By Walter Ferrer Santos, Alvaro Rittatore
This self-contained creation to geometric invariant concept hyperlinks the idea of affine algebraic teams to Mumford's thought. The authors, professors of arithmetic at Universidad de los angeles República, Uruguay, make the most the perspective of Hopf algebra concept and the speculation of comodules to simplify the various appropriate formulation and proofs. Early chapters evaluate must haves 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 word list, notations, and effects are incorporated.
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Additional resources for Actions and Invariants of Algebraic Groups
Indeed, if we consider f /g with f, g ∈ k[X] and g(x) = 0 and take Xg , it is clear that g does not vanish in Xg and then the quotient f /g represents an element in OX,x . If h ∈ OX,x is an arbitrary element, one can represent h as the quotient f /g of two polynomials f, g ∈ k[X], with g(x) = 0, in a conveniently chosen neighborhood of x. It follows that the above morphism is surjective. (2) It is clear that k[X]f injects into OX (Xf ). Consider an element g ∈ OX (Xf ), then g ∈ OX,x for all x ∈ Xf or equivalently, g ∈ k[X]M for all the ideals M corresponding to points of Xf .
Let X ⊂ An be an algebraic set. 34 we associate to each open subset U ⊂ X the algebra of regular functions OX (U ). This, together with the restriction maps, produces a sheaf of k–algebras on X, called the structure sheaf of X and denoted as OX . 1. Condition (d) follows from the local character of the definition of regular function. 4. 34 (see Exercise 24). 7. Let F and G be two presheaves of rings on a topological space X. A morphism ϕ : F → G consists of a family of ring homomorphisms ϕ(U ) : F(U ) → G(U ), U ⊂ X, U open such that whenever there is an inclusion U ⊂ V ⊂ X of open subsets, the following diagram is commutative: F(V ) ϕ(V ) ρG VU ρF VU F(U ) / G(V ) ϕ(U ) / G(U ) If F and G are sheaves, a morphism of sheaves from F to G is a morphism of presheaves.
An ) = 0, ∀ f ∈ S , where S ⊂ k[X1 , . . , Xn ]. The image of the map V is the family of closed sets of a topology of kn , called the Zariski topology. The set kn when endowed with the Zariski topology will be denoted as An and called the affine space. An algebraic set is a Zariski closed subset of An , for some n ≥ 0. If S ⊂ An is a subset, the Zariski topology of S is the topology induced by the Zariski topology of An . The above is the basic construction for developing the local theory of algebraic varieties over a field k.
Actions and Invariants of Algebraic Groups by Walter Ferrer Santos, Alvaro Rittatore