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I know two generalizations of Hilbert 90, but I don't if there is a statement that contains both:

The first statement

Let $K$ be a field, then $H^1(Gal(K), GL_n(K^{sep}))=0$.

The second statement

Let $X$ be a scheme, then $H^1(X,\mathbb{G}_m)=Pic(X)$.

Question

Is there a nice characterization of $H^1(X,GL_n)$ (where $GL_n$ is treated as a sheaf in the etale topology) for a general scheme $X$?

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One general form is that $GL_n$-torsors in the fppf (or a fortiori etale) topology coincide with Zariski $GL_n$-torsors (alias: rank $n$ vector bundles). Your first statement follows, since the left hand side computes isomorphism classes of etale $GL_n$-torsors on $\Spec(K)$, which then coincides with isomorphism classes of rank $n$ vector bundles on $\Spec(K)$, which is obviously just the one. Your second statement (which is a little imprecise, but presumably your $H^1$ is etale) is this for $n=1$. –  Moosbrugger Oct 30 '11 at 23:25

1 Answer 1

up vote 5 down vote accepted

$H^1$ computed via sheaf cohomology coincides with the Cech $H^1$, which can be interpreted as giving transition functions. In particular, $H^1(X, GL_n)$ is in bijection with the set of rank $n$ vector bundles on $X$ in the Zariski topology (Theorem 11.4 in Milne's notes on etale cohomology).

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