By Heinz Spindler

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Extra info for Algebraische Geometrie I, Edition: version 4 May 1995

Example text

R ein Ringhomomorphismus, so hei t (R; ') eine S -Algebra, und ' hei t der Strukturhomomorphismus von R. Man erhalt eine Verknupfung (skalare Multiplikation) S R ! R; (a; x) 7 ! 1 die Regeln 1) (a + b)x = ax + bx 8 a; b 2 S; x 2 R, 2) a(x + y) = ax + by 8 a 2 S; x; y 2 R, 3) (ab)x = a(bx) 8 a; b 2 S; x 2 R, 4) 1S x = x 8 x 2 R (1S = Eins in S), 5) a(xy) = (ax)y = x(ay) 8 a 2 S; x; y 2 R: Aus der Verknupfung S R ! R erhalt man den Strukturhomomorphismus zuruck: 8 a 2 S : '(a) = a 1R (1R = Eins in R): Beispiele a) R = K z1 ; : : :; zn ] ist eine K-Algebra, mit Strukturhomomorphismus ' : K !

Wir werden jedoch sehen, da sie in gewissem Sinne unabhangig von der "gewahlten\ Einbettung ist. 2 Ist f 2 K z1; : : :; zn], so ist die Polynomfunktion f : A n ! A 1 stetig (bzgl. der Zariski-Topologie auf A n und A 1 ). Beweis: Es sei V A 1 o en, also (K X] ist Hauptidealring) V = A 1 n V(g); g 2 K z]; P d g = =0 a z : P Dann ist g f = g(f) = d =0 a f 2 K z1 ; : : :; zn ] und f 1 (V ) = A n n V(g(f)) o en. 3 Sei X An a ne Varietat und f 2 K z1; : : :; zn ]: Xf := X n V(f) = fx 2 X j f(x) 6= 0g hei t ausgezeichnete o ene Menge in X (Komplement einer Hyper ache).

K 2 sei minimal gewahlt, so da jX j kn. Dann ist X der Durchschnitt von Hyper achen vom Grad k. Fall: d = kn. Induktion nach k. k = 2 ist schon bewiesen. (k 1) ! k (k 3): Es sei M = fF 2 Sk Sj F(p) = 0 8 p 2 X g. Es sei q 2 V(M). Zu zeigen ist: q 2 X. Behauptung 1: X = ki=1 Xi ; jXi j = n 8 i ) 9 i, so da q 2 Span(Xi ). 14. Sei wieder Y X minimal mit q 2 Span(Y ). Nach Behauptung 1 ist m = jY j n. Sei Y 0 X n Y beliebig mit jY 0j = n m + 1 und p 2 Y fest gewahlt. HY 0 ::= Span(Y 0 (Y n fpg)) 00 ist Hyperebene mit q 2= HY 0 .

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