By D. G. Northcott
In those notes, first released in 1980, Professor Northcott offers a self-contained creation to the speculation of affine algebraic teams for mathematicians with a easy wisdom of communicative algebra and box conception. The ebook divides into elements. the 1st 4 chapters include the entire geometry wanted for the second one half the publication which offers with affine teams. then again the 1st half offers a convinced creation to the rules of algebraic geometry. Any affine crew has an linked Lie algebra. within the final chapters, the writer reviews those algebras and indicates how, in sure vital instances, their homes might be transferred again to the teams from which they arose. those notes offer a transparent and punctiliously written creation to algebraic geometry and algebraic teams.
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Extra info for Affine Sets and Affine Groups
2. 10 Enlargement of the ground field In this section we shall examine the effect on an affine set of enlarging the ground field. 10), V and W will denote affine sets defined over K, and L will denote an extension field of K. Suppose that f E K[V]. Then f : V - K and therefore f determines a mapping of V into L. ) Thus K[V] ~ jjL (V). Denote by S the subalgebra of the Lalgebra jj L (V) that K[V] generates. Then (V , S) is a function algebra over L. We must now identify S. }. I be a base for Lover K and suppose that 1 IE Then W 46 can be represented in the form W E S.
K[V], where this is defined in the manner explained above. We shall make a fairly deep study of K-morphisms at a later stage. For the moment we shall content ourselves with some simple observations. First of all the identity mapping of V is a K- morphism and it is associated with the identity homomorphism of K[V]. Next suppose that U is a third affine set and that If> : V - Wand X: W - U are K-morphisms. /J* 0 X*. 4) Our aim is to study affine sets and their morphisms. It will now be clear that this is virtually equivalent to studying affine algebras and their homomorphisnis.
J(x». /J*(8) = 1/1*(8) now follows from Theorem 29 Cor. /J*. This yields the Corollary. There is exactly one L-morphism V(L) ... /J : V "'W. In view of this result we introduce the 51 Notation. The unique extension of a K-morphism rf>: V -W to an L-morphism V(L) _W(L) will be denoted by rf>(L). Obviously if rf> is an identity K-morphism, then rf>(L) is an identity L-morphism. Also if rf>l : V .... V' and rf>2 : V' .... V" are K-morphisms, then (rf> = rf>(L) rf> )(L) 0 2 1 2 0 rf>(L). 10. 7) 1 Lemma 16.
Affine Sets and Affine Groups by D. G. Northcott