Infinitesimal neighbourhood of a closed subscheme ($X_n=V({\mathcal{I}}^{n+1})$)

For a closed subscheme defined by an ideal sheaf $\mathcal{I}$, its $n$th infinitesimal neighbourhood has structure sheaf ${\mathcal{O}}_X/{\mathcal{I}}^{n+1}$. All these neighbourhoods have the same underlying topological space because the radicals of $\mathcal{I}$ and its positive powers agree. Their inverse limit of structure sheaves is the formal completion of a scheme.

// Target: ringed-space.bigb