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In mathematics, a '''Stanley–Reisner ring''' is a quotient of a [[polynomial ring|polynomial algebra]] over a [[field (algebra)|field]] by a square-free monomial [[ideal (ring theory)|ideal]]. Such ideals are described more geometrically in terms of finite [[simplicial complex]]es. The Stanley–Reisner ring construction is a basic tool within [[algebraic combinatorics]] and [[combinatorial commutative algebra]].<ref name=MS19>Miller & Sturmfels (2005) p.19</ref> Its properties were investigated by [[Richard P. Stanley|Richard Stanley]], [[Melvin Hochster]], and Gerald Reisner in the early 1970s.

== Definition and properties ==

Given an [[abstract simplicial complex]] ''Δ'' on the vertex set {''x''1,…,''x''''n''} and a field '''k''', the corresponding '''Stanley–Reisner ring''', or '''face ring''', denoted '''k'''[''Δ''], is obtained from the polynomial ring '''k'''[''x''1,…,''x''''n''] by quotienting out the ideal ''I''''Δ'' generated by the square-free monomials corresponding to the non-faces of ''Δ'':

: <math> 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. </math>

The ideal ''I''''Δ'' is called the '''Stanley–Reisner ideal''' or the '''face ideal''' of ''Δ''.<ref name=MS35>Miller & Sturmfels (2005) pp.3–5</ref>

=== Properties ===

* The Stanley–Reisner ring '''k'''[''Δ''] is multigraded by '''Z'''''n'', where the degree of the variable ''x''''i'' is the ''i''th standard basis vector ''e''''i'' of '''Z'''''n''.

* As a vector space over '''k''', the Stanley–Reisner ring of ''Δ'' admits a direct sum decomposition

:: <math> k[\Delta] = \bigoplus_{\sigma\in\Delta}k[\Delta]_\sigma,</math>

: 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

:: <math> H(k[\Delta]; x_1,\ldots,x_n) =
\sum_{\sigma\in\Delta}\prod_{i\in\sigma}\frac{x_i}{1-x_i}. </math>

* 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:

:: <math> H(k[\Delta]; t,\ldots,t) =
\frac{1}{(1-t)^n}\sum_{i=0}^d f_{i-1} t^i(1-t)^{n-i}, </math>

: where ''d'' = dim(''Δ'')+1 is the Krull dimension of ''Δ'' and ''f''''i'' is the number of ''i''-faces of ''Δ''. If it is written in the form

:: <math> H(k[\Delta]; t,\ldots,t) =
\frac{h_0 + h_1 t + \cdots + h_d t^d}{(1-t)^d} </math>

:then the coefficients (''h''0, …, ''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]] {''x''1,…,''x''''n''}. Then ''I''Δ is the zero ideal and

:: <math> k[\Delta]=k[x_1,\ldots,x_n] </math>

:is the polynomial algebra in ''n'' variables over '''k'''.

* The simplicial complex ''Δ'' consists of ''n'' isolated vertices {''x''1}, …, {''x''''n''}. Then

:: <math>I_\Delta=\{x_i x_j: 1\leq i < j \leq n\} </math>

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

:: <math> k[\Delta] = k\oplus\bigoplus_{1\leq i\leq n} x_i k[x_i]. </math>

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

:: <math> 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}]. </math>

* Suppose that the abstract simplicial complex ''Δ'' is a simplicial join of abstract simplicial complexes ''Δ''′ on ''x''1,…,''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''′′:

:: <math> k[\Delta]\simeq k[\Delta']\otimes_k k[\Delta'']. </math>

== 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]].<ref name=MS101>Miller & Sturmfels (2005) p.101</ref> 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 sphere]]s, 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 algebra|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:<ref name=MS101/>

: <math> \tilde{H}_{i}(\operatorname{link}_\Delta(\sigma); k)=0\quad \textrm{for\, all} \quad
i<\operatorname{dim}\, \operatorname{link}_\Delta(\sigma). </math>

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''|:

: <math> \text{For all } p\in|\Delta|\text{ and for all }
i<\operatorname{dim}\, |\Delta| = d-1, \quad
\tilde{H}_i(\operatorname |\Delta|; k) =
H_i(\operatorname |\Delta|, \operatorname |\Delta| - p; k) = 0. </math>

In particular, if the complex ''Δ'' is a [[simplicial sphere]], that is, |''Δ''| is homeomorphic to a [[n-sphere|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 ==
{{reflist}}
* [[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
* {{cite book | authorlink=Richard P. Stanley | first=Richard | last=Stanley | title=Combinatorics and commutative algebra | edition=Second | series=Progress in Mathematics | volume=41 | publisher=Birkhäuser Boston | location=Boston, MA | year=1996 | isbn=0-8176-3836-9 | zbl=0838.13008 }}
* {{cite book | first1=Winfried | last1=Bruns | first2=Jürgen | last2=Herzog | title=Cohen–Macaulay rings | series=Cambridge Studies in Advanced Mathematics | volume=39 | series=[[Cambridge University Press]] | year=1993 | isbn=0-521-41068-1 | zbl=0788.13005 }}
* {{ cite book | last1=Miller | first1=Ezra | last2=Sturmfels | first2=Bernd | author2-link=Bernd Sturmfels | title=Combinatorial commutative algebra | series=Graduate Texts in Mathematics | volume=227 | location=New York, NY | publisher=[[Springer-Verlag]] | isbn=0-387-23707-0 | year=2005 | zbl=1090.13001 }}

==External links==
*{{Springer|id=s/s130520|title=Stanley–Reisner ring}}

{{DEFAULTSORT:Stanley-Reisner ring}}
[[Category:Algebraic combinatorics]]
[[Category:Commutative algebra]]

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