Singularity Confinement and Projective Resolution of Triangulated Category

We proposed, in our previous paper, to characterize the Hirota-Miwa equation by means of the theory of triangulated category. We extend our argument in this paper to support the idea. In particular we show in detail how the singularity confinement, a phenomenon which was proposed to characterize integrable maps, can be associated with the projective resolution of the triangulated category.