The canonical trace of Stanley-Reisner rings that are Gorenstein on the punctured spectrum

TL;DR

This paper proves that nearly Gorenstein Stanley-Reisner rings of dimension at least 3 are Gorenstein, providing a complete characterization of nearly Gorenstein Stanley-Reisner rings and linking Gorenstein properties to topological features.

Contribution

It establishes that nearly Gorenstein Stanley-Reisner rings of dimension ≥3 are Gorenstein, and characterizes Gorenstein on the punctured spectrum in terms of canonical trace and topological properties.

Findings

Nearly Gorenstein Stanley-Reisner rings of dimension ≥3 are Gorenstein.

Gorenstein on the punctured spectrum characterized by nearly Gorenstein property or canonical trace conditions.

Non-orientable homology manifolds correspond to cases where the canonical trace is the square of the irrelevant maximal ideal.

Abstract

In this paper we prove that nearly Gorenstein Stanley-Reisner rings of dimension at least 3 are indeed Gorenstein. By previous work of the first author this yields a complete characterization of nearly Gorenstein Stanley-Reisner rings. We also show that a Cohen-Macaulay Stanley-Reisner ring is Gorenstein on the punctured spectrum if and only if either it is nearly Gorenstein or its canonical trace is the square of its irrelevant maximal ideal, and that the latter case happens exactly for non-orientable homology manifolds.