模间二次映射
Quadratic maps between modules
摘要：
本文引入了交换环$R$上模间的$R$ -二次映射的概念，它推广了线性代数和群论中的一些经典概念.在一个给定的模块$M$这样的地图表示为$R$ -线性映射在一定的模块$P^2_R(M)$。这个模的结构用对称张量平方$Sym^2_R(M)$、$M$上的可除幂代数的2次分量$Γ^2_R(M)$以及由元素$r^2-r$、$r\in R$生成的$R$的理想$I_2$来描述.后者表示二次导数$R$出现在理论上的模块方环。这允许将2-二项式环$R$中系数为2的幂零$R$ -群的经典概念推广到任何环$R$。我们提供了一个函式表示的$I_2$和几个确切的序列嵌入模块$P^2_R(M)$和$Γ^2_R(M)$。
英文摘要：
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 $Γ^2_R(M)$ 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 $Γ^2_R(M)$.
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