Algebraic Geometry IV: Linear Algebraic Groups Invariant by T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich

By T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich (eds.)

The difficulties being solved by means of invariant conception are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of assorted is sort of a similar factor, projective geometry. items of linear algebra or, what Invariant conception has a ISO-year heritage, which has visible alternating sessions of development and stagnation, and alterations within the formula of difficulties, equipment of resolution, and fields of program. within the final 20 years invariant concept has skilled a interval of progress, encouraged through a prior improvement of the idea of algebraic teams and commutative algebra. it really is now considered as a department of the idea of algebraic transformation teams (and less than a broader interpretation might be pointed out with this theory). we'll freely use the idea of algebraic teams, an exposition of that are chanced on, for instance, within the first article of the current quantity. we are going to additionally suppose the reader is aware the elemental recommendations and least difficult theorems of commutative algebra and algebraic geometry; while deeper effects are wanted, we'll cite them within the textual content or offer compatible references.

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One then checks that the number of different groups of block matrices also equals that number. 2. There is another useful description of parabolic subgroups. e. a homomorphism of algebraic groups k* --+ G. Define a k*-action on G by t. (tf 1 (t E k*, x E G). Proposition. (i) The set of x E G such that lim t _ o t. ) of G. The centralizer of 1m ). is a Levi subgroup of P(A). ~~ Lx = e}. ). 5. The proof comes fairly easily from the above description of parabolic subgroups. 5. Generalized Schubert Varieties.

The irreducible components of X are defined over Fs. Corollary 2. Let qJ: X -+ Y be an F-morphism. Then qJ(X) is defined over F. A. 4. Proposition. Let Y and Z be closed F-subvarieties of X. The intersection Y n Z is defined over F if one of the following conditions holds: (a) F is perfect, (b) there is a dense subset U of Y n Z such that for all x E U we have TAY n Z) = Tx Y n TxZ. Corollary. Let cp: X -+ Y be an F-morphism. If y E Y(F) the fiber cp-ly is defined over F if one of the following holds: (a) F is perfect, (b) all irreducible components of cp-ly have dimension dim X - dim Y and there exists in any such component a smooth point x E X such that the tangent map (dcp)x: TxX -+ 1'y Y is surjective.

Let H be a linear algebraic group over k. Then H is reductive if and only if any rational representation of H is fully reducible. 3, it works in arbitrary characteristic). To prove the "only if"-part one first observes (this is also elementary) that it suffices to establish: if ¢J: H -+ GL(V) is a rational representation of the reductive group H and if v E V is a non-zero fixed vector, there exists an H -stable hyperplane in V which does not contain v. 4. One establishes that if t/I is an arbitrary rational representation of H then C defines a linear map t/I(C) of the underlying vector space which commutes with t/I( G).

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