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Extra resources for Algebraic geometry : introduction to schemes

Example text

Ist x 2 X und U X o en mit x 2 U, so hei t U eine o ene Umgebung von x. Sind X und Y topologische Raume, und ist f : X ! Y eine Abbildung, x 2 X, so hei t f stetig in x : () Zu jeder o enen Umgebung V von f(x) in Y gibt es eine o ene Umgebung U von x in X, so da f(U) V . ' f r x U - r f(x) & f(U) $ V% X Y f hei t stetig, wenn f in jedem Punkt x 2 X stetig ist. Man sieht nun leicht: f : X ! Y ist genau dann stetig, wenn gilt: Ist V Y o en, so ist f 1 (V ) X o en. Beweis: Es sei f stetig und V Y o en.

Es gilt dann (X _ )_ = X fur jeden linearen Teilraum X Pn. 8) Es seien X; Y P1 lineare Teilraume. Gilt dann (a) (XY )_ = X _ \ Y _ , (b) (X \ Y )_ = X _ Y _ ? 9) Sind die Aufgaben 5b) und 6) dual zueinander? 10) Es sei F 2 R X; Y; Z] homogen vom Grad d 2, und es sei @F ; @F ; @F = ;: Sing(F) := V @X @Y @Z C = V(F) P2 sei die Kurve mit der Gleichung F. Fur p = a : b : c] 2 C ist @F (a; b; c)X + @F (a; b; c)Y + @F (a; b; c)Z Tp = V @X @Y @Z die Tangente von X in p. C _ = fTp j p 2 X g P2 hei t die duale Kurve von C.

H. R ist Integritatsbereich und jedes Ideal I in R hat die Form I = hai fur ein a 2 R. Zeige: Ist q 2 R n R ; q 6= 0 irreduzibel, so ist I = hqi ein maximales Ideal. ) Welches sind die Primideale in R? (d) R = C x; y] hy2 f i, wobei f 2 C x]: sei die Restklasse von y in R. Zeige: i. C x] ist Unterring von R. ii. R = fa + b j a; b 2 C X]g (als C x]-Modul ist R isomorph zu C x]2 ). iii. hxi = Rx ist kein Primideal in R, R hxi = C y] hy2 f(0)i. iv. Sei f(0) 6= 0 und f(0) = a2 ; a 2 C : Dann sind I1 = hx; +ai; I2 = hx; ai Primideale in R und es gilt: I1 I2 = I1 \ I2 = hxi; wobei I1 I2 das von den Produkten ; 2 I1 ; 2 I2 erzeugte Ideal in R ist.

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Algebraic geometry : introduction to schemes by I.G. Macdonald

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