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?
Best Answer
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.
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