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Let $X$ be a nice space: a manifold, a variety over a field, or so on. We know that line bundles on $X$ up to isomorphisms are given by $H^1(X,\mathcal{O}_X^*)$.

Can it be generalized to higher rankal vector bundles on $X$ by introducing a sheaf cohomology for sheaves $GL(n, \mathcal{O}_X)$?

Is it done somewhere? What are problems here? Is it useful? Where can I read about that?

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Yes, this is called non-abelian sheaf cohomology. If $X$ is a topological space and $\mathcal{G}$ is a sheaf of groups, then $H^0(X, \mathcal{G})$ is the global sections of $\mathcal{G}$, and there is also an object called $H^1(X, \mathcal{G})$. In particular, $H^1(X, GL_n)$ classifies isomorphism classes of rank $n$ vector bundles.

$H^1(X, \mathcal{G})$ is not a group, but merely a pointed set, meaning a set with an distinguished element called $0$. There are no $H^i$ for $i \geq 2$ in this case.

$H^1$ is functorial in $\mathcal{G}$. If $1 \to \mathcal{E} \to \mathcal{F} \to \mathcal{G} \to 1$ is a short exact sequence of sheaves of not-necessarily abelian groups, then we have a sequence $$1 \to H^0(X, \mathcal{E}) \to H^0(X, \mathcal{F}) \to H^0(X, \mathcal{G}) \to H^1(X, \mathcal{E}) \to H^1(X, \mathcal{F}) \to H^1(X, \mathcal{G})$$ which is exact in the sense that the image of each map is exactly the preimage of $0$ for the following map. Moreover, if $\mathcal{E}$ is abelian and central in $\mathcal{F}$, then you can further extend this sequence to $H^2(X, \mathcal{E})$.

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    $\begingroup$ Actually, you get a connecting map to $H^2(X,\mathcal{E})$ only if $\mathcal{E}$ is abelian and the extension is central. $\endgroup$
    – Donu Arapura
    Commented Mar 17, 2021 at 0:10
  • $\begingroup$ Thank you for the correction. $\endgroup$
    – David E Speyer
    Commented Mar 17, 2021 at 1:26
  • $\begingroup$ Could we use more categorical objects instead of topological spaces or abelian groups to obtain a long exact sequence whose a truncation/connected components would give rise to the above partial long exact sequence? E.g. I would like to have full long exact sequences at least for exponent SESes: $0\rightarrow \delta(n,\mathbb{Z})\rightarrow M(n,\mathbb{C})\rightarrow GL(n,\mathbb{C})\rightarrow 0$, where $\delta$ stands for diagonal matrices with integer coefficients up to a basis change. $\endgroup$
    – P. Grabowski
    Commented Mar 17, 2021 at 12:49
  • $\begingroup$ @P.Grabowski I don't know, but here are some references (which I don't understand) that seem to be discussing such a thing. mathoverflow.net/q/36466/297 ncatlab.org/nlab/show/nonabelian+cohomology $\endgroup$
    – David E Speyer
    Commented Mar 17, 2021 at 12:55
  • $\begingroup$ More mathoverflow.net/questions/130256 $\endgroup$
    – David E Speyer
    Commented Mar 17, 2021 at 13:09

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