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Emanuel Rodríguez Cirone

Publications and source records attributed to Emanuel Rodríguez Cirone.

9 recordsLinked to original sources
arXiv2026

Matrix stability and Morita invariance

Let $G$ be a group. We prove that matrix stability for either $G$-algebras or $G$-graded algebras guarantees Morita invariance. As a consequence, bivariant algebraic K-theory (either $G$-equivariant or $G$-graded) is Morita invariant. In particular, we show that if $G$ is a finite group acting freely on a finite simplicial set $X$, then $\ell^X\rtimes G$ and $\ell^{X/G}$ are kk-equivalent. Here, $\ell^Y$ denotes the $\ell$-algebra of piecewise polynomial functions on $Y$ with coefficients in the ground ring $\ell$.

math.KT↗
arXiv2025

Homotopy structures realizing algebraic kk-theory

Algebraic $kk$-theory, introduced by Cortiñas and Thom, is a bivariant $K$-theory defined on the category $\mathrm{Alg}$ of algebras over a commutative unital ring $\ell$. It consists of a triangulated category $kk$ endowed with a functor from $\mathrm{Alg}$ to $kk$ that is the universal excisive, homotopy invariant and matrix-stable homology theory. Moreover, one can recover Weibel's homotopy $K$-theory $\mathrm{KH}$ from $kk$ since we have $kk(\ell,A)=\mathrm{KH}(A)$ for any algebra $A$. We prove that $\mathrm{Alg}$ with the split surjections as fibrations and the $kk$-equivalences as weak equivalences is a stable category of fibrant objects, whose homotopy category is $kk$. As a consecuence of this, we prove that the Dwyer-Kan localization $kk_\infty$ of the $\infty$-category of algebras at the set of $kk$-equivalences is a stable infinity category whose homotopy category is $kk$.

math.KT↗
arXiv2024

Algebraic kk-theory and the KH-isomorphism conjecture

We relate the Davis-Lück homology with coefficients in Weibel's homotopy K-theory to the equivariant algebraic kk-theory using homotopy theory and adjointness theorems. We express the left hand side of the assembly map for the KH-isomorphism conjecture in terms of equivariant algebraic kk-groups.

math.KT↗
arXiv2019

Two examples of vanishing and squeezing in $K_1$

Controlled topology is one of the main tools for proving the isomorphism conjecture concerning the algebraic $K$-theory of group rings. In this article we dive into this machinery in two examples: when the group is infinite cyclic and when it is the infinite dihedral group - in both cases with the family of finite subgroups. We prove a vanishing theorem and show how to explicitly squeeze the generators of these groups in $K_1$. For the infinite cyclic group, when taking coefficients in a regular ring, we get a squeezing result for every element of $K_1$; this follows from the well-known result of Bass, Heller and Swan.

math.KT↗
arXiv2019

The loop-stable homotopy category of algebras

Let l be a commutative ring with unit. Garkusha constructed a functor from the category of l-algebras into a triangulated category D, that is a universal excisive and homotopy invariant homology theory. Later on, he provided different descriptions of D, as an application of his motivic homotopy theory of algebras. Using these, it can be shown that D is triangulated equivalent to a category, denote it by k, whose objects are pairs (A,m) with A an l-algebra and m an integer, and whose Hom-sets can be described in terms of homotopy classes of morphisms. All these computations, however, require a heavy machinery of homotopy theory. In this paper, we give a more explicit construction of the triangulated category k and prove its universal property, avoiding the homotopy-theoretic methods and using instead the ones developed by Cortiñas-Thom for defining kk-theory. Moreover, we give a new description of the composition law in k, mimicking the one in the suspension-stable homotopy category of bornological algebras defined by Cuntz-Meyer-Rosenberg. We also prove that the triangulated structure in k can be defined using either extension or mapping path triangles.

math.KT↗
arXiv2018

The homotopy groups of the simplicial mapping space between algebras

Let $\ell$ be a commutative ring with unit. To every pair of $\ell$-algebras $A$ and $B$ one can associate a simplicial set $\hom(A,B^Δ)$ so that $π_0\hom(A,B^Δ)$ equals the set of polynomial homotopy classes of morphisms from $A$ to $B$. We prove that $π_n\hom(A,B^Δ)$ is the set of homotopy classes of morphisms from $A$ to $B^{S_n}$, where $B^{S_n}$ is the ind-algebra of polynomials on the $n$-dimensional cube with coefficients in $B$ vanishing at the boundary of the cube. This is a generalization to arbitrary dimensions of a theorem of Cortiñas-Thom, which addresses the cases $n\leq 1$. As an application we give a simplified proof of a theorem of Garkusha that computes the homotopy groups of his matrix-unstable algebraic KK-theory space in terms of polynomial homotopy classes of morphisms.

math.AT↗