发表机构
Western Sydney University; Jadavpur University（西悉尼大学; 贾达普大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究高秩图的天才幺半群的素谱，证明其谱性，并建立其与Kumjian-Pask代数分次素理想空间的同胚及正则理想格同构。
AI 中文摘要
本文进一步研究了分次Grothendieck群$K_0^{\gr}$及其正锥（天才幺半群）作为区分与高秩$k$-图相关的代数结构类型的有效工具的作用。我们研究了一般交换$\Gamma$-幺半群的素谱（所有素$\Gamma$-序理想构成的、配备Zariski型拓扑的空间），并证明了：若该幺半群具有细化性质且每个$\Gamma$-序理想都是有限生成的，则该空间在Hochster意义下是谱空间。由此，我们得以证明：对于无源、顶点集有限的行有限$k$-图，其天才幺半群的素谱是谱空间。对于任何无源的行有限$k$-图$\Lambda$，我们的主要结果之一表明：Kumjian--Pask代数$\KP(\Lambda)$的所有分次素理想的空间，与天才幺半群$T_\Lambda$的所有素$\mathbb{Z}^k$-序理想的空间以及所有素$\mathbb{Z}^k$-滤子的空间都同胚。本文的另一主要结果给出了细化$\Gamma$-幺半群的正则$\Gamma$-序理想的完全拓扑描述：一个$\Gamma$-序理想$J$是正则的，当且仅当对应的闭集$V(J)$（相应地，开集$D(J)$）在素谱中是正则闭集（相应地，正则开集）。作为这些结果的应用，我们通过谱论方法，建立了天才幺半群的所有正则$\mathbb{Z}^k$-序理想的格与Kumjian--Pask代数的所有正则分次理想的格之间的格同构。
英文摘要
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.
Comments27 pages, Comments are welcome
