Stanley-Reisner ring

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