---
tag: 9186DYXE86
kind: concept
name: Stanley–Reisner ring
msc: [13F55, "05E45", "05E40"]
sources:
  - url: https://en.wikipedia.org/wiki/Stanley–Reisner_ring
same_as:
  wikidata: Q7600086
  wikipedia: Stanley–Reisner ring
---

## Summary

The Stanley–Reisner ring of a finite simplicial complex is the quotient of a polynomial ring by the ideal of products of variables over the non-faces.
It turns a combinatorial object into a commutative ring.
Algebraic invariants of the ring, such as the Hilbert series and the Krull dimension, then encode counts of faces.
Its ideal is a squarefree [[monomial ideal|4G25S7PJ22]], and every squarefree monomial ideal arises this way.
The construction is a main tool in the proof of the upper bound theorem for simplicial spheres.

## Explanations

### L1 human

A simplicial complex is a collection of vertices, edges, triangles and higher simplices glued along faces.
Give each vertex a variable.
In the Stanley–Reisner ring, a product of distinct variables is zero exactly when those vertices do not span a face.
For the boundary of a triangle with vertices $1, 2, 3$, every pair of vertices is an edge, but the whole triangle is missing, so the ring is $k[x_1, x_2, x_3]/(x_1 x_2 x_3)$.
Counting the monomials that survive in each degree recovers the numbers of vertices, edges and so on.