Stanley-Reisner ring

Stanley–Reisner face ring, face ring

The Stanley–Reisner ring of a simplicial complex over a field is the quotient ring

where are the vertices of , denotes the polynomial ring over in the variables , and is the ideal in generated by the non-faces of , i.e.,

The support of any monomial in is a face of . In particular, the square-free monomials of correspond bijectively to the faces of , and are therefore called the face-monomials

One may thus write more compactly as .

It is easy to verify that the Krull dimension of (cf. also Dimension) is one greater than the dimension of ().

Recall that the Hilbert series of a finitely-generated -graded module over a finitely-generated -algebra is defined by . The Hilbert series of may be described from the combinatorics of . Let , let , and call the -vector of . Then

where the sequence , called the -vector of , may be derived from the -vector of (and vice versa) by the equation

The mapping from to allows properties defined for rings to be naturally extended to simplicial complexes. The most well-known and useful example is Cohen–Macaulayness: A simplicial complex is defined to be Cohen–Macaulay (over the field ) when is Cohen–Macaulay (cf. also Cohen–Macaulay ring). The utility of this extension is demonstrated in the proof that if (the geometric realization of) a simplicial complex is homeomorphic to a sphere, then its -vector satisfies a condition called the upper bound conjecture (for details, see [a1], Sect. II.3,4). The statement of this result requires no algebra, but the proof relies heavily upon the Stanley–Reisner ring and Cohen–Macaulayness. Many other applications of the Stanley–Reisner ring may be found in [a1], Chaps. II, III.

Finally, there is an anti-commutative version of the Stanley–Reisner ring, called the exterior face ring or indicator algebra, in which the polynomial ring in the definition of is replaced by the exterior algebra .

References

[a1]	Richard P. Stanley, "Combinatorics and commutative algebra" , (2nd ed.) Birkhäuser (1996) ISBN 0-81764-369-9 Zbl 1157.13302 Zbl 0838.13008