arXiv subjects

Rodrigo Cuitun Coronado

Publications and source records attributed to Rodrigo Cuitun Coronado.

3 recordsLinked to original sources

arXiv2026

Bloch Groups of Rings

We give a definition of (refined) Bloch groups of general commutative rings which agrees with the standard definition in the case of local rings whose residue field has at least $4$ elements. Under appropriate conditions on a ring $A$, satisfied by any field or local ring, these groups are closely related to third homology of $\mathrm{SL}_2(A)$ and to indecomposable $K_3$ of $A$. We analyze these conditions. We calculate the Bloch groups of $\mathbb{F}_2,\mathbb{F}_3,\mathbb{Z}$ and $\mathbb{Z}[\frac{1}{2}]$.

math.KT↗

arXiv2022

Third homology of $\mathrm{SL}_{2}$ over Number fields: The norm-Euclidean quadratic imaginary case

In the article The third homology of $SL_{2}(\mathbb{Q})$, Hutchinson determined the structure of $H_{3}\left(\mathrm{SL}_{2}(\mathbb{Q}),\mathbb{Z}\left[\frac{1}{2}\right]\right)$ by expressing it in terms of $K_{3}^{\mathrm{ind}}(\mathbb{Q})\cong \mathbb{Z}/24$ and the scissor congruence group of the residue field $\mathbb{F}_{p}$ with $p$ a prime number. In this paper, we develop further the properties of the refined scissors congruence group in order to extend this result to the case of imaginary quadratic number fields whose ring of integers is a Euclidean domain with respect to the norm.

math.KT↗

arXiv2021

The third homology of $SL_{2}$ of real quadratically closed fields

For a real closed field $\mathbf{R}$, we use the theory of the refined Bloch group to give a new short proof of the isomorphisms $H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\cong K_{3}^{\mathrm{ind}}(\mathbf{R})$ and $H_{3}(SL_{n}(\mathbf{R}),\mathbb{Z})\cong H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\oplus K_{3}^{M}(\mathbf{R})^{0}$ for $n\geq3$.

math.KT↗