For any simplicial complex Δ and field K, one can associate a graded K-algebra K[Δ] (the Stanley-Reisner ring). For certain Δ and K, the Stanley-Reisner rings have a homogeneous system of parameters, Θ, such that K[Δ]/〈Θ〉 is finite-dimensional, and coefficients of its Hilbert series are the h-vector of Δ. The previous constructions of Θ were noncombinatorial. In the special case of cographic matroids, we give (for any field K) a combinatorial description of a homogeneous system of parameters in terms of the graph structure, as well as an explicit basis for the resulting quotient algebra. The results have applications to a central problem of reliability, namely the association of a multicomplex to a connected graph, such that the reliability is a simple function of the rank numbers.
- Cohen-Macaulay ring
- Gröbner basis
- Homogeneous system of parameters
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