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 on the vertex set {x1,...,xn} and a field k, the corresponding Stanley–Reisner ring, or face ring, denoted k[Δ], is obtained from the polynomial ring k[x1,...,xn] by quotienting out the ideal IΔ generated by the square-free monomials corresponding to the non-faces of Δ:
I Δ = ( x i 1 … x i r : { i 1 , … , i r } ∉ Δ ) , k [ Δ ] = k [ x 1 , … , x n ] / I Δ . {\displaystyle 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 IΔ is called the Stanley–Reisner ideal or the face ideal of Δ.
Properties The Stanley–Reisner ring k[Δ] is multigraded by Zn, where the degree of the variable xi is the ith standard basis vector ei of Zn. As a vector space over k, the Stanley–Reisner ring of Δ admits a direct sum decomposition
k [ Δ ] = ⨁ σ ∈ Δ k [ Δ ] σ , {\displaystyle 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
H ( k [ Δ ] ; x 1 , … , x n ) = ∑ σ ∈ Δ ∏ i ∈ σ x i 1 − x i . {\displaystyle 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 xi equal to 1:
H ( k [ Δ ] ; t , … , t ) = 1 ( 1 − t ) n ∑ i = 0 d f i − 1 t i ( 1 − t ) n − i , {\displaystyle 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 fi is the number of i-faces of Δ. If it is written in the form
H ( k [ Δ ] ; t , … , t ) = h 0 + h 1 t + ⋯ + h d t d ( 1 − t ) d {\displaystyle H(k[\Delta ];t,\ldots ,t)={\frac {h_{0}+h_{1}t+\cdots +h_{d}t^{d}}{(1-t)^{d}}}}
then the coefficients (h0, ..., hd) of the numerator form the h-vector of the simplicial complex Δ.
Examples It is common to assume that every vertex {xi} is a simplex in Δ. Thus none of the variables belongs to the Stanley–Reisner ideal IΔ.
Δ is a simplex {x1,...,xn}. Then IΔ is the zero ideal and
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