On Pl\"ucker properties of rings

G. B. Kleiner

Sbornik. Mathematics, Tome 13 (1971) no. 4, pp. 517-528

Résumé

Questions are considered of decomposability of $m$-vectors from $\Lambda^m(A^n)$, where $A$ is a commutative ring with $1$, and $A^n$ is the direct sum of $n$ copies of $A$. Let $A$ be a Krull ring. We shall denote by $\operatorname{div}\omega$ the greatest common divisor of the coordinates of the $m$-vector $\omega\in\Lambda^m(A^n)$. For the case where the $\operatorname{div}\omega$ is square-free in terms of the $A$-module $K_\omega=\{x\in A^n:x\land\omega=0\}$ necessary and sufficient conditions are given for decomposability of $\omega$. A characterization of factorial Plücker rings is stated, i.e. rings in which for arbitrary $n>m\geqslant2$ every $m$-vector of $\Lambda^m(A^n)$ which satisfies the Plücker condition is decomposable. Bibliography: 8 titles.