Let $X$ be a topological space, $A$ a sheaf of (unital and associative but not necessarily commutative) rings on $X$. Suppose $M$ is a simple quasicoherent $A$-module and $U$ an open subset of $X$. Is $M|\_U$ a simple $A|\_U$-module?



Is the restriction of a simple sheaf of modules simple?
Yes. Restriction is an exact functor split by its right adjoint (pushforward). If $M|\_{U}$ has a proper non-zero quotient $N$, then $i\_\*N$ receives a non-zero and thus injective map from $M$. However, this means that $M|\_U$ must map injectively to $N$, which is a contradiction.


