Antworte auf:  Grothendieck-Topologien für Schemata von xiao_shi_tou_
Forum:  Algebraische Geometrie, moderiert von: Buri Gockel

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xiao_shi_tou_
Senior
Dabei seit: 12.08.2014
Mitteilungen: 1250
Herkunft: Bonn
 Themenstart: 2019-12-11 11:58    [Diesen Beitrag zitieren]
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Hallo.
Ich habe eine etwas naive Frage.

Die Zariski Topologie erscheint auf den ersten Blick (wenn man mit Algebraischer Geometrie anfängt) die natürlichste Topologie, denn es ist die einzige Topologie die sich direkt mit den algebraischen Eigenschaften von Ringen (Primideale) definieren lässt.
Nun ist es aber so, dass die Universelle Eigenschaft von $\sp{-}$ (damit meine ich die Adjunktion mit $\Gamma(-)$) gar keine Primideale erwähnt und man kann ja auch Schemata definieren, ohne überhaupt von Primidealen zu sprechen (über Funktoren).
Die übliche Definition der Begriffe $\etale,smooth,fppf,fpqc$ usw. verwendet aber wesentlich die Existenz der Zariski Topologie. Man definiert ja erst Schemata mit der Zariski Topologie und definiert dann diese ganzen Begriffe wie $\etale$ usw.

Man kann doch diese Begriffe sicherlich auch ohne Bezug zur Zariski-topologie definieren und die Theorie so aufbauen, dass sozusagen alle diese Topologien gleichermaßen natürlich erscheinen.
Gibt es diese Voreghensweise schon?
 
Viele Grüße
XST
\(\endgroup\)

 
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