StanleyReisner

Stanley-Reisner theory, named for Richard P. Stanley and Reisner, is a framework in combinatorial commutative algebra and topological combinatorics that links simplicial complexes with square-free monomial ideals and their quotient rings. It provides a dictionary between combinatorial properties of a simplicial complex and algebraic properties of an associated monomial ideal.

Given a simplicial complex Δ on vertices {x1, ..., xn} over a field k, the Stanley-Reisner ideal IΔ in the polynomial ring k[x1, ..., xn] is generated by all square-free monomials xi1 xi2 ... xit such that {i1, ..., it} is not a face of Δ. The Stanley-Reisner ring (or face ring) is k[Δ] = k[x1, ..., xn]/IΔ. This construction encodes the combinatorial structure of Δ in algebraic form, enabling study via commutative algebra.

A central result is Reisner's criterion: Δ is Cohen–Macaulay over k if and only if the reduced homology of the links of all faces F ⊆ Δ vanishes below the top dimension of the links, with coefficients in k. This provides a purely combinatorial condition for an algebraic property of k[Δ].

Algebraic invariants of k[Δ] reflect combinatorial data: the Hilbert series of k[Δ] encodes the f-vector and h-vector of Δ, and properties such as Cohen–Macaulayness, Gorensteinness, or shellability have implications for face enumeration and topology. The theory has applications in algebraic geometry, toric varieties, and the study of simplicial complexes.

Stanley-Reisner theory remains a foundational tool in combinatorial topology and commutative algebra, bridging discrete and algebraic perspectives.