By T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich (eds.)
The difficulties being solved through invariant idea are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of assorted is nearly an identical factor, projective geometry. gadgets of linear algebra or, what Invariant thought has a ISO-year historical past, which has noticeable alternating sessions of development and stagnation, and adjustments within the formula of difficulties, equipment of answer, and fields of program. within the final twenty years invariant thought has skilled a interval of development, encouraged by means of a prior improvement of the speculation of algebraic teams and commutative algebra. it really is now seen as a department of the idea of algebraic transformation teams (and less than a broader interpretation might be pointed out with this theory). we are going to freely use the speculation of algebraic teams, an exposition of which might be stumbled on, for instance, within the first article of the current quantity. we are going to additionally imagine the reader understands the elemental strategies and easiest theorems of commutative algebra and algebraic geometry; whilst deeper effects are wanted, we are going to cite them within the textual content or supply appropriate references.
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Additional resources for Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory
Let Y be the flag variety G/B and denote by n: G -+ Y the canonical map. For any p. ) as follows. )(t- 1 )f(x) for x E n-1U, t E T, U E RuB}. ». ». Here the important result is Kempf's vanishing theorem. Theorem. If p. » = 0 for i "# O. If char(k) = 0 this is a particular case of an earlier theorem of Bott. This states that for arbitrary p. » "# O. This I. Linear Algebraic Groups 45 i can be described precisely. Bott's theorem is not true in positive characteristics. For a discussion of these matters see [J, II.
For the case of the root datum of semi-simple groups see [B4]. Another proof of the theorem can be found in [SGA 3, expo XXV]. This proof is more algebrogeometric, but perhaps less accessible. There is a counterpart of the uniqueness result for group schemes over 7L [SGA 3, expo XXIII]. It can be deduced from the result over algebraically closed fields. 5. 1. 4 one obtains a classification of quasi-simple groups. This classification is independent of the algebraically closed ground field k. It is "the same" as the Killing-Cartan classification of simple Lie algebras over 2. Let X and Y be two affine F -varieties. 1, qJ*F[X] = F[Y]. e. the homomorphisms F[X] -+ F are the Frational points of X. The set of these is denoted by X (F). If qJ: Y -+ X is an F -morphism then qJ(Y(F)) c X(F). If Y is a subvariety of X, it is said to be defined over F or to be an F-subvariety if the inclusion map Y -+ X is defined over F. 3. 3). 1. 4. The product of two affine F -varieties X, Y has a canonical F -structure, viz. F[X x Y] = F[X] ®FF[Y]. 5. If X is an affine F -variety and x E X(F) then the tangent space YxX has a canonical F -structure.
Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory by T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich (eds.)
2. Let X and Y be two affine F -varieties. 1, qJ*F[X] = F[Y]. e. the homomorphisms F[X] -+ F are the Frational points of X. The set of these is denoted by X (F). If qJ: Y -+ X is an F -morphism then qJ(Y(F)) c X(F). If Y is a subvariety of X, it is said to be defined over F or to be an F-subvariety if the inclusion map Y -+ X is defined over F. 3. 3). 1. 4. The product of two affine F -varieties X, Y has a canonical F -structure, viz. F[X x Y] = F[X] ®FF[Y]. 5. If X is an affine F -variety and x E X(F) then the tangent space YxX has a canonical F -structure.