- Group scheme
In

mathematics , a**group scheme**is agroup object in thecategory of schemes . That is, it is a scheme "G" with the equivalent properties* there is a

group law expressible as a multiplication μ and inversion map ι on "G"; or

* "G" is afunctor (as in theYoneda lemma ) mapping to thecategory of groups , rather than just sets.There are numerous examples familiar in algebra, including the

general linear group , andelliptic curve s. Those are both examples of group varieties (the scheme involved is respectively anaffine variety and aprojective variety ).In the case of

matrix multiplication , it is obvious that the multiplication formula ispolynomial in the entries. To take care ofmatrix inversion also, one should note that as a variety the "n"×"n" "invertible" matrices should be considered as having:"n"

^{2}+ 1coordinates "X

_{ij}" and "Y" subject to the equation:"det(X

_{ij})Y − 1 = 0".Then

Cramer's rule shows that matrix inversion is polynomial in the "X_{ij}" and "Y". That takes care of the first equivalent property; the second is essentially the well-known fact that matrices can have entries from anycommutative ring , and still give a ring, from which we may take the group of units.The theory of commutative group varieties occupies the place in theory that was investigated in

nineteenth century mathematics, in the search for the most generaladdition theorem s. Everyalgebraic torus andabelian variety is part of the theory, as aregroup extension s formed from both kinds of object (which have been called "quasi-abelian varieties"), used in the theory ofdifferential form s (of the "second kind" and "third kind", in classical terminology), and the geometric forms ofclass field theory .Group schemes are a source of examples of schemes that are not reduced, that is, have

nilpotent 'functions' on them (other than 0). Examples can be found overfinite field s: if "F" is a field of characteristic "p", as the kernels of endomorphisms.For example, the

**additive group**is just the functor sending a ring "R" to "R"^{+}, the underlying additive group of "R". Over "F", it is represented by:"Spec(F [X] )"

with μ the mapping coming from

:"F [X] → F [X,Y] ", "X → X + Y".

(See

spectrum of a ring for the duality here.)Now compute the kernel of the "p"-th power map, i.e.

:"x → x

^{p}"as a

fiber product in the category of schemes (dually, atensor product of R-algebras .) It turns out to be:"Spec(F [t] /t

^{p}F [t] )".The class of "t" is nilpotent, in the underlying ring.

This phenomenon was noticed in the

duality theory of abelian varieties , which in characteristic "p" was seen not to be something easily expressed in terms of variety theory alone. The finer aspects were first worked out by Pierre Cartier.**References***cite book

last = Demazure

first = Michel

authorlink =

coauthors =Alexandre Grothendieck , eds.

title = Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 1 (Lecture notes in mathematics**151**)

year = 1970

publisher = Springer-Verlag

location = Berlin; New York

language = French

pages = xv+564

*cite book

last = Demazure

first = Michel

authorlink =

coauthors =Alexandre Grothendieck , eds.

title = Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 2 (Lecture notes in mathematics**152**)

year = 1970

publisher = Springer-Verlag

location = Berlin; New York

language = French

pages = ix+654

*cite book

last = Demazure

first = Michel

authorlink =

coauthors =Alexandre Grothendieck , eds.

title = Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 3 (Lecture notes in mathematics**153**)

year = 1970

publisher = Springer-Verlag

location = Berlin; New York

language = French

pages = vii+529*

Jean-Pierre Serre , "Groupes algébriques et corps des classes".

*Wikimedia Foundation.
2010.*

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