Stanley–Reisner ring

In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combinatorial commutative algebra. Its properties were investigated by Richard Stanley, Melvin Hochster, and Gerald Reisner in the early 1970s.

Definition and properties

Given an abstract simplicial complex Δでるた on the vertex set {x₁,...,x_n} and a field k, the corresponding Stanley–Reisner ring, or face ring, denoted k[Δでるた], is obtained from the polynomial ring k[x₁,...,x_n] by quotienting out the ideal I_Δでるた generated by the square-free monomials corresponding to the non-faces of Δでるた:

$I_{\Delta }=(x_{i_{1}}\ldots x_{i_{r}}:\{i_{1},\ldots ,i_{r}\}\notin \Delta ),\quad k[\Delta ]=k[x_{1},\ldots ,x_{n}]/I_{\Delta }.$

The ideal I_Δでるた is called the Stanley–Reisner ideal or the face ideal of Δでるた.

Properties

• The Stanley–Reisner ring k[Δでるた] is multigraded by Zⁿ, where the degree of the variable x_i is the ith standard basis vector e_i of Zⁿ.
• As a vector space over k, the Stanley–Reisner ring of Δでるた admits a direct sum decomposition

$k[\Delta ]=\bigoplus _{\sigma \in \Delta }k[\Delta ]_{\sigma },$

whose summands k[Δでるた]_σしぐま have a basis of the monomials (not necessarily square-free) supported on the faces σしぐま of Δでるた.

• The Krull dimension of k[Δでるた] is one larger than the dimension of the simplicial complex Δでるた.
• The multigraded, or fine, Hilbert series of k[Δでるた] is given by the formula

$H(k[\Delta ];x_{1},\ldots ,x_{n})=\sum _{\sigma \in \Delta }\prod _{i\in \sigma }{\frac {x_{i}}{1-x_{i}}}.$

• The ordinary, or coarse, Hilbert series of k[Δでるた] is obtained from its multigraded Hilbert series by setting the degree of every variable x_i equal to 1:

$H(k[\Delta ];t,\ldots ,t)={\frac {1}{(1-t)^{n}}}\sum _{i=0}^{d}f_{i-1}t^{i}(1-t)^{n-i},$

where d = dim(Δでるた) + 1 is the Krull dimension of k[Δでるた] and f_i is the number of i-faces of Δでるた. If it is written in the form

$H(k[\Delta ];t,\ldots ,t)={\frac {h_{0}+h_{1}t+\cdots +h_{d}t^{d}}{(1-t)^{d}}}$

then the coefficients (h₀, ..., h_d) of the numerator form the h-vector of the simplicial complex Δでるた.

Examples

It is common to assume that every vertex {x_i} is a simplex in Δでるた. Thus none of the variables belongs to the Stanley–Reisner ideal I_Δでるた.

• Δでるた is a simplex {x₁,...,x_n}. Then I_Δでるた is the zero ideal and

$k[\Delta ]=k[x_{1},\ldots ,x_{n}]$

is the polynomial algebra in n variables over k.

• The simplicial complex Δでるた consists of n isolated vertices {x₁}, ..., {x_n}. Then

$I_{\Delta }=\{x_{i}x_{j}:1\leq i<j\leq n\}$

and the Stanley–Reisner ring is the following truncation of the polynomial ring in n variables over k:

$k[\Delta ]=k\oplus \bigoplus _{1\leq i\leq n}x_{i}k[x_{i}].$

• Generalizing the previous two examples, let Δでるた be the d-skeleton of the simplex {x₁,...,x_n}, thus it consists of all (d + 1)-element subsets of {x₁,...,x_n}. Then the Stanley–Reisner ring is following truncation of the polynomial ring in n variables over k:

$k[\Delta ]=k\oplus \bigoplus _{0\leq r\leq d}\bigoplus _{i_{0}<\ldots <i_{r}}x_{i_{0}}\ldots x_{i_{r}}k[x_{i_{0}},\ldots ,x_{i_{r}}].$

