Torsion Obstruction for Conclusive Posets

TL;DR

The paper refutes the CJW22 conjecture by exhibiting a finite poset (the minimal finite model of $\mathbb{R}P^2$) whose incidence-algebra cohomology depends on the base ring due to $2$-torsion in $H_1(\Delta(P),\mathbb{Z})$. It develops both a combinatorial framework and a topological perspective to characterize conclusiveness, yielding a precise criterion: a poset is conclusive iff $H_1(\Delta(P),\mathbb{Z})$ is torsion-free and equivalently $HH^1(I(P,k)) \cong H^1(\Delta(P),k)$ for all $k$. A key result is the graph-theoretic criterion $E(P) - \mathrm{rank}_k(\mathcal{M}_P) = V(P) - C(P)$ that determines when outer derivations can exist, together with a crown-based obstruction argument. The work also provides a purely combinatorial pathway (poset splitting CO18) to guarantee torsion-free homology and hence conclusiveness, supplemented by height and beat-point bounds (BM07) to yield practical tests for large posets. Together, these contributions clarify the dependence of poset cohomology on the base ring and furnish actionable criteria for determining conclusiveness without heavy homological computation.

Abstract

We give a counterexample to a conjecture made by Cigler, Jerman and Wojciechowski stating that all posets are conclusive. We also provide combinatorial characterizations for conclusiveness of finite posets and the existence of outer derivations.

Figures (2)

• Figure 1: The crown $C_n$ on $2n$ elements.
• Figure 2: The minimal finite model of $\mathbb{R}P^2$ (13 elements).

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Corollary 3.2
• Theorem 3.3
• proof
• Theorem 4.1
• proof