Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.28677

The Schur multiplier of $\rm{SL}_2$ and Dedekind zeta-functions over $S$-integers

P. H. Amorim·I. V. Picinini·B. R. Ramos·T. Verissimo
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we obtain an exact sequence connecting $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$ to $H_2(\rm{SL}_2(\mathcal{O}_{K,T}), \mathbb{Z})$, where $\mathcal{O}_{K,S}$ is a ring of $S$-integers and $T$ is a set of primes containing $S$. We apply this sequence to establish a relation between $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$ with the second $K$-group $K_2(\mathcal{O}_{K,S})$, for $S$ large enough. This leads to a description of the rank and size of the torsion of $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$. As an application, we propose a homological version of the Birch-Tate formula (conjecture) under these assumptions.

Explore related subjects

math.KTmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

P. H. Amorim, I. V. Picinini, B. R. Ramos, T. Verissimo. 2026-10-02. The Schur multiplier of $\rm{SL}_2$ and Dedekind zeta-functions over $S$-integers. https://arxiv.org/abs/2609.28677

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

K-theory and localizing invariants of large categories

In this paper we introduce and study the so-called continuous $K$-theory for a certain class of "large" stable $\infty$-categories, more precisely, for dualizable presentable categories. For compactly generated categories, the continuous $K$-theory is simply the usual (non-connective) $K$-theory of the full subcategory of compact objects. More generally, we show that any localizing invariant of small stable $\infty$-categories can be uniquely extended to a localizing invariant of dualizable categories. We compute the continuous $K$-theory for categories of sheaves on locally compact Hausdorff spaces. Using the special case for sheaves on the real line, we give an alternative proof of the theorem of Kasprowski and Winges \cite{KW19} on the commutation of $K$-theory with infinite products for small stable $\infty$-categories. We also study the general theory of dualizable categories. In particular, we give an "explicit" proof of Ramzi's theorem \cite{Ram24a} on the $ω_1$-presentability of the category of dualizable categories. Among other things, we prove that dualizability is equivalent to "flatness" in the category of presentable stable categories.

math.KT↗
arXiv2026

Some remarks on Quillen's Dévissage theorem

In this paper we give a different proof of Quillen's Dévissage theorem using Barwick's theorem of the heart. The key ingredient is a certain short exact sequence of dg categories, which is closely related to the Auslander-type construction for nilpotent extensions which was used in papers by Kuznetsov and Lunts \cite{KL15}, Land and Tamme \cite{LT19}, and the author \cite{E20}.

math.KT↗
arXiv2026

Regular boundary conditions and analytic K-homology

In their seminal work, Baum-Douglas-Taylor showed that under a certain compactness assumption, local boundary conditions for first-order elliptic operators over compact manifolds with boundary define cycles for relative K-homology. Using a systematic approach to regular boundary conditions, we generalize their result to elliptic operators of arbitrary order and to non-compact manifolds with potentially non-compact boundary. In addition, we show that under stronger regularity conditions, local boundary conditions even define cycles for absolute K-homology. In contrast to the relative classes, the resulting absolute K-homology classes are in general not independent of the choice of boundary condition.

math.KT↗