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--- blog:2023-03-12 [2023/03/13 15:05] – pzhou
+++ blog:2023-03-12 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 34: / Line 34: @@
 OK, with that in place, I believe now you can do whatever you want.
-False.
+False. You cannot just remember the top layer cover. That's the bad thing of playing too much with the Venn diagram, or matching in set. The gluing data is essential. Consider the line bundle on $\P^1$. How do we reconstruct $O(1)$ from line bundle on $\P^1$? One way of doing so is like, you remember the object (object itself, not some isomorphism class) when restricted to each patch, and ask that they glue honestly. Sure, you can say, here is "restriction of $O(1)$ to $\C^*$, and here is the restriction of $O(2)$ to $\C^*$, no, don't mistake them from one another, they had different origin. " I would say, that's only conceptually useful, but not really. What we are doing? We are considering 'sheaf of sections'. Indeed, the global section of sheaves of $\Gamma(\C^*, O(2))$ and $\Gamma(\C^*, O(1))$ are apriori non-comparable, withtout a morphism.
+When we say, glue, we mean on each local piece, there are some standard building block. But, there are interesting automorphisms. When stuff from more global places restricts to local piece, we need to specify, how do we 'standardize' it.