In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere.. Note that the cited definition that Hartshorne gives is slightly misleading. A locally Noetherian scheme is regular if all its local rings are regular, but it is not the case for schemes which are not locally Noetherian. See the cited Stacks Project page for more details. Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth..
For an example of a regular scheme that is not smooth, see .
See also Étale morphism Dimension of an algebraic variety Glossary of scheme theory Smooth completion
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