• Suppose that the abstract simplicial complex Δでるた is a simplicial join of abstract simplicial complexes Δでるた^′ on x₁,...,x_m and Δでるた^′′ on x_m+1,...,x_n. Then the Stanley–Reisner ring of Δでるた is the tensor product over k of the Stanley–Reisner rings of Δでるた^′ and Δでるた^′′:

$k[\Delta ]\simeq k[\Delta ']\otimes _{k}k[\Delta ''].$

Cohen–Macaulay condition and the upper bound conjecture

The face ring k[Δでるた] is a multigraded algebra over k all of whose components with respect to the fine grading have dimension at most 1. Consequently, its homology can be studied by combinatorial and geometric methods. An abstract simplicial complex Δでるた is called Cohen–Macaulay over k if its face ring is a Cohen–Macaulay ring. In his 1974 thesis, Gerald Reisner gave a complete characterization of such complexes. This was soon followed up by more precise homological results about face rings due to Melvin Hochster. Then Richard Stanley found a way to prove the Upper Bound Conjecture for simplicial spheres, which was open at the time, using the face ring construction and Reisner's criterion of Cohen–Macaulayness. Stanley's idea of translating difficult conjectures in algebraic combinatorics into statements from commutative algebra and proving them by means of homological techniques was the origin of the rapidly developing field of combinatorial commutative algebra.

Reisner's criterion

A simplicial complex Δでるた is Cohen–Macaulay over k if and only if for all simplices σしぐま ∈ Δでるた, all reduced simplicial homology groups of the link of σしぐま in Δでるた with coefficients in k are zero, except the top dimensional one:

${\tilde {H}}_{i}(\operatorname {link} _{\Delta }(\sigma );k)=0\quad {\text{for all}}\quad i<\dim \operatorname {link} _{\Delta }(\sigma ).$

A result due to Munkres then shows that the Cohen–Macaulayness of Δでるた over k is a topological property: it depends only on the homeomorphism class of the simplicial complex Δでるた. Namely, let |Δでるた| be the geometric realization of Δでるた. Then the vanishing of the simplicial homology groups in Reisner's criterion is equivalent to the following statement about the reduced and relative singular homology groups of |Δでるた|:

${\text{For all }}p\in |\Delta |{\text{ and for all }}i<\dim |\Delta |=d-1,\quad {\tilde {H}}_{i}(\operatorname {|} \Delta |;k)=H_{i}(\operatorname {|} \Delta |,\operatorname {|} \Delta |-p;k)=0.$

In particular, if the complex Δでるた is a simplicial sphere, that is, |Δでるた| is homeomorphic to a sphere, then it is Cohen–Macaulay over any field. This is a key step in Stanley's proof of the Upper Bound Conjecture. By contrast, there are examples of simplicial complexes whose Cohen–Macaulayness depends on the characteristic of the field k.

References

• Melvin Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes. Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), pp. 171–223. Lecture Notes in Pure and Appl. Math., Vol. 26, Dekker, New York, 1977
• Stanley, Richard (1996). Combinatorics and commutative algebra. Progress in Mathematics. Vol. 41 (Second ed.). Boston, MA: Birkhäuser Boston. ISBN 0-8176-3836-9. Zbl 0838.13008.
• Bruns, Winfried; Herzog, Jürgen (1993). Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics. Vol. 39. Cambridge University Press. ISBN 0-521-41068-1. Zbl 0788.13005.
• Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics. Vol. 227. New York, NY: Springer-Verlag. ISBN 0-387-23707-0. Zbl 1090.13001.

Further reading

• Panov, Taras E. (2008). "Cohomology of face rings, and torus actions". In Young, Nicholas; Choi, Yemon (eds.). Surveys in contemporary mathematics. London Mathematical Society Lecture Note Series. Vol. 347. Cambridge University Press. pp. 165–201. ISBN 978-0-521-70564-6. Zbl 1140.13018.

External links

• "Stanley–Reisner ring", Encyclopedia of Mathematics, EMS Press, 2001 [1994]