Stanley–Reisner ring

In mathematics, a Stanley–Reisner 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.

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:

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

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.

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. 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.

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