The partition complex: An invitation to combinatorial commutative algebra

We provide a new foundation for combinatorial commutative algebra and Stanley-Reisner theory using the partition complex introduced in [1]. One of the main advantages is that it is entirely self-contained, using only a minimal knowledge of algebra and topology. On the other hand, we also develop new techniques and results using this approach. In particular, we provide 1. A novel, self-contained method of establishing Reisner's theorem and Schenzel's formula for Buchsbaum complexes. 2. A simple new way to establish Poincaré duality for face rings of manifolds, in much greater generality and precision than previous treatments. 3. A "master-theorem" to generalize several previous results concerning the Lefschetz theorem on subdivisions. 4. Proof for a conjecture of Kühnel concerning triangulated manifolds with boundary.