Stanley–Reisner ring

Last updated February 24, 2019

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.^[1] Its properties were investigated by Richard Stanley, Melvin Hochster, and Gerald Reisner in the early 1970s.

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In algebra, a monomial ideal is an ideal generated by some monomials in a multivariate polynomial ring over a field.

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring similarly to the way that, in group theory, a normal subgroup can be used to construct a quotient group.

Definition and properties

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

In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of non-empty finite sets closed under the operation of taking non-empty subsets. In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems.

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 is called the Stanley–Reisner ideal or the face ideal of Δ.^[2]

Properties

The Stanley–Reisner ring k[Δ] 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 admits a direct sum decomposition

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

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

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

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

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] 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) + 1 is the Krull dimension of k[Δ] and f_i is the number of i-faces of Δ. 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.

Examples

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

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

Simplex generalization of the notion of a triangle or tetrahedron to arbitrary dimensions

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points $are affinely independent, which means are linearly independent. Then, the simplex determined by them is the set of points$

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

is the polynomial algebra in n variables over k.

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

In mathematics, the tensor product $V \otimes W$ of two vector spaces $V$ and $W$ is itself a vector space, endowed with the operation of bilinear composition, denoted by $\otimes$ , from ordered pairs in the Cartesian product $V \times W$ onto $V \otimes W$ in a way that generalizes the outer product. The tensor product of $V$ and $W$ is the vector space generated by the symbols $v \otimes w$ , with $v \in V$ and $w \in W$ , in which the relations of bilinearity are imposed for the product operation $\otimes$ , and no other relations are assumed to hold. The tensor product space is thus the "freest" such vector space, in the sense of having the fewest constraints.

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

Cohen–Macaulay condition and the upper bound conjecture

The face ring k[Δ] 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 is called Cohen–Macaulay over k if its face ring is a Cohen–Macaulay ring.^[3] 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 is Cohen–Macaulay over k if and only if for all simplices σ ∈ Δ, all reduced simplicial homology groups of the link of σ in Δ with coefficients in k are zero, except the top dimensional one:^[3]

{\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 over k is a topological property: it depends only on the homeomorphism class of the simplicial complex Δ. Namely, let |Δ| be the geometric realization of Δ. 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|:

{\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 is a simplicial sphere, that is, |Δ| 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.

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References

↑ Miller & Sturmfels (2005) p.19
↑ Miller & Sturmfels (2005) pp.3–5
1 2 Miller & Sturmfels (2005) p.101

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. 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. 39. Cambridge University Press. ISBN 0-521-41068-1. Zbl 0788.13005.
Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics. 227. New York, NY: Springer-Verlag. ISBN 0-387-23707-0. Zbl 1090.13001.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Stanley–Reisner ring", Encyclopedia of Mathematics , Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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[MS19-1] Miller & Sturmfels (2005) p.19

[MS35-2] Miller & Sturmfels (2005) pp.3–5

[MS101-3] 1 2 Miller & Sturmfels (2005) p.101