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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.〔Miller & Sturmfels (2005) p.19〕 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 and a field k, the corresponding Stanley–Reisner ring, or face ring, denoted k(), is obtained from the polynomial ring k() by quotienting out the ideal ''I''Δ generated by the square-free monomials corresponding to the non-faces of Δ:
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The ideal ''I''Δ is called the Stanley–Reisner ideal or the face ideal of Δ.〔Miller & Sturmfels (2005) pp.3–5〕
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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