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 $K_0^{\gr}$ 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 $Γ$-order ideals equipped with a Zariski-like topology) of a general commutative $Γ$-monoid and establish that this space is spectral in the sense of Hochster, provided the monoid has the refinement property and every $Γ$-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 $Λ$ without sources, one of our main results says that the space of all graded prime ideals of the Kumjian--Pask algebra $\KP(Λ)$ 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 $T_Λ$. Another main result of this paper provides a complete topological description of regular $Γ$-order ideals of a refinement $Γ$-monoid: a $Γ$-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.