A Laurent polynomial ring A[t,1/t] with coefficients in a unital ring A determines a category of quasi-coherent sheaves on the projective line over A; its K-theory is known to split into a direct sum of two copies of the K-theory of A. In this paper, the result is generalised to the case of an arbitrary strongly Z-graded ring R in place of the Laurent polynomial ring. The projective line associated with R is indirectly defined by specifying the corresponding category of quasi-coherent sheaves. Notions from algebraic geometry like sheaf cohomology and twisting sheaves are transferred to the new setting, and the K-theoretical splitting is established.

中文翻译：

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与强 Z 分级环相关的投影线的代数 K 理论

具有单位环 A 中系数的洛朗多项式环 A[t,1/t] 确定了 A 上投影线上的一类准相干滑轮；已知其 K 理论分裂为 A 的 K 理论的两个副本的直接和。 在本文中，结果被推广到任意强 Z 分级环 R 代替洛朗多项式环的情况. 与 R 相关的投影线是通过指定相应的准相干滑轮类别来间接定义的。来自代数几何的概念，如层上同调和扭曲层，被转移到新的设置中，并建立了 K 理论分裂。