By Walter Ferrer Santos, Alvaro Rittatore
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, make the most the point of view of Hopf algebra conception and the speculation of comodules to simplify the various appropriate formulation and proofs. Early chapters evaluate necessities in commutative algebra, algebraic geometry, and the idea of semisimple Lie algebras. insurance then progresses from Jordan decomposition via homogeneous areas and quotients. bankruptcy workouts, and a word list, notations, and effects are incorporated.
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Extra resources for Actions and Invariants of Algebraic Groups
A function of kX is a if it is the restriction R k[X1 , . . , Xn ] . We Let X ⊂ An be an algebraic subset. , if it belongs to denote the set of polynomial functions as k[X]. 24. As k[X] ⊂ kX is R k[X1 , . . , Xn ] , it follows that the algebra k[X] is isomorphic to k[X1 , . . 22). 25. If we call CZar (X) the subalgebra of kX consisting of the functions on X continuous with respect to the Zariski topology, it 20 1. ALGEBRAIC GEOMETRY is clear that k[X] ⊂ CZar (X). Notice that there exist continuous functions that are not regular.
An algebraic set X ⊂ An is irreducible if and only if I(X) is a prime ideal. In particular, An is irreducible. Proof: Let X be an irreducible algebraic subset and suppose that f, g ∈ k[X1 , . . , Xn ] are such that f g ∈ I(X). Consider the union V(f ) ∪ V(g) = V(f g). Since f g ∈ I(X), it follows that X ⊂ V(f g). Thus, either X ⊂ V(f ) or X ⊂ V(g). We suppose without loss of generality that X ⊂ V(f ). Then (f ) ⊂ I(X), and thus f ∈ I(X). Suppose now that I(X) is a prime ideal. Let X = Y ∪Z, with Y = √V(I), Z = V(J) two closed subsets.
108. 34. Let R be a noetherian regular local ring, then R is an integral domain that is also integrally closed in its field of fractions. 23] or [71, Cor. 2]. e. in the case of curves, there is an easy criterion for regularity. 35. Assume that R is a noetherian local integral domain of dimension 1. Then the following conditions are equivalent: (1) R is a discrete valuation ring; (2) R is integrally closed; (3) R is a regular local ring; (4) the maximal ideal of R is principal. Proof: See [3, Chap.
Actions and Invariants of Algebraic Groups by Walter Ferrer Santos, Alvaro Rittatore