2018/06/24 by Cortiñas, Guillermo, Montero, Diego · 1 citation
#FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1806.09204
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 ℓ. 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 ℓ, for a graph E with finitely many vertices and reduced incidence matrix AE, the structure of L(E) depends only on the isomorphism classes of the cokernels of the matrix I-AE and of its transpose, which are respectively the kk groups KH1(L(E))=kk-1(L(E),ℓ) and KH0(L(E))=kk0(ℓ,L(E)). Hence if L(E) and L(F) are unital Leavitt path algebras such that KH0(L(E))≅ KH0(L(F)) and KH1(L(E))≅ KH1(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 Künneth theorems of Rosenberg and Schochet.