6 MASAYOSHI NAGAT A A subset W of Sn i s called a k-closed set (or, more precisely , a Zariski k-closed set) if W = F(J ) fo r som e /. Suc h a W is called a n affine k-variety if W is not a proper unio n o f two closed sets . I f w e are obviousl y observin g thes e set s over k only , then k ma y b e omitted. The result s stated abov e sho w that W is a fc-variety i an d onl y i f ther e i s a prime idea l P in F suc h tha t W = V(P), or equivalently, P = I(W) i s prime. For a fc-closed seW t , we set / = I{W). Th e rin g F/J i s the se t o f function s o n W in- duced fro m element s o f F, an d F/J i s called th e coordinate ring of W over k an d is denote d by k[w]. (Fo r a very genera l treatmen t o f algebrai c varieties, it i s better t o observ e F/J without assumin g that / = I(W), bu t assumin g that W = V{J)) Jus t a s in th e cas e o f F an d Sn, w e defin e (i ) V(f) fo r a n ideal / i n k[W] an d (ii) Ik[w](T) fo r a subset T o f W. V(f) is a fc-closed subse t o f W, I k ^ W ](T) i s an ideal o f k[W] an d /^ w](*V) ) i s the radica l of/ . At thi s point, w e observe a natural correspondenc e betwee n point s o f W and certai n ideals of k[W]. Se t x( = (X( modul o I(W)). A point p = (pv . . . , pn) ( G S") lie s o n W if an d onl y if f(p) = 0 for an y /G /(HO , i. e., I(p) D I(W). Bu t p i s not uniquel y determine d by I(p). Indeed , p i s determined b y th e homomorphism \p : k[W] K suc h tha t \jjx ( = p t . Thus: (1) Eac h poin t o f W corresponds i n 1- 1 wa y t o a homomorphism o f k[W] int o K. Therefore, (2) Tw o points p = (p v . . . , pM) an d q = (# p . . . , #w) correspon d t o th e sam e ideal (i.e., I(p) I(q)) i f an d onl y i f ther e i s an isomorphism o f k\p v . . . , pn] ont o k[Q\ * ^ w ] whic h maps p. to q t (i = 1 , . . . , «). I n particular , (3) Ever y algebrai c point o f W corresponds t o a maximal idea l o f k[W] conversely , every maxima l idea l o f k[W] correspond s t o th e se t o f conjugate s o f a n algebrai c point . From no w on , we mainly observ e ^-varieties. I f W is an affine ^-variety , then I(W) i s prime an d k[W] i s an integral domain . Th e fiel d o f fraction s o f k[W] i s called th e function field o f W over k an d i s denoted b y k(W). Assume tha t U and W are Avarieties in S m an d S n , respectively . I f ther e i s given an injection o o f k(U) int o k(W), the n w e say tha t a rational mapping (or, rational correspon- dence) o f W to U is given the rea l meaning o f thi s is as follows. Writ e k[W] = k[xv . . . , xn] an d k[U] = k\yv . . . , ym]. The n conside r th e rin g B = k[xv . . . , xn, oy v . . . , oy m ]. We say that a point p o f W corresponds t o a point q i f ther e i s a homomorphism o f B int o A' whose restriction s o n k[W], k[U] defin e p, #. For on e poin t p o f W, ther e ma y b e many q, o r none, in £ / which correspon d t o p. I f all oy i ar e regular at p (i . e., o^- = g^x^h^x) wit h #f., fy G k[W], fyO?) ^ 0) , then w e say that th e rationa l mappin g is regular a t p, an d i n thi s cas e q i s unique, as is easily seen . There - fore an y rationa l mappin g is really a mapping o n th e outsid e o f a proper close d subset . If k(W) = ok(U), the n w e say tha t th e correspondenc e i s birational, because w e have rational mapping s in tw o ways . Th e correspondenc e i s biregular a t a point p o f W if an d only if
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