Division properties in exterior algebras of free modules and logarithmic residua

Abstract

Let $M$ be a free module of rank $m$ over a commutative unital ring $R$ and let $N$ be its free submodule. We consider the problem when a given element of the exterior product $\Lambda^pM$ is divisible, in a sense, over elements of the exterior product $\Lambda^r N$, $r\le p$. Precisely, we give conditions under which an element $\eta\in\Lambda^pM$ can be expressed as a finite sum of skew-products of elements of $\Lambda^r N$ and elements of $\Lambda^{p-r} M$. For a given basis $\omega_1,\dots,\omega_k$ in $N$ the elements of $\Lambda^{p-r} M$ are unique in a specified sense. Necessary and sufficient conditions for such divisibility take a simple form, provided that the submodule is embedded in $M$ with singularities having the depth larger then $p-r+1$. In the special case where $r=k=rank N$ the divisibility property means that $\eta=\Omega\wedge\gamma$ where $\Omega=\omega_1\wedge\cdots\wedge\omega_k$ and $\gamma\in\Lambda^{p-k}M$. More detailed statements of these results are then used to state criteria for existence and uniqueness of algebraic logarithmic residua when the "divisor" is defined by elements $f_1,\dots,f_k\in R$. Special cases are multidimensional logarithmic residua in complex analysis.