arXiv subjects

Guido Arnone

Publications and source records attributed to Guido Arnone.

6 recordsLinked to original sources

arXiv2026

Bredon sheaf cohomology

For a finite group $G$, we compute the algebraic $K$-theory of the category of equivariant sheaves on a locally compact Hausdorff $G$-space, generalizing a result of Efimov, and determine the equivariant $E$-theory of the $C^*$-algebra of continuous functions. These invariants admit natural descriptions in terms of a new equivariant cohomology theory, which we call Bredon sheaf cohomology. This theory recovers classical Bredon cohomology for $G$-CW complexes and ordinary sheaf cohomology when $G$ is trivial. We establish its basic structural properties and prove a strong uniqueness theorem: any functor from the category of locally compact Hausdorff $G$-spaces to a dualizable stable category satisfying equivariant open descent and cofiltered compact codescent is equivalent to Bredon sheaf cohomology, generalizing a result of Clausen.

math.KT↗

arXiv2024

Homology of Steinberg algebras

We study homological invariants of the Steinberg algebra $\mathcal{A}_k(\mathcal{G})$ of an ample groupoid $\mathcal{G}$ over a commutative ring $k$. For $\mathcal{G}$ principal or Hausdorff with ${\mathcal{G}}^{\rm{Iso}}\setminus{\mathcal{G}}^{(0)}$ discrete, we compute Hochschild and cyclic homology of $\mathcal{A}_k(\mathcal{G})$ in terms of groupoid homology. For any ample Hausdorff groupoid $\mathcal{G}$, we find that $H_*(\mathcal{G})$ is a direct summand of $HH_*(\mathcal{A}_k(\mathcal{G}))$; using this and the Dennis trace we obtain a map $\overline{D}_*:K_*(\mathcal{A}_k(\mathcal{G}))\to H_n(\mathcal{G},k)$. We study this map when $\mathcal{G}$ is the (twisted) Exel-Pardo groupoid associated to a self-similar action of a group $G$ on a graph, and compute $HH_*(\mathcal{A}_k(\mathcal{G}))$ and $H_*(\mathcal{G},k)$ in terms of the homology of $G$, and the $K$-theory of $\mathcal{A}_k(\mathcal{G})$ in terms of that of $k[G]$.

math.KT↗

arXiv2023

Graded homotopy classification of Leavitt path algebras

We show that the graded Grothendieck group classifies unital Leavitt path algebras of primitive graphs up to graded homotopy equivalence. To this end, we further develop classification techniques for Leavitt path algebras by means of (graded, bivariant) algebraic $K$-theory.

math.KT↗

arXiv2023

Lifting morphisms between graded Grothendieck groups of Leavitt path algebras

We show that any pointed, preordered module map $\mathfrak{BF}_{\mathrm{gr}}(E) \to \mathfrak{BF}_{\mathrm{gr}}(F)$ between Bowen-Franks modules of finite graphs can be lifted to a unital, graded, diagonal preserving $\ast$-homomorphism $L_\ell(E) \to L_\ell(F)$ between the corresponding Leavitt path algebras over any commutative unital ring with involution $\ell$. Specializing to the case when $\ell$ is a field, we establish the fullness part of Hazrat's conjecture about the functor from Leavitt path $\ell$-algebras of finite graphs to preordered modules with order unit that maps $L_\ell(E)$ to its graded Grothendieck group. Our construction of lifts is of combinatorial nature; we characterize the maps arising from this construction as the scalar extensions along $\ell$ of unital, graded $\ast$-homomorphisms $L_{\mathbb Z}(E) \to L_{\mathbb Z}(F)$ that preserve a sub-$\ast$-semiring introduced here.

math.RA↗

arXiv2022

Graded $K$-theory and Leavitt path algebras

Let $G$ be a group and $\ell$ a commutative unital $\ast$-ring with an element $λ\in \ell$ such that $λ+ λ^\ast = 1$. We introduce variants of hermitian bivariant $K$-theory for $\ast$-algebras equipped with a $G$-action or a $G$-grading. For any graph $E$ with finitely many vertices and any weight function $ω\colon E^1 \to G$, a distinguished triangle for $L(E)=L_\ell(E)$ in the hermitian $G$-graded bivariant $K$-theory category $kk^h_{G_{\mathrm{gr}}}$ is obtained, describing $L(E)$ as a cone of a matrix with coefficients in $\mathbb{Z}[G]$ associated to the incidence matrix of $E$ and the weight $ω$. In the particular case of the standard $\mathbb{Z}$-grading, and under mild assumptions on $\ell$, we show that the isomorphism class of $L(E)$ in $kk^h_{\mathbb{Z}_{\mathrm{gr}}}$ is determined by the graded Bowen-Franks module of $E$. We also obtain results for the graded and hermitian graded $K$-theory of $\ast$-algebras in general and Leavitt path algebras in particular which are of independent interest, including hermitian and bivariant versions of Dade's theorem and of Van den Bergh's exact sequence relating graded and ungraded $K$-theory.

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arXiv2021

Nonexistence of graded unital homomorphisms between Leavitt algebras and their Cuntz splices

Let $n\ge 2$, let $\mathcal{R}_n$ be the graph consisting of one vertex and $n$ loops and let $\mathcal{R}_{n^-}$ be its Cuntz splice. Let $L_n=L(\mathcal{R}_n)$ and $L_{n^-}=L(\mathcal{R}_{n^-})$ be the Leavitt path algebras over a unital ring $\ell$. Let $C_m$ be the cyclic group on $2\le m\le \infty$ elements. Equip $L_n$ and $L_{n^-}$ with their natural $C_m$-gradings. We show that under mild conditions on $\ell$, which are satisfied for example when $\ell$ is a field or a PID, there are no unital $C_m$-graded ring homomorphisms $L_n\to L_{n^-}$ nor in the opposite direction.

math.RA↗