Namespaces
Variants
Actions

Stanley-Reisner ring

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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=54269
This article was adapted from an original article by Art Duval (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article