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By Heiberg J.L. (ed.)
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The aim of this ebook is to illustrate that complicated numbers and geometry might be combined jointly superbly. This leads to effortless proofs and average generalizations of many theorems in aircraft geometry, akin to the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem.
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Extra info for A treatise of Archimedes: Geometrical solutions derived from mechanics
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.
A treatise of Archimedes: Geometrical solutions derived from mechanics by Heiberg J.L. (ed.)