Leavitt path algebras over a poset of fields

Abstract

Let $E$ be a finite directed graph, and let $I$ be the poset obtained as the antisymmetrization of its set of vertices with respect to a pre-order $\le$ that satisfies $v\le w$ whenever there exists a directed path from $w$ to $v$. Assuming that $I$ is a tree, we define a poset of fields over $I$ as a family $\mathbf{K}=\{K_i:i\in I\}$ of fields $K_i$ such that $K_i\subseteq K_j$ if $j\le i$. We define the concepts of a Leavitt path algebra $L_{\mathbf{K}}(E)$ and a regular algebra $Q_{\mathbf{K}}(E)$ over the poset of fields $\mathbf{K}$, and we show that $Q_{\mathbf{K}}(E)$ is a hereditary von Neumann regular ring, and that its monoid $\mathcal{V}(Q_{\mathbf{K}}(E))$ of isomorphism classes of finitely generated projective modules is canonically isomorphic to the graph monoid $M(E)$ of $E$.

12 pages, to appear in Contemporary Mathematics AMS. arXiv admin note: text overlap with arXiv:0909.0421