The algebraic K-theory of the projective line associated with a strongly Z-graded ring

A Laurent polynomial ring R=S[t, t−1] with coefficients in a unital ring determines a category of quasi-coherent sheaves on the projective line over S; its K-theory is known to split into a direct sum of two copies of the K-theory of S. 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.