Algebraische Geometrie by Claus Scheiderer

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By Claus Scheiderer

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Xn ). Also gilt (i) ⇒ (iii). Ist xdi ∈ I f¨ ur i = 0, . . , n, so ist Sm ⊂ I f¨ ur m > (n + 1)(d − 1). Das zeigt (iii) ⇒ (ii), und (ii) ⇒ (iv) ⇒ (i) sind ebenfalls klar. 8. Sei i ∈ {0, . . , n}, sei D+ (xi ) = Pn (K) Bijektion φi : D+ (xi ) → An , (x0 : · · · : xn ) → V+ (xi ). Wir betrachten die x0 xi xn xi , . . , xi , . . 11, mit Umkehrabbildung ψi : An → D+ (xi ) ⊂ Pn (K), ψi (x0 , . . , xi , . . , xn ) = (x0 : · · · : 1 : · · · : xn ). Zur Vereinfachung der Notation nehmen wir i = 0 an und schreiben φ = φ0 , ψ = ψ0 .

Der Morphismenbegriff f¨ ur funktionengeringte R¨aume ist lokal im folgenden ¨ Sinn: Ist (Ui )i∈I eine offene Uberdeckung von X , so ist eine Abbildung f : X → X genau dann ein Morphismus von (X , O ) nach (X, O), wenn f¨ ur alle i ∈ I die ¨ Einschr¨ ankung f |U ein Morphismus von (Ui , O |U ) nach (X, O) ist (Ubung). i i 6. Seien (X, O), (X , O ) in (FRSK ), und sei f : X → X eine stetige Abbildung. Sei (Ui )i∈I eine Basis offener Mengen von X. Gilt f¨ ur alle i ∈ I und alle a ∈ O(Ui ), daß f # (a) = a ◦ f f −1 (U ) ∈ O (f −1 (Ui )) ist, so ist f ein (FRSK )-Morphismus (und i ¨ umgekehrt, nat¨ urlich).

Bemerkungen. 1. F¨ ur jede Teilmenge X ⊂ Pn (K) ist I+ (X) = I(X). 1) Radikalideal von k[x0 , . . , xn ]. √ 2. 1). Es gilt V+ i Mi = i V+ (Mi ) und V+ (I1 ) ∪ V+ (I2 ) = V+ (I1 ∩ I2 ) f¨ ur homogene Ideale I1 , I2 . 3. F¨ ur beliebige Teilmengen Xi von Pn (K) (i ∈ I) gilt i Xi = i Xi = i Xi und i Xi . 4. Ist I = (1) ein homogenes Ideal, so ist V(I) = V+ (I). Denn V+ (I) ist die Menge der in V(I) enthaltenen Geraden durch 0. Da I von homogenen Elementen erzeugt wird, ist V(I) Vereinigung von Geraden durch 0.

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