Regular scheme
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In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere.[1][2] Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth.[3]
For an example of a regular scheme that is not smooth, see Geometrically regular ring § Examples.
See also
[edit | edit source]References
[edit | edit source]- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).. 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.
- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value)..
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