By Veblen O., Whitehead J. H.

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**Extra resources for A Set of Axioms for Differential Geometry**

**Example text**

A subset V (I ) of X of homomorphisms vanishing on an ideal I of O(X ) is called a closed subset. It can be identified with an affine algebraic variety Spm(A=J ), where J = rad I is the radical of I . A point x 2 X is a closed subset corresponding to the maximal ideal m x of A. Closed subsets define a topology on X , the Zariski topology. Open subsets D (f ) = X n V ((f )) f 2 A, form a basis of the topology. Each subset D (f ) can be identified with an affine algebraic variety Spm(A 1=f ]). A choice of n generators of the k -algebra O (X ) defines an isomorphism from X to a closed subset of the affine space A n .

3 to ensure that f is G-invariant. 3 Affine algebraic groups Next we observe that property (LR) from the preceding section can be stated over any algebraically closed field k . Instead of complex Lie groups, we will be dealing with affine algebraic groups G over k . Definition. An affine algebraic group G over a field k is an affine algebraic variety over k with the structure of a group on its set of points such that the multiplication G and the inversion map : G G are regular maps. map : G G !

5) on A satisfies A i = 0 for i < 0 and A0 = k . Such a grading is called a geometric grading and the corresponding action is called a good G m -action. In this case, the ideal m0 = i>0 Ai is a maximal ideal of A and hence defines a point p0 of X , called the vertex. We set P X = Spm (A) = Spm(A) n fp0g: The group G m acts on the open set X ; the quotient set is denoted by Projm(A) and is called the projective spectrum of A. Assume that A is a finitely generated k-algebra with a geometric grading.