2. Affin e Space s and Projective Spaces The structur e o f affin e space s seems to b e simple r tha n tha t o f projective spaces . But , in orde r t o develo p geometry , o r t o understand geometri c natur e better , projectiv e space s help us quite a lot. We shall begin wit h tw o typica l difference s betwee n affin e an d projectiv e spaces . On e example i s Bezout theorem . I f w e look a t tw o plane curves , of degree s m an d n, respectively , and having no common component , the n i n the projectiv e case , we have a fine result tha t th e total numbe r o f commo n points , counted wit h multiplicities define d b y th e degree s o f con - tacts, is equal to mn. Thi s is generalized als o to th e higher dimensiona l case , i.e., th e Bezou t theorem. O n the contrary , w e cannot hav e such an equality i n th e affin e case . O f course , since an affin e plan e i s imbedded i n a projective plan e naturally , w e have a n inequality i n th e affine case . Another exampl e i s the completenes s o f projectiv e varieties . Fo r th e discussion , let u s begin with affin e varieties . Let k b e a field an d conside r a n algebraically close d fiel d K containin g k f th e polyno - mial ring F = k[Xv . . . , Xn], an d th e ^-dimensional affin e spac e Sn ove r K. I f w e fix a coordinate syste m o n Sn, the n eac h element f{Xv . . . , Xn) o f F define s a function o n S n , so that eac h point (a v . . . , an) o f S n i s mapped t o th e elemen t f(ax, . . . , an) o f K. I f a subset / o f F i s given, then w e define th e zero-point set V(J) o f / t o b e th e se t o f point s p = (p v . . . , pn) G Sn suc h tha t f(p) = 0 for an y /G / . Obviously , V(J) coincides wit h V(J*) if/ * i s the idea l o f F generate d b y /. Thu s in orde r t o observ e set s of th e for m V(J), it suffice s t o observ e th e cas e where / i s an ideal. T o th e contrar y direction , if we have a subset T o f Sn, the n w e defin e th e ideal 7(7) fo r T to b e the se t o f polynomials / G F suc h that f(p) = 0 for an y p e T. (The n 7(7) become s a n ideal o f F.) Under th e notation , w e see easily th e followin g thre e assertions : (1) I f J i s an ideal i n F, the n J C I(V(J)). (2) If / an d / ar e ideal s in F having the sam e radical , then V(J) = V(f). (3) If/j , . . . , Jm ar e ideals in F, the n (i) F(/ , •• Jm) = Wt n n/M) = v(J t ) u ••• u v(J m ), (ii) v(J t + ••• + j m ) = F(/, ) n - n v(J m ). A little mor e difficul t resul t i s that th e algebrai c closednes s o f K implie s tha t (4) If / i s an ideal o f F, the n 7(F(/)) coincide s wit h y/J. Or , more precisely , if / (G F) vanishes a t ever y algebrai c poin t p = (pv . . . , pn) withi n V(J) (i . e., p G V(J) an d every pt i s algebraic ove r k), the n som e power o f / is i n /. This (4) follow s fro m Theore m 1. 4 an d i s another for m o f th e Hilber t zero-poin t theorem. 5
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