Schur multiplier of $\mathrm{SL}_2$ over finite commutative rings
Behrooz Mirzaii, Abraham Rojas Vega
TL;DR
The paper determines the Schur multiplier $H_2({\rm SL}_2(A),\mathbb{Z})$ for finite commutative rings by reducing to finite local rings and employing a refined unimodular complex together with a spectral sequence. Under the condition of odd residue characteristic and $|A/\mathfrak{m}_A|\notin\{3,5,9\}$, it proves $H_2({\rm SL}_2(A),\mathbb{Z})\cong K_2(A)$, and for Galois/quasi-Galois rings with the same restriction, the multiplier vanishes; for principal finite local rings with odd $p$, $H_2({\rm SL}_2(A),\mathbb{Z})$ is a finite cyclic $p$-group. The work connects $H_2$ with $K$-theory, refined scissors congruence groups $\mathcal{RP}_1(A)$ and Grothendieck–Witt invariants, yielding explicit results in classical cases (finite fields, $\mathbb{Z}/p^n$, local rings with $X^2$-type quotients) and providing a unified framework for computations via GAP in exceptional small cases. It also extends to a detailed analysis of $H_3({\rm SL}_2(A),\mathbb{Z})$ using refined Bloch groups, linking third homology to refined scissors congruence and Tor phenomena.
Abstract
In this article, we investigate the Schur multiplier of the special linear group $\mathrm{SL}_2(A)$ over finite commutative local rings $A$. We prove that the Schur multiplier of these groups is isomorphic to the $K$-group $K_2(A)$ whenever the residue field $A/\mathfrak{m}_A$ has odd characteristic and satisfies $|A/\mathfrak{m}_A| \neq 3,5,9$. As an application, we show that if $A$ is either the Galois ring $\mathrm{GR}(p^l,m)$ or the quasi-Galois ring $A(p^m,n)$ with residue field of odd characteristic and $|A/\mathfrak{m}_A| \neq 3,5,9$, then the Schur multiplier of $\mathrm{SL}_2(A)$ is trivial.