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

Abstract

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 $\mathbb{Z}$-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.