关于分次代数的满同态

On Epimorphisms of Graded Algebras

首先将向量空间上的多项式函数概念扩展到无挠交换群上,并自然地引出可去集的概念,以类比代数几何中的代数集.接着提炼出一类非常一般的由无挠交换群分次的非结合代数,称之为拟Block代数.然后利用可去集的基本性质和"有限VS无限"的组合技巧,借助几何直观,证明拟Block代数之间的满同态总是"接近于"分次.最后以实例演示该结论的应用.

Firstly we extend the notion of polynomial functions on vector spaces to polynomial functions on abelian groups. Then the notion of negligible sets is naturally defined, as an analogy of the algebraic sets in algebraic geometry. Upon this foundation, we introduce a class of non-associative algebras which are graded by torsion-free abelian groups. These algebras are called quasi-Block algebras in the present paper, and we claim that any epimorphism between them is ＂nearly＂ graded. This is demonstrated by a ＂finitude VS infinity＂ trick, which is combinatorial in nature. Among others, the negligible sets, along with some geometric intuition, play a prominent role in the proof. As an application of this general result, some examples are given.