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''Stanley–Reisner face ring, face ring''
 
''Stanley–Reisner face ring, face ring''
  
The Stanley–Reisner ring of a [[Simplicial complex|simplicial complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s1305201.png" /> over a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s1305202.png" /> is the quotient ring
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The Stanley–Reisner ring of a [[simplicial complex]] $\Delta$ over a [[field]] $k$ is the quotient ring
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$$
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k[\Delta] = k[x_1,\ldots,x_n]/I_\Delta
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$$
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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.,
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$$
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I_\Delta = \left\langle{ x_{i_1}\cdots x_{i_j} : \{i_1,\ldots,i_j\} \not\in \Delta }\right\rangle \ .
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$$
  
<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/s1305203.png" /></td> </tr></table>
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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
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s1305204.png" /> are the vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s1305205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s1305206.png" /> denotes the polynomial ring over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s1305207.png" /> in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s1305208.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s1305209.png" /> is the [[Ideal|ideal]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052010.png" /> generated by the non-faces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052011.png" />, i.e.,
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x^F = \prod_{x_i\in F} x_i \ .
 
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$$
<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/s13052012.png" /></td> </tr></table>
 
 
 
The support of any monomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052013.png" /> is a face of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052014.png" />. In particular, the square-free monomials of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052015.png" /> correspond bijectively to the faces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130520/s13052016.png" />, and are therefore called the face-monomials
 
 
 
<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/s13052017.png" /></td> </tr></table>
 
  
 
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" />.
 
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" />.
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<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>
 
<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>
 
</table>
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{{TEX|part}}

Revision as of 20:00, 22 January 2018

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 more compactly as .

It is easy to verify that the Krull dimension of (cf. also Dimension) is one greater than the dimension of ().

Recall that the Hilbert series of a finitely-generated -graded module over a finitely-generated -algebra is defined by . The Hilbert series of may be described from the combinatorics of . Let , let , and call the -vector of . Then

where the sequence , called the -vector of , may be derived from the -vector of (and vice versa) by the equation

The mapping from to 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 is defined to be Cohen–Macaulay (over the field ) when is Cohen–Macaulay (cf. also 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 -vector satisfies a condition called the upper bound conjecture (for details, see [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 [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 in the definition of is replaced by the exterior algebra .

References

[a1] Richard P. Stanley, "Combinatorics and commutative algebra" , (2nd ed.) Birkhäuser (1996) ISBN 0-81764-369-9 Zbl 1157.13302 Zbl 0838.13008
How to Cite This Entry:
Stanley-Reisner ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stanley-Reisner_ring&oldid=42756
This article was adapted from an original article by Art Duval (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article