Algebraische Geometrie I by Heinz Spindler

Algebraic Geometry

By Heinz Spindler

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D) Ist X = V(F1; : : :; Fs); deg(Fi) = di, und ist d 2 N beliebig mit d di 8 i, so gibt es homogene Polynome F~1 ; : : :; F~r 2 Sd , so da X = V(F~1; : : :; F~r ). Beweis: Anstelle von Fi wahle man die Polynome Z0 Znn Fi mit 0 + + n + di = d. h. ist k n + 1 und sind a10 : : : : : a1n]; : : :; ak0 : : : : : akn] verschiedene Punkte von X, so ist der Rang der Matrix (aij ) ji ;:::;k maximal, also gleich k. 14 Ist X Pn endlich, in allgemeiner Lage und gilt X = d 2n, so gibt es quadratische S2 , so da X = V(Q1; : : :; Qs).

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 !

Beweis: 1) m ist Ideal: a c = ad cb 2 m , wenn a; c 2 p; b; d 2 R n p: b d bd 2) Rp n m = Menge der Einheiten in Rp . x 2 Rp n m ) x = ab fur ein a 2 R n p; b 2 Rp ) y = ab 2 Rp mit xy = 1: Umgekehrt: ab dc = 1 , 9 e 2 R n p : e(ac bd) = 0 ) eac = abd 2 R n p: Da p Ideal, folgt a 2 R n p, also ab 2 Rp n m: 3) Ist nun I Rp Ideal mit I 6= Rp , so ist I \ (R | p{zn m}) = ;, also I m: 2 c) Ist f 2 R, so ist N Einheiten n = ff j n 2 Ng naturlich multiplikativ. Rf := N 1 R = fan j a 2 R; n 2 N ; ' : R !

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