# Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles

Let n be a positive integer and X a scheme. Then for all the Grothendieck topologies I know (Zariski, etale, fppf) the set $H^1(X,GL_n)$ is the set of (isomorphism classes of) rank $n$ vector bundles.

Is there a Grothendieck topology (something not too trivial...) so that the corresponding H^1 is different? (Of course, you will need GL_n to still be a sheaf.)

I'm thinking about Nisnevich/Voevodksy type topologies for example.

• You mean the set of isomorphism classes. – Qiaochu Yuan Jul 17 '14 at 19:02
• The Nisnevich or cd topology lies between Zariski and etale, so it works. In Voevodsky's cdh topology the structure sheaf and $GL_n$ are not sheaves. But you could still ask about their cohomologies (ie, of their sheafifications). – Ben Wieland Jul 18 '14 at 3:53
• I think the cdh cohomology is the same for a scheme and its reduced subscheme, but they do not have the same vector bundles. In characteristic zero, the answer is probably the vector bundles on the reduced subscheme. – Ben Wieland Jul 20 '14 at 0:40