In algebraic geometry, a regular scheme is a scheme whose local rings are regular everywhere.^{[1]} Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth.^{[2]}
For an example of a regular scheme that is not smooth, see Geometrically regular ring#Examples.
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References
 ^ Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, p. 238, ISBN 9780387902449.
 ^ Demazure, Michel (1980), Introduction to algebraic geometry and algebraic groups, NorthHolland Mathematics Studies, 39, NorthHolland, Proposition 3.2, p. 168, ISBN 9780080871509.