By Walter Ferrer Santos, Alvaro Rittatore
This self-contained advent to geometric invariant thought hyperlinks the idea 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 idea and the idea of comodules to simplify a number of the proper formulation and proofs. Early chapters assessment 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 incorporated.
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Extra resources for Actions and Invariants of Algebraic Groups
If there is no danger of confusion A is denoted as k[X], or OX (X), and the affine variety (X, A, ϕ) is written as X, k[X] or even as X. A morphism of affine algebraic varieties with domain (X, A, ϕ) and codomain (Y, B, ψ) is a pair (f, f # ), where f : X → Y is a continuous map ∗ and f # : B → A is a morphism of k–algebras such that f # : Spm(A) → Spm(B) makes the diagram below commutative X f /Y ϕ Spm(A) ψ f# ∗ / Spm(B) In accordance with the standard notations, we denote ϕ(x) = Mx . 18. Assume that (X, A, ϕ) is an affine algebraic variety and Y a closed subset of X.
8 (Hilbert’s Nullstellensatz). √ nomial ring I ⊂ k[X1 , . . , Xn ], then I = I V(I) . 20) √ I= M ⊂ k[X1 , . . , Xn ] : I ⊂ M, M maximal ideal . If M is maximal, then M = X1 −a1 , . . , Xn −an for some a1 , . . 17). Clearly, I ⊂ X1 − a1 , . . , Xn − an if and only if f (a1 , . . e. if and only if (a1 , . . , an ) ∈ V(I). Thus, we conclude that √ I= X1 − a1 , . . , Xn − an ⊂ k[X1 , . . , Xn ] : (a1 , . . , an ) ∈ V(I) . ,an )∈V(I) X1 − a1 , . . , Xn − an . It is √ then evident that I = I V(I) .
If M is maximal, then M = X1 −a1 , . . , Xn −an for some a1 , . . 17). Clearly, I ⊂ X1 − a1 , . . , Xn − an if and only if f (a1 , . . e. if and only if (a1 , . . , an ) ∈ V(I). Thus, we conclude that √ I= X1 − a1 , . . , Xn − an ⊂ k[X1 , . . , Xn ] : (a1 , . . , an ) ∈ V(I) . ,an )∈V(I) X1 − a1 , . . , Xn − an . It is √ then evident that I = I V(I) . 9. If we fix n and restrict the domain of the map I to the family of algebraic subsets of An and the domain of V to the family of radical ideals of k[X1 , .
Actions and Invariants of Algebraic Groups by Walter Ferrer Santos, Alvaro Rittatore