Namespaces
Variants
Actions

Difference between revisions of "Stanley-Reisner ring"

From Encyclopedia of Mathematics
Jump to: navigation, search
(→‎References: expand bibliodata)
 
(6 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
{{TEX|done}}{{MSC|13F55|05E45}}
 +
 
''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
+
The Stanley–Reisner ring of a [[simplicial complex]] $\Delta$ over a [[field]] $k$ is the quotient ring
 
+
$$
<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>
+
k[\Delta] = k[x_1,\ldots,x_n]/I_\Delta
 
+
$$
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.,
+
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.,
 
+
$$
<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>
+
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 <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" />.
 
 
 
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
+
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
 +
$$
 +
x^F = \prod_{x_i\in F} x_i \ .
 +
$$
  
<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>
+
One may thus define $I_\Delta$ more compactly as $I_\Delta = \left\langle{ x^F : F \not\in \Delta }\right\rangle$.
  
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
+
It is easy to verify that the Krull dimension of $k[\Delta]$ (cf. also [[Dimension]]) is one greater than the dimension of $\Delta$ ($\dim k[\Delta] = (\dim \Delta) + 1$).
  
<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>
+
Recall that the ''Hilbert series'' of a finitely-generated $\mathbf{Z}$-graded module $M$ over a finitely-generated $k$-algebra is defined by
 +
$$
 +
F(M,\lambda) = \sum_{i\in\mathbf{Z}} \dim_k M_i \, \lambda^i
 +
$$
 +
The Hilbert series of $k[\Delta]$ may be described from the combinatorics of $\Delta$. Let $\dim \Delta = d-1$, let $f_i = \vert\{ F\in \Delta : \dim F = i\}\vert$, and call $(f_{-1},f_0,\ldots,f_{d-1})$ the $f$-vector of $\Delta$. Then
 +
$$
 +
F(f[\Delta],\lambda) = \sum_{i=-1}^{d-1} \frac{f_i \lambda^{i+1}}{(1-\lambda)^{i+1}} = \frac{h_0 + \cdots + h_d\lambda^d}{(1-\lambda)^d}
 +
$$
 +
where the sequence $(h_0,\ldots,h_d)$, called the $h$-vector of $\Delta$, may be derived from the $f$-vector of $\Delta$ (and vice versa) by the equation
 +
$$
 +
\sum_{i=0}^d h_i x^{d-i} = \sum_{i=0}^d f_{i-1} (x-1)^{d-i} \ .
 +
$$
  
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.
+
The mapping from $\Delta$ to $k[\Delta]$ 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 $\Delta$ is defined to be Cohen–Macaulay (over the field $k$) when $k[\Delta]$ 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 $f$-vector satisfies a condition called the upper bound conjecture (for details, see {{Cite|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 {{Cite|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" />.
+
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 $k[x_1,\ldots,x_n]$ in the definition of $k[\Delta]$ is replaced by the [[exterior algebra]] $k\langle x_1,\ldots,x_n \rangle$.
  
 
====References====
 
====References====
<table>
+
* {{Ref|a1}} Richard P. Stanley, "Combinatorics and commutative algebra", (2nd ed.) Birkhäuser (1996) {{ISBN|0-81764-369-9}} {{ZBL|1157.13302|}}  {{ZBL|0838.13008}}
<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 {{ZBL1157.13302|}}  {{ZBL|0838.13008}}</TD></TR>
 
</table>
 

Latest revision as of 20:16, 8 November 2023

2020 Mathematics Subject Classification: Primary: 13F55 Secondary: 05E45 [MSN][ZBL]

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 define $I_\Delta$ more compactly as $I_\Delta = \left\langle{ x^F : F \not\in \Delta }\right\rangle$.

It is easy to verify that the Krull dimension of $k[\Delta]$ (cf. also Dimension) is one greater than the dimension of $\Delta$ ($\dim k[\Delta] = (\dim \Delta) + 1$).

Recall that the Hilbert series of a finitely-generated $\mathbf{Z}$-graded module $M$ over a finitely-generated $k$-algebra is defined by $$ F(M,\lambda) = \sum_{i\in\mathbf{Z}} \dim_k M_i \, \lambda^i $$ The Hilbert series of $k[\Delta]$ may be described from the combinatorics of $\Delta$. Let $\dim \Delta = d-1$, let $f_i = \vert\{ F\in \Delta : \dim F = i\}\vert$, and call $(f_{-1},f_0,\ldots,f_{d-1})$ the $f$-vector of $\Delta$. Then $$ F(f[\Delta],\lambda) = \sum_{i=-1}^{d-1} \frac{f_i \lambda^{i+1}}{(1-\lambda)^{i+1}} = \frac{h_0 + \cdots + h_d\lambda^d}{(1-\lambda)^d} $$ where the sequence $(h_0,\ldots,h_d)$, called the $h$-vector of $\Delta$, may be derived from the $f$-vector of $\Delta$ (and vice versa) by the equation $$ \sum_{i=0}^d h_i x^{d-i} = \sum_{i=0}^d f_{i-1} (x-1)^{d-i} \ . $$

The mapping from $\Delta$ to $k[\Delta]$ 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 $\Delta$ is defined to be Cohen–Macaulay (over the field $k$) when $k[\Delta]$ 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 $f$-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 $k[x_1,\ldots,x_n]$ in the definition of $k[\Delta]$ is replaced by the exterior algebra $k\langle x_1,\ldots,x_n \rangle$.

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=42755
This article was adapted from an original article by Art Duval (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article