Confusion about how the first cohomology classifies torsors - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:17:14Z http://mathoverflow.net/feeds/question/22907 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22907/confusion-about-how-the-first-cohomology-classifies-torsors Confusion about how the first cohomology classifies torsors Makhalan Duff 2010-04-28T23:11:56Z 2010-04-29T01:59:59Z <p>This question is inspired by, but is independent of: <a href="http://mathoverflow.net/questions/2414/sheaf-description-of-g-bundles" rel="nofollow">http://mathoverflow.net/questions/2414/sheaf-description-of-g-bundles</a></p> <p>Line bundles are classified by $H^1(X,\mathcal{O}^\times_X)$. We also know that in general that $H^1(X,G)$, where $G$ is a sheaf from open sets in $X$ to $Grps$, classifies $G$-torsors over X.</p> <p>With this insight in mind: $\mathcal{O}^\times_X$-torsors should correspond to line bundles. Indeed, if $P$ is one, then $\mathcal{O} _X \times _{\mathcal{O} _X^\times}P$ gives the desired line bundle, and all line bundles are achieved this way (see the question I linked to).</p> <p>My question is about the more mundane $H^1(X,\mathbb{C})$, which can be thought of as $H^1(X,\underline{\mathbb{C}})$ where $\underline{\mathbb{C}}$ is the constant sheaf $\mathbb{C}$ (which I think of as a sheaf going to $Grps$). These should supposedly correspond to $\mathbb{C}$-bundles over $X$. Which appears to be line bundles. But of course there's no reason for $H^1(X,\mathbb{C})$ to equal $H^1(X,\mathcal{O}^\times_X)$... My intuition is that this should correspond to the more naive version of fiber bundles that doesn't involve a structure group. Do you have any thoughts?</p> http://mathoverflow.net/questions/22907/confusion-about-how-the-first-cohomology-classifies-torsors/22911#22911 Answer by Mike Skirvin for Confusion about how the first cohomology classifies torsors Mike Skirvin 2010-04-28T23:57:11Z 2010-04-28T23:57:11Z <p>Your confusion seems to stem from the difference between topological bundles and algebraic/holomorphic bundles.</p> <p>For a scheme $(X, \mathcal{O}_X),$ you say that $H^1(X, \mathcal{O}_X^{\times})$ classifies line bundles on $X$. This is true, as long as you mean algebraic line bundles. If, for example, $X$ is something like a complex algebraic variety with the analytic topology (which seems to be how you're thinking about it, perhaps), then there can certainly be plenty of topological line bundles on $X$ which aren't algebraic.</p> <p>Also, as Lucas Culler notes in the comments to your question, a line bundle is not a torsor under the additive group $\mathbb{C}$. Instead, it is actually a $\mathbb{C}^{\times}$ torsor, if you remove the zero section. Algebraically, a torsor for the additive group $\mathbb{C}$ (which, by the way, is often denoted $\mathbb{G}_a$ to avoid confusion) is classified by all extensions of $\mathcal{O}_X$ by $\mathcal{O}_X$.</p> <p>Anyway, it is true that (maybe with some mild assumptions on your space $X$) that $H^1(X,G)$ classifies topological $G$-bundles on $X$.</p> http://mathoverflow.net/questions/22907/confusion-about-how-the-first-cohomology-classifies-torsors/22921#22921 Answer by VA for Confusion about how the first cohomology classifies torsors VA 2010-04-29T01:29:09Z 2010-04-29T01:29:09Z <p>The general principle is: if you have some objects which are locally trivial but globally possibly not trivial then the isomorphism classes of such objects are classified by $H^1(X,\underline{Aut})$, where $\underline{Aut}$ is the sheaf of automorphisms of your objects.</p> <p>So, if your objects locally are $U\times \mathbb A^n$ (i.e. vector bundles) or they are $\mathcal O_U^{\oplus n}$ (i.e. locally free sheaf) then either of these are classifed by $H^1(X, GL(n,\mathcal O))$. For $n=1$, you get $GL(1,\mathcal O)=\mathcal O^*$. </p> <p>Now what is $\mathbb C$ an automorphism group of? Certainly not of line bundles (zero has to go to zero).</p> http://mathoverflow.net/questions/22907/confusion-about-how-the-first-cohomology-classifies-torsors/22925#22925 Answer by Evgeny Shinder for Confusion about how the first cohomology classifies torsors Evgeny Shinder 2010-04-29T01:54:32Z 2010-04-29T01:59:59Z <p>I think the right thing to look at is $H^1(X, \mathbb C^*)$. This classifies line bundles with a flat connection, or equivalently, line bundles with locally constant transition functions.</p> <p>Now the natural embedding $\mathbb C^* \to \mathbb O_X^*$ induces on a map on cohomology $H^1(X, \mathbb C^*) \to H^1(X, \mathbb O_X^*)$ which is forgetting the flat connection.</p>