Stanley–Reisner rings and simplicial complexes

The Stanley–Reisner ideal is generated by the squarefree product of variables in every nonface; minimal nonfaces suffice. The quotient k[Δ] retains a monomial exactly when its support is a face. Powers themselves are not forbidden: if vertex i is present, all positive powers of xi survive. Thus this ring is not the finite-dimensional face vector space whose basis is only squarefree faces. It is an infinite-dimensional graded algebra with a finite combinatorial support rule.

For a face F, monomials with support exactly F have positive exponents on its vertices and exponent zero elsewhere. Their generating series is (t/(1−t)) to the size of F, with the empty face contributing one. Summing over faces gives the Hilbert series. This counts an explicit monomial basis and works over every field, so the resulting face numbers and Hilbert coefficients are characteristic-independent even though depth and homology can later depend on characteristic.

Write d for the maximum face size, so the complex has dimension d−1 and its face ring has Krull dimension d. Put f−1=1 for the empty face and fi−1 for the number of i-element faces. Clearing the denominator of the support formula defines the h-polynomial of degree at most d. This change of coordinates between f and h is purely algebraic and does not require purity, shellability or Cohen–Macaulayness. The resulting h-coefficients need not be nonnegative for an arbitrary complex.

When k is infinite and k[Δ] is Cohen–Macaulay, a linear system of parameters is a regular sequence. Quotienting by those d linear forms produces a finite-dimensional graded algebra whose Hilbert series is the h-polynomial. Hence every h-coefficient is a nonnegative integer. These hypotheses explain the positivity, rather than allowing one to infer it from the rational-form Hilbert series alone. Over a finite field, one may extend scalars to an infinite field without changing the relevant Hilbert numbers.

For a face F, its link consists of faces G disjoint from F such that their union remains a face. Reisner’s criterion says that a finite complex is Cohen–Macaulay over k exactly when every link, including the link of the empty face, has vanishing reduced k-homology below its own dimension. This is stated as a theorem, not proved from the elementary support count. The empty face test matters: inspecting only vertex links can miss a disconnected complex.

Homology over k can depend on characteristic because integral torsion behaves differently after changing coefficients. Consequently the face ring may be Cohen–Macaulay over one field and fail over another, although its monomial Hilbert series stays the same. Even under Cohen–Macaulayness, h-positivity alone does not imply unimodality, log-concavity or real-rootedness. Each stronger property needs its own assumptions or structure. Keeping these distinctions prevents an attractive translation between algebra and combinatorics from becoming an unjustified chain of implications.