Consider a pure finite abstract simplicial complex $\Delta$. Define its ***diameter*** as the maximal distance between any two facets, i.e., between any two faces of maximal dimension $d-1$. The ***distance*** between two facets is the shortest path between them. A ***path*** of length $k$ in $\Delta$ is just a sequence of facets $(F_1, \ldots, F_{k+1})$ with $F_i \in \Delta$ such that $F_i \cap F_{i+1}$ is a ridge, i.e., a $(d-2)$-dimensional face.

For the class of simplicial complexes I am interested in, we can assume that such a path always exists.

To $\Delta$ one can associate a ring, the so-called Stanley-Reisner ring. It is defined as the quotient of a polynomial ring. Suppose $x_1, \ldots, x_n$ are the vertices of $\Delta$. Then its ***Stanley-Reisner ideal*** $I_{\Delta}$ is generated by the non-faces:

$I_{\Delta} = (x_{i_1} \cdots x_{i_s} : \lbrace x_{i_1}, \ldots, x_{i_s} \rbrace \not\in \Delta)$

Here $x_i$ denotes a vertex in $\Delta$ and at the same time also a variable in $k[x_1, \ldots, x_n]$, where $k$ is a field. The ***Stanley-Reisner ring*** is then defined as $k[\Delta] = k[x_1, \ldots, x_n]/I_{\Delta}$. These rings provide a nice bridge between combinatorics and geometry on the one hand and commutative algebra on the other.

So much for the setting. Now, what I am wondering about is if the diameter of the simplicial complex is represented by some property of $k[\Delta]$. Or is it maybe known that the diameter cannot be extracted from the ring?