代数双变K理论与Leavitt路径代数。

Guillermo Cortiñas, Diego Montero
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引用次数: 11

摘要

本文是两篇文章中的第一篇,在这两篇文章中,我们研究了在交换地环上图$E$和图$F$的同伦不变、完备和矩阵稳定同伦理论在多大程度上帮助人们区分levitt路径代数$L(E)$和$L(F)$。在第一篇文章中,我们考虑一般地环上一般图的Leavitt路径代数;第二篇文章将主要关注域上的纯无限简单单莱维特路径代数。二元代数$K$-理论$kk$是具有上述性质的全称同调理论;证明了$kk$中一元莱维特路径代数的一个结构定理。我们证明了在$\ell$的非常温和的假设下,对于具有有限多个顶点和简化关联矩阵$A_E$的图$E$, $L(E)$的结构仅取决于矩阵$I-A_E$及其转置的核的同构类,它们分别是$kk$群$KH^1(L(E))=kk_{-1}(L(E),\ell)$和$KH_0(L(E))=kk_0(\ell,L(E))$。因此,如果$L(E)$和$L(F)$是一元莱维特路径代数,使得$KH_0(L(E))\cong KH_0(L(F))$和$KH^1(L(E))\cong KH^1(L(F))$,则没有具有上述性质的同调理论可以区分它们。我们还证明了对于Leavitt路径代数,$kk$具有与$C^*$图代数的Kasparov双变$K$-理论相似的几个性质,包括Rosenberg和Schochet的普适系数和Kunneth定理的类似物。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Algebraic bivariant $K$-theory and Leavitt path algebras.
This article is the first of two where we investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras $L(E)$ and $L(F)$ of graphs $E$ and $F$ over a commutative ground ring $\ell$. In this first article we consider Leavitt path algebras of general graphs over general ground rings; the second article will focus mostly on purely infinite simple unital Leavitt path algebras over a field. Bivariant algebraic $K$-theory $kk$ is the universal homology theory with the properties above; we prove a structure theorem for unital Leavitt path algebras in $kk$. We show that under very mild assumptions on $\ell$, for a graph $E$ with finitely many vertices and reduced incidence matrix $A_E$, the structure of $L(E)$ depends only on the isomorphism classes of the cokernels of the matrix $I-A_E$ and of its transpose, which are respectively the $kk$ groups $KH^1(L(E))=kk_{-1}(L(E),\ell)$ and $KH_0(L(E))=kk_0(\ell,L(E))$. Hence if $L(E)$ and $L(F)$ are unital Leavitt path algebras such that $KH_0(L(E))\cong KH_0(L(F))$ and $KH^1(L(E))\cong KH^1(L(F))$ then no homology theory with the above properties can distinguish them. We also prove that for Leavitt path algebras, $kk$ has several properties similar to those that Kasparov's bivariant $K$-theory has for $C^*$-graph algebras, including analogues of the Universal coefficient and Kunneth theorems of Rosenberg and Schochet.
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