Is there a name for the property of a sheaf $\mathcal F$ such that the restriction maps $\mathcal F(V) \to \mathcal F(U)$ are injective when $V$ is a connected open and $U$ is a nonempty open?
In other words, this is a sheaf which satisfies the "identity theorem" of complex analysis (taking for $\mathcal F$ the sheaf of holomorphic functions).
Many sheaves in algebraic geometry satisfy this property, but I've never heard a name for it. Perhaps a "rigid" sheaf, in contrast to a "flasque" sheaf? (This "identity theorem" property seems to be the one which gave rigid analytic geometry its name, but I have never seen the words "rigid sheaf" written down.)
Thank you.