I am reading section 3.1 of this article by Bezrukavnikov and Kaledin and in the very beginning of this section there is this paragraph where they use a notion of (formal) germs around a closed subscheme:
Well, I am trying to guess what is this "(formal) germs of functions near a closed subscheme". My try:
Let $X$ be a scheme and $i: Z \hookrightarrow X$ a closed subscheme, maybe i want the germs to be $i^{-1} O_X$, but is this naturally a $O_Z-$algebra? And if I understand it well, I want it to be local as well so I could take the completion in the adic topology.
If $Z$ is a closed point (and the base field $k$ here is always algebraically closed) then $k(Z) \simeq k$ and $i^{-1} O_{X} = O_{X,Z}$, so the answer is yes. I don't know what is going on in general.