The normalizer of the elementary linear group of a~module arising under extension of the base ring

N. H. T. Nhat ; T. N. Hoi

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 31, Tome 455 (2017), pp. 122-129

Résumé

Let $S$ be a commutative ring with $1$ and $R$ a unital subring. Let $M$ be a free $S$-module of rank $n\geq3$. In [1], V. A. Koibaev described the normalizer of $\operatorname{Aut}_S(M)$ in the group $\operatorname{Aut}_R(M)$. In this paper, we show that in $\operatorname{Aut}_R(M)$ the normalizer of the elementary linear group $E_\mathfrak B(M)$ coincides with the one of $\operatorname{Aut}_S(M)$, namely, $N_{\operatorname{Aut}_R(M)}(E_\mathfrak B(M))=\operatorname{Aut}(S/R)\ltimes\operatorname{Aut}_S(M)$. If $S$ is free of rank $m$ as an $R$-module, then $N_{\operatorname{GL}(mn,R)}(E(n,S))=\operatorname{Aut}(S/R)\ltimes\operatorname{GL}(n,S)$, moreover, for any proper ideal $A$ of $R$, we have $$ N_{\operatorname{GL}(mn, R)}(E(n,S)E(mn,R,A))=\rho_A^{-1}(N_{\operatorname{GL}(mn,R/A)}(E(n,S/SA))). $$