Determinants in net subgroups

Z. I. Borevich ; N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Modules and algebraic groups, Tome 114 (1982), pp. 37-49

Résumé

Suppose $R$ is a commutative ring with 1, $\sigma=(\sigma_{ij})$ is a fixed $D$-net of ideals of $R$ of order $n$, and $G(\sigma)$ is the corresponding net subgroup of the general linear group $GL(n,R)$. There is constructed for $\sigma$ a homomorphism $\det_\sigma$ of the subgroup $G(\sigma)$ into a certain Abelian group $\Phi(\sigma)$. Let $I$ be the index set $\{1,\dots,n\}$. For each subset $\alpha\subseteq I$ let $\sigma(\alpha)=\sum\sigma_{ij}\sigma_{ji}$, where $i$, ranges over all indices in $\alpha$ and $j$ independently over the indices in the complement $I\backslash\alpha$ ($\sigma(I)$ is the zero ideal). Let $\det_\alpha(a)$ denote the principal minor of order $|\alpha|\leqslant n$ of the matrix $a\in G(\sigma)$ corresponding to the indices in $\alpha$, and let $\Phi(\sigma)$ be the Cartesian product of the multiplicative groups of the quotient rings $R/\sigma(\alpha)$ over all subsets $\alpha\subseteq I$. The homomorphism $\det_\sigma$ is defined as follows: $$ \det_\sigma(a)=(\det_\alpha(a)\mod\sigma(\alpha))_\alpha\in\Phi(\sigma). $$ It is proved that if $R$ is a semilocal commutative Bezout ring, then the kernel $\operatorname{Ker}\det_\sigma$ coincides with the subgroup $E(\sigma)$ generated by all transvections in $G(\sigma)$. For these $R$ is also defined $\operatorname{Im}\det_\sigma$.