The prime spectrum of the talented monoid of a higher-rank graph and applications
Abstract
In this paper, we further investigate the role of the graded Grothendieck group K0gr 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 ℤk-order ideals and the space of all prime ℤk-filters of the talented monoid TΛ. 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 ℤk-order ideals of the talented monoid and the lattice of all regular graded ideals of the Kumjian--Pask algebra via a spectrum-theoretic approach.
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BibTeX
@misc{hazrat2026prime,
title = {{The prime spectrum of the talented monoid of a higher-rank graph and applications}},
author = {Roozbeh Hazrat and Promit Mukherjee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16632},
eprint = {2609.16632},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16632},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The prime spectrum of the talented monoid of a higher-rank graph and applications
AU - Roozbeh Hazrat
AU - Promit Mukherjee
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16632
UR - https://arxiv.org/abs/2609.16632
AB - 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 ℤ^k-order ideals and the space of all prime 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 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.
ER -