By Walter Ferrer Santos, Alvaro Rittatore

ISBN-10: 082475896X

ISBN-13: 9780824758967

ISBN-10: 1420030795

ISBN-13: 9781420030792

This self-contained creation to geometric invariant thought hyperlinks the idea of affine algebraic teams to Mumford's conception. The authors, professors of arithmetic at Universidad de l. a. República, Uruguay, take advantage of the point of view of Hopf algebra idea and the speculation of comodules to simplify the various correct formulation and proofs. Early chapters evaluation must haves in commutative algebra, algebraic geometry, and the idea of semisimple Lie algebras. assurance then progresses from Jordan decomposition via homogeneous areas and quotients. bankruptcy workouts, and a thesaurus, notations, and effects are integrated.

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If X is affine, then an open subvariety of X is called a quasi–affine variety. 13. 37. 10). In other words the map f # given by composition with f is a morphism f # : OY → f∗ (OX ) of sheaves on Y . An invertible morphism is an isomorphism. 38. (1) In particular, if f : X → Y is an isomorphism of algebraic varieties then the map f is a homeomorphism, and the map f # is an isomorphism of sheaves. 38 1. ALGEBRAIC GEOMETRY (2) Notice that the conditions that f is a morphism of algebraic varieties and a homeomorphism of the underlying spaces do not guarantee that it is an isomorphism of algebraic varieties.

Using a standard result on general algebraic geometry (see Exercise 27), we conclude that X is affine. 50. A projective variety is a closed subvariety of a projective space. A quasi–projective variety is an open subvariety of a projective variety. 51. Since An can be identified with an open subset of P , it follows that any quasi–affine, and hence any affine, variety is quasi– projective. However, the converse is not true (see Exercise 40). 52. Consider V = An as a vector space. The flag variety is the set F(V ) of maximal chains of subspaces — full flags — V0 = {0} V1 ··· Vn = V .

12. Let A be a commutative ring and call X = Sp(A). If f ∈ A we define Xf = P ∈ Sp(A) : f ∈ P . If Y = Spm(A), we define Yf = Xf ∩ Y = Xf = M ∈ Spm(A) : f ∈ M . The proof of the theorem that follows is an easy exercise in commutative algebra. 13. Let A be a commutative ring and X = Sp(A) or X = Spm(A). 12, is the basis of a topology of X that is called the Zariski topology. A subset Y ⊂ X is closed in this topology if and only if Y = Q ∈ X : Q ⊃ I , where I is an ideal of A. 4. 14. (1) The assignment A → Sp(A) can be extended to a contravariant functor from the category of commutative rings to the category of topological spaces.

### Actions and Invariants of Algebraic Groups by Walter Ferrer Santos, Alvaro Rittatore

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