Division properties in exterior algebras of free modules and logarithmic residua

TL;DR

This paper investigates conditions under which elements in exterior algebras of free modules can be decomposed over submodules, with applications to algebraic logarithmic residua and complex analysis.

Contribution

It provides new criteria for divisibility in exterior algebras and applies these to the theory of logarithmic residua, including cases with singularities.

Findings

Derived simple divisibility conditions for exterior algebra elements.

Established criteria for existence and uniqueness of algebraic logarithmic residua.

Connected algebraic divisibility to complex analysis applications.

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.