Mathematics – Rings and Algebras
Scientific paper
2010-05-24
Mathematics
Rings and Algebras
17 pages. Updated version, in which an erroneous proof (of "not primitive implies not rational" in the main theorem) has been
Scientific paper
Let $K$ be a field, let $E$ be a finite directed graph, and let $L_K(E)$ be the Leavitt path algebra of $E$ over $K$. We show that for a prime ideal $P$ in $L_K(E)$, the following are equivalent: \begin{enumerate} \item $P$ is primitive; \item $P$ is rational; \item $P$ is locally closed in ${\rm Spec}(L_K(E))$. \end{enumerate} We show that the prime spectrum ${\rm Spec}(L_K(E))$ decomposes into a finite disjoint union of subsets, each of which is homeomorphic to ${\rm Spec}(K)$ or to ${\rm Spec}(K[x,x^{-1}])$. In the case that $K$ is infinite, we show that $L_K(E)$ has a rational $K^{\times}$-action, and that the indicated decomposition of ${\rm Spec}(L_K(E))$ is induced by this action.
Abrams Gene
Bell Jason P.
Rangaswamy Kulumani M.
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