加法范畴的回路范畴与η-扩张

Loop Categories of Additive Categories and η-extensions

设A是加法范畴,在定义范畴A的η-扩张基础上,得到加法范畴的η-扩张的不变性,进而得到范畴A的回路范畴与η-扩张交换性,即有加法范畴等价Ω(A(η))≌(ΩA)(η→).最后将主定理应用到环R上的模范畴,得到环R与其扩张环的K_1群的一个结果.

Let A be a category. Based on the definition of an η-extension of A it obtains the additive invariant on its η-extension, further, gets an exchange property between an -extension and a loop category, that is, there is an equivalenceΩ（A（η））≌（ΩA）（η→） as additive catego- ries. At the end, it applies the main theorem to module categories of ring R, obtains a result on Ki groups of R and its extension ring.