Cohomology of the general linear group on punctured spectra of 2-dimensional power series rings

Question

$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Quot{Quot}\DeclareMathOperator\GL{GL}\DeclareMathOperator\char{char}$Let $(A,\mathfrak{m})=k[[x,y]]$ with $\char(k)=0$ and $K=\Quot(A)$. Set $X=\Spec(A)$, $U=\Spec(A)\backslash \lbrace \mathfrak{m} \rbrace$ the pointed spectrum. Furthermore given an $A$-algebra $B$, which can be embedded in $C=\mathrm{M}_n(A)$, where $B$ is free $A$-module of rank $n^2$. One can see the algebra as a sheaf on $X$ resp. $U$ and $B^{\times}$ denotes its group of units.

Now we get an exact sequence $0\rightarrow B^{\times} \rightarrow C^{\times}=\GL_n(A) \rightarrow F \rightarrow 0$ and $F$ is supported on $Y=\lbrace x=0 \rbrace$, where it can be identified with some flag space.

Now in the article i'm reading, there are 3 facts i don't quite understand (now it is just one):

a) $H^0(X,C^{\times})=H^0(U,C^{\times})$ because $C$ is a free $A$-module

b) $H^0(X,C^{\times}) \rightarrow H^0(X,F)$ is surjective, because $X$ is local (see vytas comment)

c) $H^1(U,C^{\times})=0$ because $C$ and $A$ are Morita equivalent and this holds for $A$ (here one has to use "reflexive" Morita equivalence, then this follows from the fact that every reflexive A-module is free)

My question is: a) why does this follow from the freeness. If it was just $C$ i would believe this, because sections extend uniquely to codimension 2 points, which works here because $\dim(A)=2$. But why should this be true for the group/sheaf of units? More generally this should be true if we replace $A$ by a free $A$-algebra $D$, i.e $H^0(X,\GL_n(D))=H^0(U,\GL_n(D))$. I tried using the Cech complex for $U$, as suggested in the comments, but it didn't help.

The article/text i'm referring to is this. It is called "Stable orders and the Riemann-Roch Theorem". It is Lemma 3.1.9 on page 62 ( Page 3 in this document ).

Answer 1

a): An element of $C^\times$ can be thought of as a pair $(a,b)$ of elements of $C$ with $ab=1$. This gives a) by applying of existence extension to $a$ and $b$ and unicity to $ab$ and $1$.

b): The relation $F=C^\times/B^\times$ is a relation of a sheaf quotient for the flat topology (even if one defines $F$ as schematic quotient as $B^\times$ is flat. Hence by general nonsensen, the obstruction to lifting a section of $F$ to one of $C^\times$ is a $B^\times$-torsor. This can also be interpreted as as a locally free $B$-module of rank $1$ which is trivial as $B$ is finite over a local base. Hence the obstruiction vanishes and the section lifts.

c): As you say this follows from Morita equivalence (it would actually be enough that $C$ be an Azumaya algebra).