Stanley Reisner Ring
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 c...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 on the vertex set {x1, ,xn} and a field k, the corresponding Stanley Reisner ring, or face ring, denoted k[ ], is obtained from the polynomial ring k[x1, ,xn] by quotienting out the ideal I generated by the square-free monomials corresponding to the non-faces of : I_Delta=(x_{i_1}ldots x_{i_r}: {i_1,ldots,i_r}notinDelta), quad k[Delta]=k[x_1,ldots,x_n]/I_Delta. The ideal I is called the Stanley Reisner ideal or the face ideal of .
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