Diameter of simplicial complex mirrored in property of Stanley-Reisner ring?

Question

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?

Answer 1

Interestingly enough, there have been quite a few recent attempts at answering your question. First, some relevant background. The diameter that you defined is known as the diameter of the dual graph of the Stanley-Reisner ring $R=k[\Delta]$. The dual graph $G(X)$ can be defined over any scheme $X$ as follows: the vertices are the irreducible components of $X$, and two of them are connected by an edge if the dimension of the intersection is $\dim X-1$). In the local commutative algebra community, this is also sometimes referred to as the Hochster-Huneke graph, thanks to this paper. In combinatorics, it is also called the ridge graph.

It follows from a classical result by Hartshorne that if $R$ satisfies Serre's condition $(S_2)$ and $\dim R\geq 2$, then the dual graph of $Spec(R)$ is connected. So, if $R$ is Cohen-Macaulay, Goresntein (which all imply $(S_2)$) and $\dim R\geq 2$, it makes sense to ask whether one can bound the diameter of $G(R)$. The elephant in the room is:

(Polynomial Hirsch Conjecture, algebraic version): If $R=k[\Delta]$ is Gorenstein, the diameter of $G(R)$ is bounded above by a polynomial $f(c)$, here $c=n-\dim R$ is the codimension.

Note that Santos' counter-example to the original Hirsch conjecture implies that $f(c)=c$ does not work.

Now here are some sources of recent references:

1) See this paper for some introduction on the polynomial Hirsch conjecture.

2) A lot of good introductions and recent results on bounding $G(R)$, even for general standard-graded algebras, can be found on Matteo Varbaro's website and the references in his papers on dual graphs. For example, this recent one establishes the Hirsch bound for certain class of Gorenstein ideals.

3) Another question is: suppose we assume $R$ is just $(S_2)$, the minimal algebraic condition to guarantee that the diameter is finite (in the combinatorics community, this is sometimes known as $\Delta$ being normal). Can we bound the diameter?

Brent Holmes, a PhD student of mine has been working on this question, and he found some exponential bounds that slightly improve the known ones in the literature, as well as compute the best bounds for small $n$ and $d$. You can find the most updated version of his work on his website (the results I mentioned can be found in the paper "On the Diameter of Dual Graphs of Stanley-Reisner Rings...").