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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] | R. Stanley, "Combinatorics and commutative algebra" , Birkhäuser (1996) (Edition: Second) |
Stanley-Reisner ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stanley-Reisner_ring&oldid=23037