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?



$H^1(X, GL(n, \mathcal{O}_X))$ and Vector Bundles
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 $H^2(X, \mathcal{E})$.


