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 Zap. Nauchn. Sem. LOMI, 1982, Volume 114, Pages 37–49 (Mi znsl1765)

Determinants in net subgroups

Z. I. Borevich, N. A. Vavilov

Abstract: 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 $\{1,…,n\}$. 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 $\operatorname{Im}\det_\sigma$.

Full text: PDF file (1112 kB)

English version:
Journal of Soviet Mathematics, 1984, 27:4, 2855–2865

Bibliographic databases:

UDC: 519.46

Citation: Z. I. Borevich, N. A. Vavilov, “Determinants in net subgroups”, Modules and algebraic groups, Zap. Nauchn. Sem. LOMI, 114, "Nauka", Leningrad. Otdel., Leningrad, 1982, 37–49; J. Soviet Math., 27:4 (1984), 2855–2865

Citation in format AMSBIB
\Bibitem{BorVav82} \by Z.~I.~Borevich, N.~A.~Vavilov \paper Determinants in net subgroups \inbook Modules and algebraic groups \serial Zap. Nauchn. Sem. LOMI \yr 1982 \vol 114 \pages 37--49 \publ "Nauka", Leningrad. Otdel. \publaddr Leningrad \mathnet{http://mi.mathnet.ru/znsl1765} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=669558} \zmath{https://zbmath.org/?q=an:0552.15002|0496.15008} \transl \jour J. Soviet Math. \yr 1984 \vol 27 \issue 4 \pages 2855--2865 \crossref{https://doi.org/10.1007/BF01410739}