By Walter Ferrer Santos, Alvaro Rittatore

This self-contained creation to geometric invariant idea hyperlinks the idea of affine algebraic teams to Mumford's thought. 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 some of the correct formulation and proofs. Early chapters evaluate 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 word list, notations, and effects are integrated.

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2) Usually — and the motivation for this abuse of notation will become clear in what follows — if U ⊂ V and s ∈ F (V ), we write s|U = ρV U (s). 24 1. ALGEBRAIC GEOMETRY (3) If X is a topological space a more formal definition of a presheaf on X would be the following. Consider the topology T as a category — viewing it as an ordered set. A presheaf on X is a contravariant functor from the topology into the category of rings. In this interpretation sheaves are functors satisfying certain equalization properties.

Assume that A and B are commutative k–algebras. The maximal ideals of A ⊗ B are of the form M ⊗ B + A ⊗ N for M and N maximal ideals of A and B respectively. 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).

If Z is an irreducible component of X we have that Z = i=1 (Xi ∩ Z), then, using the irreducibility of Z we conclude that for some 1 ≤ i ≤ n, Z = Z ∩ Xi . Then, Z ⊂ Xi and hence, Z = Xi . 19. The algebraic subset V(XY ) ⊂ k2 (the union of the two coordinate axes) is reducible, with irreducible components V(XY ) = 3. ALGEBRAIC SUBSETS OF THE AFFINE SPACE 19 (0, b) : b ∈ k ∪ (a, 0) : a ∈ k . It is very easy to see that the lines (0, b) : b ∈ k and (a, 0) : a ∈ k are irreducible. The irreducibility of an algebraic set can be completely characterized in terms of the corresponding ideal.