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...y in [[algebra]], an '''étale group scheme''' is a certain kind of [[group scheme]]. ...a [[field (mathematics)|field]] <math>K</math> is called an ''étale group scheme'' if it is represented by an [[étale algebra|étale K-algebra]] <math>\mathf ...

...example, a [[smooth variety]] is a variety that is geometrically [[regular scheme|regular]]. Given a scheme ''X'' that is of finite type over a field ''k'', the following are equivale ...

{{Short description|Generalization of equivalence classes to scheme theory}} {{about|a generalization to [[category theory]], used in [[scheme theory]]|the common meaning|Equivalence class}} ...

...Spec}(A)</math> is an [[affine scheme|affine]] [[group-scheme action|''G''-scheme]]. [[Category:Representation theory]] ...

...heory of Error-Correcting Codes'', Elsevier, New York, 1978.</ref> In this scheme <math>X=\mathcal{F}^n,</math> the set of binary vectors of length <math>n,< ...ics)|relation]] <math>R_i.</math> The <math>c_{ijk}</math> in a '''Hamming scheme''' are given by ...

...ry of group schemes based on the notion of group functor instead of scheme theory. == Group functor as a generalization of a group scheme == ...

...y, given a group ''S''-scheme ''G'', a '''left action of ''G'' on an ''S''-scheme ''X''''' is an ''S''-morphism ...d with a left or right action of a group scheme ''G'' is called a '''''G''-scheme'''. An [[equivariant morphism]] between ''G''-schemes is a [[morphism of sc ...

...eorem of absolute (cohomological) purity''' is an important theorem in the theory of [[étale cohomology]]. It states:<ref>A version of the theorem is stated *a [[regular scheme]] ''X'' over some base scheme, ...

By definition, ''X'' is a '''separated scheme''' over ''S'' (<math>p: X \to S</math> is a '''separated morphism''') if th ...n. Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism <math>X \rightarrow \textrm{Spec} (\ ...

...09}}</ref> as the main tool in the construction of the [[fundamental group scheme]]. Even if the definition is not intuitive there is a nice characterizatio Let <math>X</math> be a scheme and <math>V</math> a vector bundle on <math>X</math>. For <math>f = a_0 + a ...

[[File:Kinetic scheme.jpg|thumb|right| Figure 1. A kinetic scheme with 18 states]] ...esses generalized kinetic schemes are used. Figure 1 illustrates a kinetic scheme. ...

...bundle]] modulo some singularity. The notion is important both in [[scheme theory]] and [[complex algebraic geometry]]. ...domain|integral]] [[Noetherian scheme|noetherian]] [[Scheme (mathematics)|scheme]]. ...

In [[algebraic geometry]], the '''Behrend function''' of a scheme ''X'', introduced by [[Kai Behrend]], is a [[constructible function (mathem ...a quasi-projective proper moduli scheme carrying a [[symmetric obstruction theory]], then the '''weighted Euler characteristic''' ...

| thesis_title = The fundamental group-scheme ...= https://www.researchgate.net/publication/227215050_The_fundamental_group-scheme ...

...cal quotient for [[algebraic variety|varieties]] or [[scheme (mathematics)|scheme]]s. ...'' to be the category of varieties or the category of schemes over a fixed scheme. A categorical quotient <math>\pi</math> is a '''universal categorical quot ...

...dle]]s, whose naturally associated group scheme is the [[fundamental group scheme]] <math>\pi_1(X,x)</math>. Let <math>X</math> be a scheme over a field <math>k</math> and <math>V</math> a vector bundle on <math>X</ ...

...e ''S''-group schemes to itself. If ''G'' is a constant commutative group scheme, then its Cartier dual is the diagonalizable group ''D''(''G''), and vice v A finite commutative group scheme over a field corresponds to a finite dimensional commutative cocommutative ...

...[[moduli spaces]], in [[deformation theory]] and Fontaine's [[p-adic Hodge theory]], among others. The idea is to study some [[algebraic variety]] (or [[scheme (mathematics)|scheme]]) ''U'' which is [[smooth morphism|smooth]] but not necessarily [[proper m ...

Roughly speaking, for a variety or scheme ''X'', the idea is to consider étale coverings <math>U \rightarrow X</math> ...o the étale topology, in particular the [[étale fundamental group]] of the scheme and the [[étale cohomology]] of locally constant étale [[sheaf (mathematics ...

...dying [[algebraic curve]]s, one can study families of curves over any base scheme ''Y''. Indeed, the two approaches enrich each other. ...field ''k'' means an [[glossary of algebraic geometry|integral separated]] scheme of [[finite morphism#Morphisms of finite type|finite type]] over ''k''.<ref ...