A splitting result for the algebraic K-theory of projective toric schemes

Suppose X is a projective toric scheme defined over a ring R and equipped with an ample line bundle L . We prove that its K-theory has a direct summand of the form K(R)(k+1) where k = 0 is minimal such that L?(-k-1) is not acyclic. Using a combinatorial description of quasi-coherent sheaves we interpret and prove this result for a ring R which is either commutative, or else left noetherian.