By Walter Ferrer Santos, Alvaro Rittatore

ISBN-10: 082475896X

ISBN-13: 9780824758967

ISBN-10: 1420030795

ISBN-13: 9781420030792

This self-contained advent to geometric invariant idea hyperlinks the speculation of affine algebraic teams to Mumford's thought. The authors, professors of arithmetic at Universidad de los angeles República, Uruguay, take advantage of the perspective of Hopf algebra idea and the speculation of comodules to simplify the various suitable formulation and proofs. Early chapters assessment necessities in commutative algebra, algebraic geometry, and the idea of semisimple Lie algebras. assurance then progresses from Jordan decomposition via homogeneous areas and quotients. bankruptcy routines, and a word list, notations, and effects are incorporated.

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**Walter Ferrer Santos, Alvaro Rittatore's Actions and Invariants of Algebraic Groups PDF**

This self-contained advent to geometric invariant concept 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 concept and the idea of comodules to simplify some of the correct formulation and proofs.

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**Extra resources for Actions and Invariants of Algebraic Groups**

**Example text**

21. 43. Let X be an algebraic k–variety and assume that Y ⊂ X is a closed subset endowed with the induced topology. 42, Y is said to be a closed subvariety of X, or simply a subvariety. 44. 37 and makes of Y a closed subvariety of X. See Exercise 31. 45. Let X, Y, Z be algebraic varieties, and let f : X → Z, g : Y → Z be morphisms. We define the fibered product of X and Y over Z as a triple (X ×Z Y, pX , pY ) where X ×Z Y is an algebraic variety and 40 1. ALGEBRAIC GEOMETRY pX : X ×Z Y → X, pY : X ×Z Y → Y are morphisms of varieties such that f ◦pX = g ◦pY , and satisfying the following universal property: For an arbitrary triple (W, q1 , q2 ) with W an algebraic variety and q1 : W → X and q2 : W → Y morphisms such that f ◦q1 = g ◦q2 , there exists a unique morphism h : W → X ×Z Y such that q1 = pX ◦h and q2 = pY ◦h.

We call ιX (A) = k[X]. (4) Viewing the k–algebra A as a subalgebra of kX as before, the map f # can be visualized as the composition by f or in other words, the diagram below is commutative. 22. Hence, in this situation the map f # is determined by f . (5) It follows from the previous definitions above that an affine algebraic variety is isomorphic to the affine algebraic variety associated to an algebraic subset of some An . Indeed, if we have a triple (X, A, ϕ) the affine k–algebra A is isomorphic to a quotient k[X1 , .

8) If I is an ideal in k[X1 , . . , Xn ], then I ⊂ √ I ⊂ I V(I) . (9) The image of I consists of radical ideals. 5. 4 parts (7) and (8), if X ⊂ An , then X ⊂ V I(X) and if I ⊂ k[x1 , . . , xn ] is an ideal, then I ⊂ I V(I) . These inclusions are not necessarily equalities: take for example X = k \ {0} ⊂ k, and I = x2 ⊂ k[X] and perform the explicit computations. 6. If X ⊂ An is an arbitrary subset of the affine space and X denotes its closure, then X = V I(X) . 16 1. ALGEBRAIC GEOMETRY Proof: The proof of this lemma is left as an exercise (see Exercise 6).

### Actions and Invariants of Algebraic Groups by Walter Ferrer Santos, Alvaro Rittatore

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