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...

points on a wave-like function Smooth scheme Smooth number – Integer having only small prime factors (number theory) Smoothing – Fitting an approximating...

point is said to be non-singular or smooth. The concept is generalized to smooth schemes in the modern language of scheme theory. A plane curve defined by...

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,...

is a scheme X that is locally of finite type and regular over k. 3.  A smooth scheme over a field k is a scheme X that is geometrically smooth: X × k...

regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite...

cycles is isomorphic to H2i(X,Z(i)) when X is smooth over k. The motivic cohomology Hi(X, Z(j)) of a smooth scheme 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 smooth schemes in Sch / S {\displaystyle {\text{Sch}}/S} . Using the exact triangle...

that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality...

following case: suppose we have an embedding of a scheme X {\displaystyle X} into a smooth scheme Y {\displaystyle Y} i : X ↪ Y {\displaystyle i:X\hookrightarrow...

version works for schemes with mild singularities, Cohen–Macaulay schemes, not just smooth schemes. Namely, for a Cohen–Macaulay scheme X of pure dimension...

functor from the category S m S {\displaystyle Sm_{S}} of smooth S {\displaystyle S} -schemes towards an infinity category which satisfies Nisnevich descent...

When X {\displaystyle X} is smooth over k {\displaystyle k} (or more generally, a locally Noetherian normal factorial scheme ), this is isomorphic to the...

p-forms Ω X p ( log ⁡ D ) {\displaystyle \Omega _{X}^{p}(\log D)} on a smooth scheme X over a field, with respect to a divisor D = ∑ D j {\displaystyle D=\sum...

cohomology only works well for smooth proper schemes. Rigid cohomology extends it to more general schemes. For schemes in characteristic p, crystalline...

properties are much the same as in the special case of smooth schemes. For a Gorenstein scheme X of finite type over a field, f: X → Spec(k), the dualizing...

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...

rank n {\displaystyle n} . If Y {\displaystyle Y} is a smooth closed subscheme of a smooth scheme X {\displaystyle X} over k {\displaystyle k} , then there...

scheme Scheme Éléments de géométrie algébrique Grothendieck's Séminaire de géométrie algébrique Fiber product of schemes Flat morphism Smooth scheme Finite...

comparison isomorphisms in arithmetic and complex geometry: If X is a proper smooth scheme over C, there is a classical comparison isomorphism between the algebraic...

generated k-algebra A is 0-smooth over k if and only if Spec A is a smooth scheme over k. A separable algebraic field extension L of k is 0-étale over...

generalization to projective, but not necessarily smooth schemes is known as Verdier duality. For a (smooth projective) curve X, H2 and higher vanish for...

Levine. In more precise terms, a theorem of Voevodsky implies: for a smooth scheme X over a field and integers p, q, there is a natural isomorphism H p...

to algebraic vector bundles over a smooth scheme. But, there is an alternative construction for any Noetherian scheme X {\displaystyle X} . If we look at...

In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking...

of smooth curves in the Hilbert scheme. "Hilbert scheme" sometimes refers to the punctual Hilbert scheme of 0-dimensional subschemes on a scheme. Informally...

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 smooth scheme U of dimension...

D_{\mathbb {A} _{R}^{n}/R}} . More generally, let X {\displaystyle X} be a smooth scheme over a ring R {\displaystyle R} . Locally, X → R {\displaystyle X\to...

where f is a proper map between two smooth schemes. Generalisations of the theorem can be made to the non-smooth case by considering an appropriate generalisation...

descent results mentioned imply that a scheme X over k is smooth over k if and only if the base change XE is smooth over E. The same goes for properness...