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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 anarbitrary strongly $\mathbb{Z}$-graded ring $R$ in place of the Laurentpolynomial ring. The projective line associated with $R$ is indirectly definedby specifying the corresponding category of quasi-coherent sheaves. Notionsfrom algebraic geometry like sheaf cohomology and twisting sheaves aretransferred to the new setting, and the $K$-theoretical splitting isestablished.

The algebraic $K$-theory of the projective line associated with a strongly $\mathbb{Z}$-graded ring