Stanley–Reisner ring

Examples

It is common to assume that every vertex {x_i} is a simplex in Δ. Thus none of the variables belongs to the Stanley–Reisner ideal I_Δ.

• Δ is a simplex {x₁,...,x_n}. Then I_Δ is the zero ideal and

${\displaystyle k[\mathrm{\Delta }]=k[x_1,\dots ,x_n]}$

is the polynomial algebra in n variables over k.

• The simplicial complex Δ consists of n isolated vertices {x₁}, ..., {x_n}. Then

${\displaystyle I_{\mathrm{\Delta }}=\{x_i{x}_j:1\le i<j\le n\}}$

and the Stanley–Reisner ring is the following truncation of the polynomial ring in n variables over k:

${\displaystyle k[\mathrm{\Delta }]=k\oplus \underset{1\le i\le n}{⨁}{x}_{i}k[x_i].}$

• Generalizing the previous two examples, let Δ be the d-skeleton of the simplex {x₁,...,x_n}, thus it consists of all (d + 1)-element subsets of {x₁,...,x_n}. Then the Stanley–Reisner ring is following truncation of the polynomial ring in n variables over k:

${\displaystyle k[\mathrm{\Delta }]=k\oplus \underset{0\le r\le d}{⨁}\underset{i_0<\dots <i_r}{⨁}{x}_{i_0}\dots x_{i_r}k[x_{i_0},\dots ,x_{i_r}].}$

• Suppose that the abstract simplicial complex Δ is a simplicial join of abstract simplicial complexes Δ^′ on x₁,...,x_m and Δ^′′ on x_m+1,...,x_n. Then the Stanley–Reisner ring of Δ is the tensor product over k of the Stanley–Reisner rings of Δ^′ and Δ^′′:

${\displaystyle k[\mathrm{\Delta }]\simeq k[{\mathrm{\Delta }}^{\prime }]{\otimes }_{k}k[{\mathrm{\Delta }}^{″}].}$

Cohen–Macaulay condition and the upper bound conjecture

The face ring k[Δ] is a multigraded algebra over k all of whose components with respect to the fine grading have dimension at most 1. Consequently, its homology can be studied by combinatorial and geometric methods. An abstract simplicial complex Δ is called Cohen–Macaulay over k if its face ring is a Cohen–Macaulay ring. In his 1974 thesis, Gerald Reisner gave a complete characterization of such complexes. This was soon followed up by more precise homological results about face rings due to Melvin Hochster. Then Richard Stanley found a way to prove the Upper Bound Conjecture for simplicial spheres, which was open at the time, using the face ring construction and Reisner's criterion of Cohen–Macaulayness. Stanley's idea of translating difficult conjectures in algebraic combinatorics into statements from commutative algebra and proving them by means of homological techniques was the origin of the rapidly developing field of combinatorial commutative algebra.

Reisner's criterion

A simplicial complex Δ is Cohen–Macaulay over k if and only if for all simplices σ ∈ Δ, all reduced simplicial homology groups of the link of σ in Δ with coefficients in k are zero, except the top dimensional one:

${\displaystyle {\tilde{H}}_i({\mathrm{link}}_{\mathrm{\Delta }}(\sigma );k)=0\text{for all}i<\dim {\mathrm{link}}_{\mathrm{\Delta }}(\sigma ).}$

A result due to Munkres then shows that the Cohen–Macaulayness of Δ over k is a topological property: it depends only on the homeomorphism class of the simplicial complex Δ. Namely, let |Δ| be the geometric realization of Δ. Then the vanishing of the simplicial homology groups in Reisner's criterion is equivalent to the following statement about the reduced and relative singular homology groups of |Δ|:

${\displaystyle \text{For all }p\in |\mathrm{\Delta }|\text{ and for all }i<\dim |\mathrm{\Delta }|=d-1,{\tilde{H}}_i(|\mathrm{\Delta }|;k)=H_i(|\mathrm{\Delta }|,|\mathrm{\Delta }|-p;k)=0.}$

In particular, if the complex Δ is a simplicial sphere, that is, |Δ| is homeomorphic to a sphere, then it is Cohen–Macaulay over any field. This is a key step in Stanley's proof of the Upper Bound Conjecture. By contrast, there are examples of simplicial complexes whose Cohen–Macaulayness depends on the characteristic of the field k.