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 $R$?



Connection: locally free - locally projective
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.


