Download Algebraische Geometrie I by Heinz Spindler PDF

By Heinz Spindler

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Example text

4) Es seien R1; R2 Integritatsbereiche. R = R1 R2 sei der Produktring ((a; b) + (c; d) = (a + c; b + d); (a; b) (c; d) = (ac; bd)), und p = f(a; b) 2 R j b = 0g. Zeige: p ist Primideal in R. Berechne den Restklassenring R p und die Lokalisierung Rp von R in p. 5) Sei R ein Ring und p ein Primideal. m sei das (einzige) maximale Ideal in Rp . Zeige: Die kanonische Abbildung ' : R ! Rp (a 7 ! a1 ) induziert einen injektiven Ringhomomorphismus ' : R p ! Rp m und ' induziert einen Isomorphismus K ! Rp m, wobei K der Quotientenkorper von R p ist.

18 Es sei R ein Ring. Ein R-Modul M ist eine (additiv geschriebene) abelsche Gruppe (M; +) mit einer Verknupfung R M ! M; (a; x) 7 ! ax, so da gilt: a(x + y) = ax + ay; (a + b)x = ax + bx; (ab)x = a(bx); 1x = x fur alle a; b 2 R; x; y 2 M. Beispiele a) b) c) d) Rn ist ein R-Modul. Jede R-Algebra S ist insbesondere ein R-Modul (man vergesse die Multiplikation in S). Ist K ein Korper, so ist M genau dann K-Vektorraum, wenn M ein K-Modul ist. Abelsche Gruppen sind Z-Moduln. 19 Es sei M ein R-Modul.

Beweis: Ubung. Ubungen P @F . 1) Beweise: Ist F 2 K Z0 ; : : :; Zn] homogen vom Grad m, so gilt m F = n=0 Z @Z @F ; @F ; @F 2) Fur F 2 R Z0; Z1 ; Z2] sei Sing(F) := V @Z P2. ' : P2 n V(Z0) ! R2 sei die @Z @Z Bijektion '( Z0 : Z1 : Z2 ]) = ZZ ; ZZ = (z1 ; z2). f 2 R z1; z2 ] sei die Dehomogenisierung von F bzgl. Z0 . Zeige: (a) '(Sing(F)) = Sing(f). (b) Ist p = a] 2 V(F) n V(Z0) und p 62 Sing(F), so ist @F (a)Z + @F (a)Z + @F (a)Z 2 T = V @Z 0 @Z1 1 @Z2 2 P 0 der projektive Abschlu der (a nen) Tangente von V(f) im Punkt '(p).

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