The prime spectrum of the talented monoid of a higher-rank graph and applications

In this paper, we further investigate the role of the graded Grothendieck group $K_0^{\text{gr}}$ and its positive cone (the talented monoid) as an effective tool for distinguishing structural types of algebras associated to higher-rank $k$-graphs. We study the prime spectrum (the space of all prime $\mathrm{\Gamma }$-order ideals equipped with a Zariski-like topology) of a general commutative $\mathrm{\Gamma }$-monoid and establish that this space is spectral in the sense of Hochster, provided the monoid has the refinement property and every $\mathrm{\Gamma }$-order ideal is finitely generated. As a result we are able to show that the prime spectrum of the talented monoid of a row-finite $k$-graph without sources and with a finite set of vertices is spectral. For any row-finite $k$-graph $\mathrm{\Lambda }$ without sources, one of our main results says that the space of all graded prime ideals of the Kumjian–Pask algebra $\text{KP}(\mathrm{\Lambda })$ is homeomorphic to both the space of all prime ${\mathbb{Z}}^k$-order ideals and the space of all prime ${\mathbb{Z}}^k$-filters of the talented monoid $T_{\mathrm{\Lambda }}$. Another main result of this paper provides a complete topological description of regular $\mathrm{\Gamma }$-order ideals of a refinement $\mathrm{\Gamma }$-monoid: a $\mathrm{\Gamma }$-order ideal $J$ is regular if and only if the corresponding closed (resp., open) set $V(J)$ (resp., $D(J)$) is regular closed (resp., regular open) in the prime spectrum. As an application of these results, we establish a lattice isomorphism between the lattice of all regular ${\mathbb{Z}}^k$-order ideals of the talented monoid and the lattice of all regular graded ideals of the Kumjian–Pask algebra via a spectrum-theoretic approach.