By G. Kempf

ISBN-10: 0521426138

ISBN-13: 9780521426138

During this ebook, Professor Kempf supplies an creation to the idea of algebraic forms from a sheaf theoretic perspective. by way of taking this view he's capable of provide a fresh and lucid account of the topic, as a way to be simply obtainable to all rookies to algebraic types.

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**Example text**

To prove (a) we may assume by replacing X by a smaller open set that X is affine and dimz X = dim X. Let Ji, ... ,f m be regular functions on X which vanish at x such that the differentials dfilz, ... ,d/mlz span Cotz:X. We want to show that m ~ dim X. Let Y = zeroes(/1, ... , f m) in X. Then x is in Y. Let I be tbe ideal sheaf of {x} in X. Then (/i, ... , f m) C I. By assumption (/1, ... , f m) lz_. Ilz is surjective. Thus (/i, ... ,/m) = I in a neigbborhood of x. e. component) of Y. By the corollary to the principal ideal theorem dim( X) - m ~ dim {x} = O.

We have the old sheaf M on C(X). Let = Mlc(X)-o· 1 Let U be an open subset oí X. Then M(7r- U) is graded. By definition M(U) =· (M(11'- 1 U))degree o· Explicitly if f is a homogeneous element of k(C(X)] then Mio(/) = M(l)-:;;ree 0 where D(/) =Spec k[C(X))(J) degree o· Thus Mis quasi-coherent on X and it is coherent ií M is finitely generated. 1. shea11e1 on X have the form M. s a finitely generated graded k[C(X))-module. Remark. One must be careful because M does not determine M. sz. M =O. 62 .

S'lxlz = Cotz(X). {1) /*(a· w) =/*(a)· /*(w) for a E k[Y) and w E Sl[Y). F\irthermore, from the definition, one easily verifies the equation (2) Proof. We may assume that X is affine and nis a maximal ideal of x in k(X). Then we want a natural isomorphism O[X) ®A:[X) k(X)/n ::::s n/n 2. dx(/*a) = /*(dya) for a E k[Y). ) - t O[X}. The next lemma explain (in a particular case) how /' may be used to compute O(X) for an arbitrary affine X as X is isomorphic to a closed subvariety of an affine space.

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