Formal completion of a complex normal bundle along the zero section The total space of a complex normal bundle over a submanifold $Y$ in a complex manifold $X$ can be seen as an analytic scheme. If one blindly uses  Hartshorne's definition about "formal schemes" from his book "Algebraic Geometry" to formally complete the normal bundle $N$ along the zero section $\epsilon $,  the formal completion of $N$ would be of the form
$(\epsilon, \lim_{\leftarrow}\mathcal{O}_N/\mathcal{I}^n)$
 where $\mathcal{I}=\{f\in\mathcal{O}(N): f|_\epsilon=0\}$ is the ideal sheaf. 
What worries me is that Hartshorne's definition in the category of algebraic schemes is specifically said to be valid for noetherian schemes, which the complex manifold N isn't. Is that a problem? Do we need the noetherian condition in the complex analytic case at all? How can we adapt the algebraic definition of "formal scheme" for analytic schemes?
Thank you for your help. I hope I was able to formulate clearly my problem.
 A: Instead of trying to adapt Hartshorne's definition for formal completion of noetherian schemes to the difficult case of analytic schemes, which are never noetherian, we could instead consider the holomorphic symmetric algebra bundle $\eta:Sym^{\bullet}(N^*)\rightarrow Y$ of the conormal bundle $N^*\rightarrow Y$ and formally complete this bundle.
According section 3 of the article "Formality theorems for Hochschild chains in the Lie algebroid setting" by D. Calaque, V. Dolgushev and G. Halbout one can formally complete the holomorphic symmetric algebra bundle $Sym^{\bullet}(N^*)$ of the complex conormal bundle $N^*$. The local sections of the completion $\widehat Sym^{\bullet}(N^*)$ are then given in local coordinates by $f=\sum_{l\geq0}f_{i_1\dots i_l}(y)w^{i_1}\dots w^{i_l}$, where $y_1, \dots, y_l$ are some local coordinates on $Y$ and $w_1, \dots, w_m$, m=codim(Y) are the local coordinates on the fibres of $N$. 
This bundle, I think, can be interpreted as the formal completion of the complex-analytic locally free quasi-coherent sheaf $\mathcal{Sym^{\bullet}(N^*)}$. Moreover, comparing Hartshorne with this, it seems that $\mathcal{\widehat Sym^{\bullet}(N^*)}$ is indeed the analytic version of the usual formal completion of an algebraic bundle $N$ along its zero section. It is an exercise to show that in the algebraic case $\mathcal{\widehat Sym^{\bullet}(N^*)}=\mathcal O_{\hat N}$ which is precisely Hartshorne's definition of formal completion of a noetherian scheme. I am not fooling myself, do I?
