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I am trying to understand the fact that vector bundles of rank $r$ over a space $X$ are classified by the Cech cohomology group $\check{H}^{1}\big(X, GL_{r}(\mathcal{O}_{X})\big)$. I believe this should work in any of the usual categories, so I wont specify smooth, holomorphic, etc. I understand broadly how this goes, but there are a few key details tripping me up.

So if we have a Cech 1-cocycle $g = \{g_{\alpha \beta}\} \in \check{H}^{1}\big(X, GL_{r}(\mathcal{O}_{X})\big)$ with respect to some open cover $\{U_{\alpha}\}$, then we know:

$$(dg)_{\alpha \beta \gamma} = g_{\beta \gamma} \, g_{\alpha \gamma}^{-1} \, g_{\alpha \beta} =1$$

where we write everything multiplicatively, since that's the group operation on sections of $GL_{r}(\mathcal{O}_{X})$. So this equation above is obviously the cocycle condition satisfied by vector bundle transition functions. But for bundles, we also require that $g_{\alpha \alpha} =1$. Is this latter condition true in general for Cech cohomology, or is it somehow an extra requirement in this case?

My second confusion is the statement that isomorphic bundles define cohomologous cocycles. If we have a 0-cochain $\lambda = \{\lambda_{\alpha}\} \in \mathcal{C}^{0}(GL_{r}(\mathcal{O}_{X}))$, then applying the differential we get

$$(d\lambda)_{\alpha \beta} = \lambda_{\beta} \, \lambda_{\alpha}^{-1}$$

So I would be inclined to say that the condition that two 1-cocycles $\{g\}$ and $\{g'\}$ are cohomologous is

$$g_{\alpha \beta} \, (g_{\alpha \beta}')^{-1} = \lambda_{\beta} \, \lambda_{\alpha}^{-1}.$$

However, I know that two bundles are equivalent when their transition functions satisfy

$$g_{\alpha \beta} = \lambda_{\alpha} g_{\alpha \beta}' \lambda_{\beta}^{-1}$$

and things are clearly in the wrong order (for all ranks larger than 1) to be compatible with the previous equation. So where are the flaws in my understanding?

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The condition you suspected, namely $g_{\alpha\beta}(g'_{\alpha\beta})^{-1} = \lambda_{\beta}\lambda_{\alpha}^{-1}$, is the correct condition if the coefficient group is abelian. The latter condition, namely $g_{\alpha\beta} = \lambda_{\alpha}g'_{\alpha\beta}\lambda_{\beta}^{-1}$, is the correct one for non-abelian coefficient groups; see chapter $4$, section $4.1$, equation $4$-$2$ of Brylinski's Loop Spaces, Characteristic Classes and Geometric Quantization. Note, if the coefficient group is abelian, the two conditions are equivalent.

You may be interested in this question and its answers.

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  • $\begingroup$ Thanks for the reference, that was helpful. I understand that equation 4-2 in Brylinski is the condition that two bundles are isomorphic. I guess what I'm confused about is where that condition comes from in cohomology. I mean, Brylinski seems to define this as cohomologous, but since we already know Cech cohomology, why do we need another definition of this? If we understand Cech cohomology for all sheaves, why should it matter if it's a sheaf of non-abelian groups? $\endgroup$ – Benighted Mar 25 at 0:30
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    $\begingroup$ @Benighted: Do you have a reference for the cohomologous condition for Cech cohomology of a sheaf of groups? All the references I know only define this for sheaves of abelian groups, hence the need for Brylinski's definition. $\endgroup$ – Michael Albanese Mar 25 at 0:38
  • $\begingroup$ Ah sorry, I understand now. I overlooked the fact you point out here; we need a sheaf of abelian groups to apply the basic Cech cohomology. Thanks! As for the requirement that $g_{\alpha \alpha} =1$, do you know if somehow this is a general requirement for Cech 1-cocycles? I certainly see why we impose this condition for vector bundles, I just don't see where it comes from in cohomology. $\endgroup$ – Benighted Mar 25 at 0:48
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    $\begingroup$ Well, the cocycle condition $(dg)_{\alpha\beta\gamma} = 1$ reduces to $g_{\alpha\alpha} = 1$ when $\alpha = \beta = \gamma$. $\endgroup$ – Michael Albanese Mar 25 at 16:04

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