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--- blog:2024-12-04 [2024/12/05 08:33] – [Twisted Sheaves] pzhou
+++ blog:2024-12-04 [2024/12/05 09:51] (current) – [Twisted Sheaves] pzhou
@@ Line 16: / Line 16: @@
 It still might make sense, maybe the trivialness of $\alpha$ corresponds to a locally constant deformation. Just like a flat connection gives zero curvature, but still flat connection is useful.
-So, what is the connection? We can try to do resolution, but no.
+How to take a sheaf of categories and take the global section? Or given a diagram of categories, how to take the limit? Suppose we are trying to get $Coh(\P^1)$ twisted by a complex number $c$
+$$ Coh(\P^1 \RM \infty ) \xto{res \otimes O(c)} Coh(\C^*) \gets Coh(\P^1 \RM 0) $$
-So, what is connection? Locally it is one form, but it is an affine space over the one-form space, so that $d$ makes sense.
+Now, what is $O(c)$ on $\C^*$? I want to say it is a holomorphic line bundle that does not have any global section (so probably shouldn't be called a coherent sheaf), it is constructed by considering $\C^* \cong \C / \Z$, we consider line bundle given by the quotient $(\C \times \C) / \Z$, where the $\Z$ action is
+$$(z, \eta) \mapsto (z+ 2\pi i , \eta e^{c z})$$
+Indeed, we have $e^z = y$ (hmm, if $c=1$, I am supposed to still get a trivial bundle)
+OK, maybe we use analytic topology on $\C^*$.