By Walter Ferrer Santos, Alvaro Rittatore
This self-contained creation to geometric invariant idea hyperlinks the speculation of affine algebraic teams to Mumford's thought. The authors, professors of arithmetic at Universidad de l. a. República, Uruguay, make the most the perspective of Hopf algebra concept and the speculation of comodules to simplify the various correct formulation and proofs. Early chapters overview necessities in commutative algebra, algebraic geometry, and the speculation of semisimple Lie algebras. insurance then progresses from Jordan decomposition via homogeneous areas and quotients. bankruptcy workouts, and a thesaurus, notations, and effects are incorporated.
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Extra info for Actions and Invariants of Algebraic Groups
Hence, one does not have a natural way to view Spm as a functor in all the category of commutative rings. 15. Let A and B be commutative finitely generated k–algebras, α : A → B a morphism of k–algebras and M ∈ Spm(B). Then α−1 (M ) ∈ Spm(A). In other words, α∗ Spm(B) ⊂ Spm(A). Proof: Let M be a maximal ideal in B, consider M = α−1 (M ) and the map α : A/M → B/M . As B is a quotient of a polynomial algebra the Nullstellensatz guarantees that B/M coincides with the base field k. Then, as α is k–linear and injective, we conclude that A/M is also the field k and hence that M is a maximal ideal.
Suppose X, Y, Z are affine. Then we define X ×Z Y as the affine variety corresponding to the k–algebra k[X]⊗k[Z] k[Y ]. In the general case, we consider an atlas Ui , i ∈ I, covering Z and atlases Vj , j ∈ J, of X and Wk , k ∈ K, of Y such that for all j ∈ J, f (Vj ) ⊂ Ui for some i ∈ I, and for all k ∈ K, g(Wk ) ⊂ Ui for some i ∈ I. To obtain X ×Z Y we consider the affine varieties Vj ×Ui Wk and glue them together. Next we look at the correspondence between closed subsets and ideals of the ring of global regular functions in the case of quasi–affine varieties.
But, the localization map establishes a bijective correspondence between the set of maximal ideals of k[X] that do not contain f and the set of maximal ideals of k[X]f . Moreover, as k[X]M = (k[X]f )Mf we conclude that (k[X]f )M OX (Xf ) = f : M ⊂ k[X]f , M is maximal = k[X]f . 15. 37. If U ⊂ V ⊂ X are open subsets, the restriction of functions from V to U induces a morphism of k–algebras ρV U : OX (V ) → OX (U ). Given two open subsets U, V ⊂ X, f ∈ OX (U ) and g ∈ OX (V ), such that f |U ∩V = g|U ∩V , the function h : U ∪ V → k defined as: h(x) = f (x) if x ∈ U , h(x) = g(x) if x ∈ V , belongs to OX (U ∪ V ).
Actions and Invariants of Algebraic Groups by Walter Ferrer Santos, Alvaro Rittatore