Elements de Mathematique. Algebre. Chapitre 9 by N. Bourbaki

By N. Bourbaki

Formes sesquilin?©aires et formes quadratiques.

Les ?‰l?©ments de math?©matique de Nicolas BOURBAKI ont pour objet une pr?©sentation rigoureuse, syst?©matique et sans pr?©requis des math?©matiques depuis leurs fondements.

Ce neuvi??me chapitre du Livre d Alg??bre, deuxi??me Livre du trait?©, est consacr?© aux formes quadratiques, symplectiques ou hermitiennes et aux groupes associ?©s.

Il contient ?©galement une observe historique.

Ce quantity est une r?©impression de l ?©dition de 1959.

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Si−1 /Si , Si+1 /Si , . . , Sn /Si ), which is clearly a homogeneous polynomial in S0 , . . , Sn . Now, if X in An is the zero set of polynomials f1 , . . , fm ∈ k[T0 , . . , Ti−1 , Ti+1 , . . , Tn ], then α−1 (X) = Ui ∩ Z(fˆ1 , . . , fˆm ). We note in passing, that Z(fˆ1 , . . ). Conversely, to each homogeneous polynomial g(S0 , . . , Sn ) we associate the polynomial g¯(T0 , . . , Ti−1 , Ti+1 , . . , Tn ) := g(T0 , . . , Ti−1 , 1, Ti+1 , . . , Tn ). Now α(Z(g1 , . . , gl ) ∩ Ui ) = Z(¯ g1 , .

A subset of a topological space is called locally closed if it is an intersection of an open set and a closed set. It follows from above that a locally closed subset of a prevariety is again a prevariety. We will refer to the locally closed subsets as subprevarieties. 4 (Affine Criterion) Let X, Y be prevarieties, and ϕ : X → Y be a map. Assume that there is an affine open covering Y = ∪i∈I Vi and an open covering X = ∪i∈I Ui such that (i) ϕ(Ui ) ⊂ Vi for each i ∈ I; (ii) f ◦ ϕ ∈ OX (Ui ) whenever f ∈ OY (Vi ).

Ir − 1, . . , in ), which is true by induction. in ui00 . . uinn be a form of degree m and H be a hypersurface in Pn defined by the equation F = 0. in = 0. Let us now concentrate on the special case α3 : P1 → P3 : (a0 : a1 ) → (a30 : a20 a1 : a0 a21 : a31 ). The corresponding Veronese variety C is called the twisted cubic. 13) where 2 F0 = v30 v12 − v21 , F1 = v21 v12 − v30 v03 , 2 F2 = v21 v03 − v12 . The twisted cubic consists of all points of the form (1 : c : c2 : c3 ) for c ∈ k together with the point (0 : 0 : 0 : 1).

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