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.
^Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer, p. 238, ISBN 9780387902449. 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.
^Demazure, Michel (1980), Introduction to algebraic geometry and algebraic groups, North-Holland Mathematics Studies, vol. 39, North-Holland, Proposition 3.2, p. 168, ISBN 9780080871509.
{O}}_{X,x}} of germs at x is regular. (See also: regularscheme.) Regular local rings are not related to von Neumann regular rings. For Noetherian local...
match the regular expression. The picture shows the NFA scheme N(s*) obtained from the regular expression s*, where s denotes a simpler regular expression...
particular, a scheme X is smooth over a field k if and only if the morphism X → Spec k is smooth. A smooth scheme over a field is regular and hence normal...
result is true for schemes. So, for example, every normal curve is regular. Any reduced scheme X has a unique normalization: a normal scheme Y with an integral...
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simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism. Contents: ...
base field. Geometrically regular schemes are defined in a similar way. In older terminology, points with regular local rings were called simple points...
These characters appear in the American animated television series Regular Show, created by J. G. Quintel for Cartoon Network. The series revolves around...
closed immersion i : X ↪ Y {\displaystyle i:X\hookrightarrow Y} of schemes is a regular embedding of codimension r if each point x in X has an open affine...
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birational map f : Y → X {\displaystyle f\colon Y\rightarrow X} from a regularscheme Y {\displaystyle Y} such that the higher direct images of f ∗ {\displaystyle...
It states: given a regularscheme X over some base scheme, i : Z → X {\displaystyle i:Z\to X} a closed immersion of a regularscheme of pure codimension...
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in other countries. This is done by granting some exemptions in the regularscheme of examination and training. ICAI currently has MRAs with following...
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divisorial schemes is quite large: it includes affine schemes, separated regular (noetherian) schemes and subschemes of a divisorial scheme (such as projective...
\mathbb {P} _{U}^{n}} and transition automorphisms are linear. Over a regularscheme S such as a smooth variety, every projective bundle is of the form P...
incorporated more of Jomala territory. Mariehamn was built according to a very regularscheme which is well-preserved. One of the oldest streets is Södragatan where...
library for regular expressions (regex), and an interface for Portable Operating System Interface (POSIX). It is supported by the portable Scheme library...
In geometric language, if X is an affine scheme and r1, ..., rd is a regular sequence in the ring of regular functions on X, then we say that the closed...
The Robodebt scheme was an unlawful method of automated debt assessment and recovery implemented under the Liberal-National Coalition governments of Tony...
Old Pension Scheme (OPS) in India was abolished as a part of pension reforms by Union Government. Repealed from 1 January 2004, it had a defined-benefit...