Suppose that $\left(M, \mathcal{O}\right)$ is an analytic space and $A$ is a subspace with defining coherent ideal sheaf $J$. One can construct from any coherent sheaf $\mathcal{F}$ on $M$ the linear fiber space $V\left(\mathcal{F}\right)$.
Moreover, there exist the tangent fiber spaces $TM$ and $TA$ and there is an embedding $\iota\colon TA \to TM\times_M A$. To me the generalization of the "normal vector bundle" would be the cokernel of $\iota$. However, Fischer defines the normal space of $A$ as the fiber space $V\left(J\right)$ associated to $J$.
My question is now: Is there an isomorphism $\text{coker}\left(\iota\right) \to V\left(J\right)\times_M A$?
I vaguely remember such a comment in the manifold case, but can not recall the source or the argument. I have not been able to come up with a proof, since I do not see a connection between the two fiber spaces.