---
tag: N8YJAFVQRW
kind: definition
name: Stanley–Reisner ring
msc: [13F55, "05E45", "05E40"]
defines: 9186DYXE86
uses: [04C5ZBG1WT, 0SDC934R52, 4G25S7PJ22, 0076HWHGHQ, MVGBS3J1P7]
sources:
  - url: https://en.wikipedia.org/wiki/Stanley–Reisner_ring
---

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $n \ge 1$.
Let $\Delta$ be a simplicial complex on the vertex set $[n] = \set{1, \ldots, n}$.
So $\Delta$ is a family of subsets of $[n]$ such that every subset of a member is a member.
For $F \subseteq [n]$, let $x_F = \prod_{i \in F} x_i$ in the [[polynomial ring|0SDC934R52]] $S = k[x_1, \ldots, x_n]$.
The *Stanley–Reisner ideal* of $\Delta$ is
$$I_\Delta = (x_F : F \subseteq [n], \; F \notin \Delta).$$
The *Stanley–Reisner ring*, or face ring, of $\Delta$ is the [[quotient ring|0076HWHGHQ]] $k[\Delta] = S/I_\Delta$.

## Remarks

The ideal $I_\Delta$ is already generated by the $x_F$ for the minimal non-faces $F$.
It is a squarefree [[monomial ideal|4G25S7PJ22]], and $\Delta \mapsto I_\Delta$ is a bijection from simplicial complexes on $[n]$ to squarefree monomial ideals of $S$.
The [[Krull dimension|MVGBS3J1P7]] of $k[\Delta]$ is $\dim \Delta + 1$, the largest size of a face.

Example: let $\Delta$ be the boundary of a triangle, with faces all subsets of $\set{1, 2, 3}$ except $\set{1, 2, 3}$.
The only non-face is $\set{1, 2, 3}$, so $I_\Delta = (x_1 x_2 x_3)$ and $k[\Delta] = k[x_1, x_2, x_3]/(x_1 x_2 x_3)$, of Krull dimension $2$.

Non-example: the ideal $(x_1^2)$ of $k[x_1, x_2]$ is a monomial ideal, but it is not of the form $I_\Delta$, because it is not generated by squarefree monomials.
Its quotient ring has the nonzero nilpotent element $x_1$, while every $k[\Delta]$ is reduced.