Download PDF by Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen, Visit: An Invitation to Algebraic Geometry

By Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen, Visit Amazon's William Traves Page, search results, Learn about Author Central, William Traves,

ISBN-10: 8181282655

ISBN-13: 9788181282651

This can be a description of the underlying ideas of algebraic geometry, a few of its vital advancements within the 20th century, and a few of the issues that occupy its practitioners this day. it's meant for the operating or the aspiring mathematician who's unusual with algebraic geometry yet needs to realize an appreciation of its foundations and its pursuits with at least necessities. Few algebraic must haves are presumed past a uncomplicated direction in linear algebra.

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Mit O(U) werde die Menge aller in U regularen Funktionen bezeichnet. O(U) = ff : U ! 7 O(U) ist Unterring von C(U; K). Beweis: Ist f : U ! K regular in p, so ist f auch stetig in p. Sei dazu W go ene Umgebung von f(p) in K = A 1 ; W = A 1 n V(q); q 2 K z] mit q(f(p)) 6= 0. Sei f = h in einer o enen Umgebung V von p in U. Dann ist g(x) = 6 0 : f 1 (W) \ V = x 2 V j q h(x) q= d X =0 az X ) q hg = h1d a hd g : | {z } q~2K z1 ;:::;zn ] Also gilt: U~ = f 1 (W) \ V = V n V(~q) ist o ene Umgebung von p in U mit f(U~ ) V .

8: Unter der kanonischen Abbildung A(X) ! O(Xf ) wird f auf die nirgends verschwindende Funktion f jXf abgebildet; f jXf ist Einheit in O(Xf ). Die universelle Eigenschaft fur A(X) f1 ] gibt uns den Homomorphismus : A(X) 1f ] ! O(Xf ): Wir zeigen zunachst: a) ist injektiv. Es sei g = fhn 2 A(X) f1 ] und g(x) = 0 fur alle x 2 Xf . h. fh = 0 in A(X), also g = 0 in A(X) 1f ]: Das war trivial. Tiefliegender ist die Surjektivitat: b) Es sei r 2 O(Xf ) eine regulare Funktion auf Xf .

Dann gilt: Der kanonische Ringhomomorphismus A(X) ! O(Xf ); g + I(X) 7 ! gjXf induziert einen Ringisomorphismus A(X) f1 ] ! O(Xf ): Wir schreiben O(Xf ) = A(X) f1 ] = fgn j g 2 A(X); n 2 N : Insbesondere (wahle f 1) gilt O(X) = A(X): Weiter gilt: Fur jede ausgezeichnete o ene Menge U = Xf ist O(U) noetherscher Ring. Sind f1 ; : : :; fm 2 K z1 ; : : :; zn] Erzeuger des Ideals I(X) und ist fm+1 2 K z1 ; : : :; zn] Reprasentant von f 2 A(X), so ist O(Xf ) = K z1 ; : : :; zn+1] hf1 ; : : :; fm ; 1 zn+1 fm+1 i: Xf kann also als abgeschlossene Untervarietat von A n+1 aufgefa t werden!

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An Invitation to Algebraic Geometry by Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen, Visit Amazon's William Traves Page, search results, Learn about Author Central, William Traves,


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