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By Heinz Spindler

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Extra resources for Algebraische Geometrie I

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Ist f nilpotent, so ist Rf = 0. Ist f kein Nullteiler von R, also fa = 0 , a = 0 , so ist R Unterring von Rf . Es sei T eine Unbestimmte und R f1 ] := R T] (fT 1); : R ! R f1 ] die kanonische Abbildung : R R T] q / / R T] (fT 1); T / q(T): (f)q(T) = q(fT ) = q(1) = 1 ) q(T) ist das inverse Element von (f): (f) ist also eine Einheit. Damit gibt es genau einen eindeutig bestimmten Homomorphismus ~ : Rf ! R 1 ] mit f a ~ n = (a) (f n ) 1 f = (a)q(T n ) = q(aT n ): Man kann leicht zeigen: ~ ist Isomorphismus (Ubung).

A) N M hei t R-Untermodul () N= 6 ;; x; y 2 N ) x y 2 N; x 2 N; a 2 R ) ax 2 N: b) Ist N M R-Untermodul, so ist M N = fx+N j x 2 M g R-Modul mit den Operationen (x + N) + (y + N) = (x + y) + N; a(x + N) = ax + N: M N hei t der Quotientenmodul von M nach N. 20 ' : M ! M 0 hei t Homomorphismus () '(ax + by) = a'(x) + b'(y) 8 a; b 2 R; x; y 2 M: Es gilt: ker ' und im' sind Untermoduln von M bzw. M 0 und M ker ' = im' induziert von x mod ker ' 7 ! '(x) . 21 M ist endlich erzeugter R-Modul () 9 x1; : : :; xm 2 M; so da M = m nX i=1 o ai xi j ai 2 R = hx1 ; : : :; xmi () 9 surjektiver R-Modulhomomorphismus ' : Rm !

Wenn ja, bestimme eine Parametrisierung ' : P1 ! C und eine Gleichung F 2 K Z0 ; Z1; Z2 ] des Kegelschnitts C P2 durch diese funf Punkte. 2) Seien X = fp1; : : :; pd+3 g, X 0 = fp01; : : :; p0d+3 g Pd in allgemeiner Lage. C und C 0 seien die rationalen Normkurven in Pd mit X C, X 0 C 0 . ' : P1 ! C, '0 : P1 ! C 0 seien Parametrisierungen, und es sei qi = ' 1(pi ), qi0 = '0 1 (p0i). Zeige: Es gibt genau dann eine Projektivitat : Pd ! Pd mit (pi ) = p0i 8 i, wenn fur alle i = 4; : : :; d + 3 gilt DV (q1 ; q2; q3; qi) = DV (q10 ; q20 ; q30 ; qi0 ).

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