Level complexes and barycentric subdivisions (Q2277032)

If \(\Delta\) is a finite simplicial complex, you can associate to it its Stanley-Reisner ring K[\(\Delta\) ]. A ring property is called topological if \(K[\Delta_ 1]\) has the property if and only if \(K[\Delta_ 2]\) has it when the geometric realization \(| \Delta_ 1|\) and \(| \Delta_ 2|\) are homeomorphic. Suppose \(\Delta\subset \{1,2,...,n\}\), denote \(A=K[x_ 1,...,x_ n]\) and let \(...\to F_ 2\to F_ 1\to F_ 0\to R\to 0\) a minimal free resolution of \(R=K[\Delta]\) as an A-module. \(\Delta\) is called a level complex if \(F_{n-d}\) is generated by \((F_{n-d})_ i\) for some i, d\(=\dim (R)\). The author shows that level is not a topological property. However, if \(\Delta\) is a level complex so is its barycentric subdivision.