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''Stanley–Reisner face ring, face ring''

The Stanley–Reisner ring of a [[simplicial complex]] $\Delta$ over a [[field]] $k$ is the quotient ring
$$
k[\Delta] = k[x_1,\ldots,x_n]/I_\Delta
$$
where $\{x_1,\ldots,x_n\}$ are the vertices of $\Delta$, $k[x_1,\ldots,x_n]$ denotes the polynomial ring over $k$ in the variables $\{x_1,\ldots,x_n\}$, and $I_\Delta$ is the [[ideal]] in $k[x_1,\ldots,x_n]$  generated by the non-faces of $\Delta$, i.e.,
$$
I_\Delta = \left\langle{ x_{i_1}\cdots x_{i_j} : \{i_1,\ldots,i_j\} \not\in \Delta }\right\rangle \ .
$$

The support of any monomial in $k[\Delta]$ is a face of $\Delta$. In particular, the square-free monomials of $k[\Delta]$ correspond bijectively to the faces of $\Delta$, and are therefore called the face-monomials
$$
x^F = \prod_{x_i\in F} x_i \ .
$$

One may thus write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052018.png" /> more compactly as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052019.png" />.

It is easy to verify that the Krull dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052020.png" /> (cf. also [[Dimension|Dimension]]) is one greater than the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052021.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052022.png" />).

Recall that the Hilbert series of a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052023.png" />-graded module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052024.png" /> over a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052025.png" />-algebra is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052026.png" />. The Hilbert series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052027.png" /> may be described from the combinatorics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052028.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052029.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052030.png" />, and call <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052031.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052033.png" />-vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052034.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052035.png" /></td> </tr></table>

where the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052036.png" />, called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052038.png" />-vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052039.png" />, may be derived from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052040.png" />-vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052041.png" /> (and vice versa) by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052042.png" /></td> </tr></table>

The mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052043.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052044.png" /> allows properties defined for rings to be naturally extended to simplicial complexes. The most well-known and useful example is Cohen–Macaulayness: A simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052045.png" /> is defined to be Cohen–Macaulay (over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052046.png" />) when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052047.png" /> is Cohen–Macaulay (cf. also [[Cohen–Macaulay ring|Cohen–Macaulay ring]]). The utility of this extension is demonstrated in the proof that if (the geometric realization of) a simplicial complex is homeomorphic to a sphere, then its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052048.png" />-vector satisfies a condition called the upper bound conjecture (for details, see [[#References|[a1]]], Sect. II.3,4). The statement of this result requires no algebra, but the proof relies heavily upon the Stanley–Reisner ring and Cohen–Macaulayness. Many other applications of the Stanley–Reisner ring may be found in [[#References|[a1]]], Chaps. II, III.

Finally, there is an anti-commutative version of the Stanley–Reisner ring, called the exterior face ring or indicator algebra, in which the polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052049.png" /> in the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052050.png" /> is replaced by the [[Exterior algebra|exterior algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052051.png" />.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">Richard P. Stanley,   "Combinatorics and commutative algebra" , (2nd ed.) Birkhäuser  (1996) ISBN 0-81764-369-9  {{ZBL|1157.13302|}}  {{ZBL|0838.13008}}</TD></TR>
</table>

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