Why is the sheaf $\mathcal{O}_X(n)$ called the "twisting sheaf" (where $X=\operatorname{Proj}(S)$ for a graded ring $S$)?

Asked Jul 03 '15 at 00:36

Active Jul 03 '15 at 16:00

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Basically my question is why the sheaf $\mathcal{O}_X(n)$ is called the twisting sheaf, here $X=\operatorname{Proj}(S)$ and $S$ any graded ring.

edited Jul 03 '15 at 00:45

Zev Chonoles

asked Jul 03 '15 at 00:36

sti9111

Why it was downvoted? – sti9111 Jul 03 '15 at 00:42
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Because the $O(1)$ of $P^1$ is morally a Möbius band, I imagine. – Mariano Suárez-Álvarez Jul 03 '15 at 00:49
Ohhh this is a nice, identification, Thanks. – sti9111 Jul 03 '15 at 00:51
@MarianoSuárez-Alvarez: Could you explain how $\mathscr O(1)$ is a Möbius band (or provide a link)? I'm not sure if I see the connection. – Fredrik Meyer Jul 03 '15 at 07:19
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@FredrikMeyer He's thinking about the real case, and actually $O(-1)$ would correspond to a Mobius band (although in this case I suppose the metric would give an isomorphism to the dual). This is an easy exercise, if you can't solve it see the beginning Milnor and Stasheff's book on characteristic classes. – user40276 Jul 05 '15 at 11:56
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I think it's called twisting sheaf because locally you just twist the ring by shifting the grading. If you know about blow-ups, see the similarities between the two constructions. – user40276 Jul 05 '15 at 11:57

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