
.
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
Subject classifications
