Timeline for Intersection of open affines is affine
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 9, 2019 at 19:05 | comment | added | Qfwfq | @Aknazar Kazhymurat: it was just me not remembering a well known and easy proof. Could have been moved to MathStackExchange (but now the question is too old to migrate, I'm afraid) | |
| Apr 9, 2019 at 10:43 | comment | added | user74900 | @DinakarMuthiah how do you know? | |
| Mar 24, 2010 at 19:53 | comment | added | Zoran Skoda | The asnwer is in textbooks as quoted. But some comments. Thomason and Trobaugh have introduced a notion of semi-separatedness, which means that this scheme has an affine cover with affine double intersections. Of course, this is weaker than separatedness. In noncommutative geometry almost all interesting schemes are not semi-separated (principal example: quantum flag varieties). | |
| Mar 18, 2010 at 20:22 | comment | added | Qfwfq | Muthiah was right, it's easy: just take the diagonal inside the product of the two open affines etc. | |
| Mar 18, 2010 at 20:18 | comment | added | Qfwfq | Ok, found! It is spelled out in Mumford's book. Before I was only looking in Hartshorne's. | |
| Mar 18, 2010 at 19:50 | comment | added | Shizhuo Zhang | @llya: this is true and it is equivalent to say any algebraic variety is semi-separated. | |
| Mar 18, 2010 at 19:44 | comment | added | Ilya Grigoriev | Duh! I was confused... For schemes, this property is the most important consequence of separatedness. For non-separated schemes, it'll be wildly unture (there is a weaker notion of quasiseparatedness which means that intersections of open affines are finite unions of open affines, but there are schemes that don't even have this property.) | |
| Mar 18, 2010 at 19:36 | comment | added | Ilya Grigoriev | Nice question! Form Ravi's old notes (link follows): "Another nice property of varieties: the intersection of any two affine opens is another affine open. I don't foresee using this, so I won't prove it, but you can find a proof in Mumford (p. 55) or Hartshorne (Exercise II.4.4)." I might look there or think about it. math.stanford.edu/~vakil/725/class12.pdf | |
| Mar 18, 2010 at 19:33 | comment | added | Dinakar Muthiah | This is homework | |
| Mar 18, 2010 at 18:20 | history | asked | Qfwfq | CC BY-SA 2.5 |