Schur multiplier of $\mathrm{SL}_2$ over finite commutative rings

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.