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 ${\Lambda }^{p}M$ 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 $\eta \in {\Lambda }^{p}M$ 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 $r=k=rankN$ 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.

arXiv comment16 pages