In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology.
algebraic geometry, a smoothscheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making...
regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smoothscheme is regular, and every regular scheme of finite...
points on a wave-like function Smoothscheme – type of schemePages displaying wikidata descriptions as a fallback Smooth number – Integer having only small...
following case: suppose we have an embedding of a scheme X {\displaystyle X} into a smoothscheme Y {\displaystyle Y} i : X ↪ Y {\displaystyle i:X\hookrightarrow...
geometry, a morphism f : X → S {\displaystyle f:X\to S} between schemes is said to be smooth if (i) it is locally of finite presentation (ii) it is flat,...
cycles is isomorphic to H2i(X,Z(i)) when X is smooth over k. The motivic cohomology Hi(X, Z(j)) of a smoothscheme X over k is the cohomology of X in the Zariski...
differentials. Let i : X → Y {\displaystyle i:X\to Y} be a closed embedding of smoothschemes in Sch / S {\displaystyle {\text{Sch}}/S} . Using the exact triangle...
properties are much the same as in the special case of smoothschemes. For a Gorenstein scheme X of finite type over a field, f: X → Spec(k), the dualizing...
Resolution of singularities Singular point of a curve Singularity theory Smoothscheme Zariski tangent space Hartshorne, Robin (1977). Algebraic Geometry....
that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality...
In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data...
When X {\displaystyle X} is smooth over k {\displaystyle k} (or more generally, a locally Noetherian normal factorial scheme ), this is isomorphic to the...
cohomology only works well for smooth proper schemes. Rigid cohomology extends it to more general schemes. For schemes in characteristic p, crystalline...
rank n {\displaystyle n} . If Y {\displaystyle Y} is a smooth closed subscheme of a smoothscheme X {\displaystyle X} over k {\displaystyle k} , then there...
generated k-algebra A is 0-smooth over k if and only if Spec A is a smoothscheme over k. A separable algebraic field extension L of k is 0-étale over...
of smooth curves in the Hilbert scheme. "Hilbert scheme" sometimes refers to the punctual Hilbert scheme of 0-dimensional subschemes on a scheme. Informally...
version works for schemes with mild singularities, Cohen–Macaulay schemes, not just smoothschemes. Namely, for a Cohen–Macaulay scheme X of pure dimension...
to be smooth of dimension n - j if there is a smooth presentation U → X {\displaystyle U\to X} of relative dimension j for some smoothscheme U of dimension...
p-forms Ω X p ( log D ) {\displaystyle \Omega _{X}^{p}(\log D)} on a smoothscheme X over a field, with respect to a divisor D = ∑ D j {\displaystyle D=\sum...
the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally...
generalization to projective, but not necessarily smoothschemes is known as Verdier duality. For a (smooth projective) curve X, H2 and higher vanish for...
discrete or continuous: Continuous: Color schemes that have a smooth color gradient. Continuous color schemes are intended to display sets of continuous...
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