Given a smooth and projective surface $S$ over an algebraically closed field $k$ and a sheaf of Azumaya algebras $R$, i.e. $R$ is a locally free $O\_S$-module of finite rank. Let $M$ be a coherent and torsion free $O\_S$-module, which is also a left $R$-module, such that generically $M\_\eta$ is a simple $R\_\eta$-module. Then we have $Hom\_R(M,M)=k$.


Now $M^\\*:=Hom\_{O\_S}(M,O\_S)$ is a right $R$-module and $M^{\\*\\*}$ is a left $R$-module. We have the canonical map $\iota: M \rightarrow M^{\\*\\*}$.


Is it true that $Hom\_R(M,M^{\\*\\*})$ just consists of the muliples of $\iota$, i.e. is it a one dimensional $k$-vector space?


I tried to use the sequence $0\rightarrow M\rightarrow M^{\\*\\*} \rightarrow Q\rightarrow 0$. Since $M$ is torsion free $Q$ has support in codimension 2. Then apply $Hom\_R(M, - )$, which is left exact, so we get, with $Hom\_R(M,M)=k$: $0\rightarrow k\rightarrow Hom\_R(M,M^{\\*\\*}) \rightarrow Hom\_R(M,Q)$. But here i am stuck.


Or is this assertion wrong, i.e. are there more morphisms? If it is right, can it be generalized to a bigger class of algebras $R$?



Morphisms of a simple sheaf over an algebra to its double dual
Any $R$-homomorphism (in fact any $\mathcal O\_S$-homomorphism) $M \to M^{\*\*}$ extends to a morphism $M^{\*\*}\to M^{\*\*}$ (as $M$ is locally free in codimension $1$ and $M^{\*\*}$ is the maximal extension from outside codimension $2$. This gives what you want. as $Hom\_R(M^{\*\*},M^{\*\*})=k$ for the same reason as it is true of $M$.


