A short computation of the Rouquier dimension for a cycle of projective lines

Given a dg category 𝒞, we introduce a new class of objects (weakly product bimodules) in 𝒞^op ⊗ 𝒞 generalizing product bimodules. We show that the minimal generation time of the diagonal by weakly product bimodules provides an upper bound for the Rouquier dimension of 𝒞. As an application, we give a purely algebro-geometric proof of a result of Burban and Drozd that the Rouquier dimension of the derived category of coherent sheaves on an n-cycle of projective lines is one. Our approach explicitly gives the generator realizing the minimal generation time.