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Why does $H^1(X, \mathcal{O}_X^*)$ classify isomorphism classes of line bundles on a scheme/manifold $X$?

I heard this somewhere. Can someone recommend a reference?

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    $\begingroup$ Hartshorne exercise III.4.5 if you'd like to prove it yourself. It has a hint, as well. $\endgroup$ – KReiser Sep 6 '15 at 1:30
  • $\begingroup$ It's also likely in Hirzebruch's topological methods in algebraic geometry, but I don't have a copy with me, so I can't check. $\endgroup$ – Michael Burr Sep 6 '15 at 1:39
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    $\begingroup$ As suggested above, Hartshorne exercise III.4.5 is quite general and works for any ringed spaces $(X,\mathscr{O}_X)$, setting up an isomorphism between H^1 calculated using the derived functor formalism and cech cohomology. For integral schemes, the situation is also easily seen by applying the global section functor to the SES $0 \to \mathscr{O}_X^{*} \to K_X^{*} \to \mathscr{K}^{*}/\mathscr{O}_X^{*} \to 0$, obtaining a part of the long exact sequence $\cdots \to K_X^* \to \Gamma(X, K^{*}/\mathscr{O}_X^{*}) \to H^1(X,O_X^{*}) \to 0$. Middle mod out image of $K^{*}$ is Cartier divisors. $\endgroup$ – MS-DOS Sep 6 '15 at 5:53
  • $\begingroup$ See also here. $\endgroup$ – Zhen Lin Sep 6 '15 at 13:20
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This is more as a general remark (it'd be a comment if I had room), since I don't want to write enough words to make this precise. All of this can be found, somewhere, in the stacks project though.

In general, the group $H^1(X,G)$ classifies so-called '$G$-torsors' on $X$. Now, this is a slightly complicated notion, but it's the abstraction of the notion of principal $G$-bundles from algebraic topology (although the naive definition only works in the étale, say, $G$ is affine and smooth).

But, in the case that $G$ happens to be $\mathrm{Aut}_\mathcal{C}(F)$ for $\mathcal{C}$ some category of objects 'on $X$' and $F$ an object of that category, then the group $H^1(X,G)$, in nice situations, has a nice reinterpretation. Namely, $H^1(X,G)$ classifies isomorphism classes of objects of $\mathcal{C}$ which are, locally on $X$, isomorphic to $F$—the so-called twists of $F$.

We can be even more explicit. Namely, given a cover $\mathcal{U}$ of $X$ the group (or set depending on the commutativity of $G$) $\check{H}^1(\mathcal{U},G)$ is the group (or set) of isomorphism classes of objects of $\mathcal{C}$ which become isomorphic to $F$ on each element of $\mathcal{U}$. The general remark is then recovered by recalling that for any reasonable set up one has

$$H^1(X,G)=\varinjlim_{\mathcal{U}} \check{H}^1(\mathcal{U},F)$$

(note that Cech cohomology and derived cohomology 'always' agree for $H^1$).


Remark: In fact, this gives one an intuition about why Cech cohomology and agree for $H^1$ but not higher $H^i$. Namely, it becomes the tautological statement that an object which is 'locally trivial' is trivial on some open cover. One can try to extend this to higher $H^i$, but one runs into obvious trouble with $H^2$ (i.e. gerbes) and for even higher $H^i$ there is no interpretation (I know of) for cohomology classes in this generality.


In particular, taking $X$ to be a scheme, or a manifold, taking $\mathcal{C}$ to be the category of $\mathcal{O}_X$-modules, and taking $F=\mathcal{O}_X$, we see that the group

$$H^1(X,\mathrm{Aut}_{\mathcal{C}}(\mathcal{O}_X))=H^1(X,\mathcal{O}_X^\times)$$

classifies the group of isomorphism classes of $\mathcal{O}_X$-modules locally isomorphic to $\mathcal{O}_X$. Or, said differently, it classifies the isomorphism classes of line bundles on $X$.

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I don't know about schemes, but for complex manifolds this is discussed (admittedly, not in complete detail) in Griffiths and Harris' Principles of Algebraic Geometry, chapter $1$, section $1$. Another reference for complex manifolds is Huybrechts' Complex Geometry: An Introduction, Corollary $2.2.10$.

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