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By Heiberg J.L. (ed.)

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5. Let a geometrically reductive algebraic group G act rationally on a k-algebra A leaving an ideal I invariant. Consider A G=I AG as a subalgebra of (A=I )G by means of the injective homomorphism induced by the inclusion A G A. For any a (A=I )G there exists d > 0 such that ad AG=I AG. If G is linearly reductive then d can be taken to be 1. \ 2 2 \ Proof. Let a be a nonzero element from (A=I )G , let a be its representative in A and let (a) = i i ai : Let V be the G-invariant subspace of A spanned by the G-translates of a.

For example, any surjective homomorphism of graded algebras : A B preserving the grading (the latter will be always assumed) defines a closed embedding : Spm(B ) Spm(A) whose restriction to any subset D (f ) is a closed embedding of affine varieties. It corresponds to the homomorphism : A 1=f ] B 1= (f )]. This defines a closed embedding from D(f )+ to D( (f ))+ and a morProjm(A). , an isomorphism onto a closed subset of the target space). 6) that any projective algebraic variety is isomorphic to some Projm(A).

The set of elements a i is a spanning set. Note that not every homomorphism of groups rational action of G on X . G! 1. Let G = G m act on an affine algebraic variety X = Spm(A). Let : A ! O(G) A = k T T ;1] A be the corresponding coaction homomorphism. 5) ! 7! , pi (ai ) = ai . 6) i2Z This defines a grading on A. Conversely, given a grading of A, we define by i (a) = i2Z T ai , where ai is the ith graded part of a. This gives a geometric interpretation of a grading of a commutative k -algebra.