Connection: locally free - locally projective Given a smooth projective variety $X$ over some algebraically closed field $k$
and a locally free sheaf $R$ of $O_X$-algebras, e.g. central simple algebras or orders.
If $M$ is a left $R$-module which is locally free over $O_X$, is it true that $M$ is locally projective over $R$? For example if $X$ is a curve a torsion free $O_X$-module would be locally projective over $R$. Or do we need more conditions for $R$ and $M$ for this to be true?
Why can one compute $Ext_R^1(M,M)$ via $H^1(\mathcal{H}om_R(M,M))$ only in the case $M$ is locally projective over $R$?
 A: The answer to your first question is a resounding no. An example (among many) is
given by $X=\mathrm{Spec} k$, $R=k[x]/(x^2)$ and $M=k$ considered as an
$R$-module through the $k$-algebra homomorphism given by $x\mapsto 0$.
As for the second question, the reason is that in general there are correction
terms to this formula coming from the failure of $M$ being locally
projective. In fact there is a long exact sequence
$$
0\rightarrow H^1(X,\mathcal{H}\mathrm{om}_R(M,M))\rightarrow
\mathrm{Ext}^1_R(M,M)\rightarrow H^0(X,\mathcal{E}\mathrm{xt}^1_R(M,M)),
$$
where the $\mathcal{E}\mathrm{xt}^i_R(M,M)$ are the sheaves of Ext-classes
(their stalks at $x$ are the $\mathrm{Ext}^i_{R_x}(M_x,M_x)$). Hence, you need
something like (possibly  something a little bit weaker) the vanishing
$\mathrm{Ext}^i_{R_x}(M_x,M_x)$ for $i=1$ which in turn are implied by (though not
implying) the local $R$-projectivity of $M$. This sequence is most easily obtained by the right hand map taking a sequence to the isomorphism clases of local extensions and the second map is obtained by twisting the trivial sequence by a torsor over its automorphism group. the exactness in the middle comes from the fact that any locally trivial sequence comes from such a twisting.
Addendum: Corrected a typo (stacks) and changed the exact sequence to be correct (one way to get it is from the local to global spectral sequence and I had, as I unfortunately do too often, flipped it 45 degrees).
A: It is true for locally free sheaves of algebras that are central simple algebras at every point, though. These are known as sheaves of Azumaya algebras; I mention them since they were brought up in the question. I don't have a reference here, but the proof is not hard.
A: Let $R$ be an Azumaya algebra and $M$ an $R$-module which is locally free over $O_X$. Note that $R$ is a locally projective $R^{opp}\otimes_{O_X} R$-module, hence a direct summand of $R^{opp}\otimes_{O_X} R$. It follows that $M = M \otimes_R R$ is a direct summand of $M\otimes_R (R^{opp}\otimes_{O_X} R) = M\otimes_{O_X} R$ which is a locally free $R$-module.
Being a direct summand of a locally free $R$-module, it is certainly locally projective. I guess this argument can be found in the Miln book on etale cohomology.
