By Walter Ferrer Santos, Alvaro Rittatore

This self-contained creation to geometric invariant thought hyperlinks the idea of affine algebraic teams to Mumford's concept. The authors, professors of arithmetic at Universidad de l. a. República, Uruguay, make the most the perspective of Hopf algebra concept and the idea of comodules to simplify a number of the proper formulation and proofs. Early chapters evaluate must haves in commutative algebra, algebraic geometry, and the speculation of semisimple Lie algebras. assurance then progresses from Jordan decomposition via homogeneous areas and quotients. bankruptcy routines, and a thesaurus, notations, and effects are incorporated.

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**Actions and Invariants of Algebraic Groups**

This self-contained creation to geometric invariant thought 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 conception and the idea of comodules to simplify the various suitable formulation and proofs.

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**Example text**

Xn ]/I correspond to the ideals of k[X1 , . . , Xn ] that contain I, the closed subsets of X in the Zariski topology correspond to the ideals in k[X]. In particular, the points in X correspond to the maximal ideals of k[X]. 11 is Xf : f ∈ k[X] . 27. In the case that X and Y are abstract sets and F : X → Y is a function, define a k–algebra homomorphism F # : kY → kX as F # (f ) = f ◦F . The following definition of morphism between algebraic sets generalizes and is motivated by the construction of k[X].

26. 16. Assume that X is an algebraic subset of An and consider Spm k[X] as defined before. Then the map ιX : X → Spm k[X] defined as ιX (a1 , . . , an ) = X1 − a1 , . . , Xn − an + I(X) ⊂ k[X1 , . . , Xn ]/ I(X) , is a natural homeomorphism when we endow the domain and codomain with the corresponding Zariski topologies. Proof: The proof is a direct consequence of the theory developed so far. We only verify the assertions concerning the topology. Consider f ∈ k[X]; then ιX (Xf ) = X1 − a1 , .

Let k be an algebraically closed field. An affine variety over k consists of a triple (X, A, ϕ), where X is a topological space — the underlying topological space of the affine variety — A is an affine k–algebra — the algebra of regular functions of the affine variety — and ϕ : X → Spm(A) is a homeomorphism. 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 .