Elementary nets (carpets) over a discrete valuation ring

Vladimir A. Koibaev

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 6, pp. 728-735

Résumé

Elementary net (carpet) $\sigma = (\sigma_{ij})$ is called closed (admissible) if the elementary net (carpet) group $E(\sigma)$ does not contain a new elementary transvections. The work is related to the question of V. M. Levchuk 15.46 from the Kourovka notebook( closedness (admissibility) of the elementary net (carpet)over a field). Let $R$ be a discrete valuation ring, $K$ be the field of fractions of $R$, $\sigma = (\sigma_{ij})$ be an elementary net of order $n$ over $R$, $\omega=(\omega_{ij})$ be a derivative net for $\sigma$, and $\omega_{ij}$ is ideals of the ring $R$. It is proved that if $K$ is a field of odd characteristic, then for the closedness (admissibility) of the net $\sigma$, the closedness (admissibility) of each pair $(\sigma_{ij}, \sigma_{ji})$ is sufficient for all $i\neq j$.

Détail

Keywords: nets, carpets, elementary net, closed net, derivative net, elementary net group, discrete valuation ring.
Mots-clés : transvections