Homology groups of simplicial complements: A new proof of Hochster theorem

TL;DR

This paper introduces a new proof of Hochster's theorem by exploring homology groups derived from the exterior algebra of simplicial complements, revealing dualities with simplicial cohomology via Čech homology and Alexander duality.

Contribution

It provides a novel proof of Hochster's theorem using Čech homology and Alexander duality, connecting homology groups of simplicial complements to cohomology of subcomplexes.

Findings

Homology groups are isomorphic to Tor-groups of the face ring.

Homology groups have dualities with cohomology of subcomplexes.

New proof of Hochster's theorem established.

Abstract

In this paper, we consider homology groups induced by the exterior algebra generated by a simplicial compliment of a simplicial complex $K$. These homology groups are isomorphic to the Tor-groups $\mathrm{Tor}_{i, J}^{\mathbf{k}[m]}(\mathbf{k}(K),\mathbf{k})$ of the face ring $\mathbf{k}(K)$, which is very useful and much studied in toric topology. By using $\check{C}ech$ homology theory and Alexander duality theorem, we prove that these homology groups have dualities with the simplicial cohomology groups of the full subcomplexes of $K$. Then we give a new proof of Hochster's theorem.