Silting and Calabi–Yau reductions are important processes in representation theory to construct new triangulated categories from given ones, which similar Verdier quotient. In this paper, first we introduce a reduction process of category, is analogous the silting (Calabi–Yau) reduction. For category T $\mathcal {T}$ with pre-simple-minded collection (pre-SMC) R {R}$ , U {U}$ such that SMCs bij...

For a suitable triangulated category $\mathcal{T}$ with Serre functor $S$ and full precovering subcategory $\mathcal{C}$ closed under summands extensions, an indecomposable object $C$ in is called Ext-projective if Ext$^1(C,\mathcal{C})=0$. Then there no Auslander-Reiten triangle end term $C$. In this paper, we show that if, for such $C$, minimal right almost split morphism $\beta:B\rightarrow ...

We prove the relation between the triangulated categories of graded B-branes on simple singularities and the derived categories of representations of Dynkin quivers. The proof is based on the theory of weighted projective lines by Geigle and Lenzing and Orlov’s theorem on the relation between triangulated categories of graded B-branes and the derived category of coherent sheaves.