Difference between revisions of "Stanley-Reisner ring"
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''Stanley–Reisner face ring, face ring'' | ''Stanley–Reisner face ring, face ring'' | ||
| − | The Stanley–Reisner ring of a [[ | + | 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 | |
| − | + | $$ | |
| − | + | x^F = \prod_{x_i\in F} x_i \ . | |
| − | + | $$ | |
| − | |||
| − | |||
| − | The support of any monomial in | ||
| − | |||
| − | |||
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> | ||
| + | |||
| + | {{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 |
Stanley-Reisner ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stanley-Reisner_ring&oldid=42756