Sequences of groups, exact not just in étale but also in the Zariski topology

Question

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?

Answer 1

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