Let $(A,\mathfrak{m}_A)$ be a local Artinian $k$-algebra with residue field $k$. Then the scheme $\mathrm{Spec}(A)$ can be loosely seen as a "fat point", or an "infinitesimal neighbourhood" of a point $\mathrm{Spec}(k)$. If the maximal ideal satisfies the condition $\mathfrak{m}^k=0$, then I think the spec of $A$ can be seen as a "$(k-1)$-th order infinitesimal neighbourhood" of the closed point.
Recall: a small extension of $A$ is an extension of local Artinian $k$-algebras $0\to I \to B\to A \to 0$ such that $\mathfrak{m}_B\cdot I =0$.
Does the concept of a small extension have any geometric interpretation?
Does it mean that the closed embedding $\mathrm{Spec}(A)\hookrightarrow \mathrm{Spec}(B)$ is such that it only "fattens" already existent "directions" of $\mathrm{Spec}(A)$, making it into a "higher order" fattening of the closed point but without adding new "directions" of fattening? Is my suggestion totally misleading? Edit: as comments point out, my interpretation was not correct.
