  We provide a new foundation for combinatorial commutative algebra and
Stanley-Reisner theory using the partition complex introduced in [Adi18]. 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
  - A novel, self-contained method of establishing Reisner's theorem and
Schenzel's formula for Buchsbaum complexes.
  - A simple new way to establish Poincar\'e duality for face rings of
manifolds, in much greater generality and precision than previous treatments.
  - A "master-theorem" to generalize several previous results concerning the
Lefschetz theorem on subdivisions.
  - Proof for a conjecture of K\"uhnel concerning triangulated manifolds with
boundary.
