Let $A=k[[x,y]]$, with say $k=\mathbb{C}$ and $X = \operatorname{Spec}(A)$. Let $B$ denote a free $A$-algebra of rank $e^2$; actually, we have $B=A[u,v]$ with $u^e=x$, $v^e=y$ and $uv = \xi\_e vu$, where $\xi\_e$ is an $e$-th root of unity. Now we have $M\_n(B)$ and the subalgebra $D$, where entries under the diagonal are in $uB$ and otherwise in $B$.


This gives us an exact sequence of noncommutative groups, where $i: \lbrace x=0 \rbrace \hookrightarrow X$:
$$ 1\rightarrow D^{\times} \rightarrow Gl\_n(B) \rightarrow i\_{\ast}F \rightarrow 1 \, .$$
Why is this an exact sequence of sheaves in the Zariski topology on $X$? That is: why can i look at the induced exact sequence in Zariski cohomology?


This is stated on page 3 of <http://www.math.lsa.umich.edu/courses/711/ordpages60-85.ps>, around the $3\times 3$ diagram. Or do we need the whole diagram to see this?


I mean even in the simple example $1\rightarrow \mu\_n \rightarrow \mathbb{G}\_m \rightarrow \mathbb{G}\_m \rightarrow 1$, where $\mathbb{G}\_m \rightarrow \mathbb{G}\_m$ is $x \mapsto x^n$, this sequence is not exact when we use the Zariski topology. 


If it is easier, one can replace $B$ with $A$ and $uB$ with $xA$. The goal is to see that $H^1(U,D^{\times})=0$, i.e. every fractional reflexive left $D$-ideal is free. Here $U = X\backslash \lbrace \mathfrak{m} \rbrace$. Maybe there is an easier way to see this?



Sequences of groups, exact not just in étale but also in the Zariski topology
Let me elaborate on Torstens comment.


The map $GL\_n(B) \to i\_\*F$ in your example is an fppf $D^\times$-torsor over $i\_\*F$,
a fact which can be checked easily directly from the definition of torsor which you can find in
any of the referenced texts. A priori it is a torsor for the fppf topology, but it can be
shown that any $D^\times$-torsor for the fppf topology is also a torsor for the
Zariski-topology.


There are only a few groups with this property. They were studied by Serre and
Grothendieck [GS58] who classified all such groups over algebraically closed fields.
They called them "special groups" (probably because of the joy of saying things like:
"the general linear group is special" or better still: "the special orthogonal group is not special").
If I recall correctly, the complete list is $SL\_n$, $Sp\_{2n}$, all connected
affine solvable groups and extensions thereof (in particular, $GL\_n$ is special). Your group is defined over a more general base,
so the result doesn't apply directly.
However, specialness for $D^\times$ can be proven with exactly the same methods as for instance for $GL\_n$.


This seems to be a folklore result, and I don't know of any reference for a proof (although it is mentioned for instance in [Joy07, Definition 2.1]), but the
idea is as follows. The key is to use flat descent for quasi-coherent sheaves.
The group $D^\times$ is the automorphism group for $D$ viewed as a module
over itself. Given any module $M$ which is fppf-locally isomorphic to $D$, which is the same as locally free of rank 1 by flat descent, you get a corresponding $D^\times$-torsor by taking the
sheaf of $D$-module isomorphisms $Isom(M, D)$. Note that this sheaf has a natural action
on the right by $Aut(D) \simeq D^\times$. This establishes a one-to-one
correspondance between isomorphism classes of locally free $D$-modules of rank 1 and isomorphism classes of torsors [Gir70 III 2.5]. Since any locally free $D$-module 
allows a Zariski-trivialisation, so does any $D^\times$-torsor.


[GS58] Séminaire C. Chevalley; 2e année: 1958. Anneaux de Chow et applications


[Gir70] J. Giraud, Cohomologie non abélienne


[Vis05] Fantechi et al. FGA explained, Part 1


[Mil80] Milne, Étale cohomology


[Joy07] D. Joyce, Motivic invariants of Artin stacks and 'stack functions'. <http://xxx.lanl.gov/abs/math.AG/0509722>


