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.

analytic Zariskitopology? $\endgroup$ – Qfwfq Jan 31 '17 at 19:08