By Walter Ferrer Santos, Alvaro Rittatore

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

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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.

36. Let X be an algebraic variety. 35. 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.

Hence, as (abstract) sets Spm(A ⊗ B) and Spm(A) × Spm(B) are isomorphic. See Appendix, Section 3. Let X and Y be affine varieties. Then X × Y, k[X] ⊗ k[Y ] is an affine variety, when we endow the set X × Y with the topology induced by the isomorphism X×Y = Spm k[X]⊗k[Y ] . This topology in general is not the product topology (see Exercise 19). Moreover, if X is an algebraic subset of An and Y of Am , we can consider in a natural way X × Y as a subset of An+m and as such it is also an affine algebraic set.