Quadratic maps between modules

TL;DR

This paper introduces a generalized concept of quadratic maps between modules over a commutative ring, linking classical algebraic notions with new structures involving symmetric tensors, divided powers, and quadratic derivations.

Contribution

It defines $R$-quadratic maps, describes their representing modules, and extends classical nilpotent group concepts to arbitrary rings using new algebraic structures.

Findings

Representation of quadratic maps via $P^2_R(M)$

Description of $P^2_R(M)$ using symmetric tensors and divided powers

Extension of nilpotent $R$-group theory to general rings

Abstract

We introduce a notion of $R$-quadratic maps between modules over a commutative ring $R$ which generalizes several classical notions arising in linear algebra and group theory. On a given module $M$ such maps are represented by $R$-linear maps on a certain module $P^2_R(M)$. The structure of this module is described in term of the symmetric tensor square $Sym^2_R(M)$, the degree 2 component $\Gamma^2_R(M)$ of the divided power algebra over $M$, and the ideal $I_2$ of $R$ generated by the elements $r^2-r$, $r\in R$. The latter is shown to represent quadratic derivations on $R$ which arise in the theory of modules over square rings. This allows to extend the classical notion of nilpotent $R$-group of class 2 with coefficients in a 2-binomial ring $R$ to any ring $R$. We provide a functorial presentation of $I_2$ and several exact sequences embedding the modules $P^2_R(M)$ and $\Gamma^2_R(M)$.