论文标题
代数$ k $的“基本定理” - 强烈$ \ mathbb {z} $ - 分级戒指
The "fundamental theorem" for the algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings
论文作者
论文摘要
代数$ k $的“基本定理”表达了laurent多项式环$ l [t，t，t^{ - 1} $的$ k $ groups，作为$ k $ -groups $ l $的两个副本的直接总和（一份副本，一份副本），以及某些$ l $ l $ l $ l $。这里显示，此结果的修改版本大概是$ \ mathbb {z} $ - 分级环；分裂不是$ l $的代数$ k $ - 组，涉及与来自分级结构的$ l $模型类别相关的群体。 （在经典案例中，这些动作是微不足道的）。 NIL基团通过减少的$ K $ - 同型nilpotent扭曲的内态理论，并建立了Mayer-Vietoris的类似物和定位序列。
The "fundamental theorem" for algebraic $K$-theory expresses the $K$-groups of a Laurent polynomial ring $L[t,t^{-1}]$ as a direct sum of two copies of the $K$-groups of $L$ (with a degree shift in one copy), and certain "nil" groups of $L$. It is shown here that a modified version of this result generalises to strongly $\mathbb{Z}$-graded rings; rather than the algebraic $K$-groups of $L$, the splitting involves groups related to the shift actions on the category of $L$-modules coming from the graded structure. (These action are trivial in the classical case). The nil groups are identified with the reduced $K$-theory of homotopy nilpotent twisted endomorphisms, and analogues of Mayer-Vietoris and localisation sequences are established.
