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

Given a dg category




C

\mathcal C



, we introduce a new class of objects (weakly product bimodules) in






C


o
p



⊗



C


\mathcal C^{op}\otimes \mathcal 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

\mathcal 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
n



-cycle of projective lines is one. Our approach explicitly gives the generator realizing the minimal generation time.