From objects finitely presented by a rigid object in a triangulated category to 2-term complexes

Abstract

For a rigid object $M$ in an algebraic triangulated category $\mathcal{T}$, a functor pr$(M)\to\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ is constructed, which essentially takes an object to its `presentation', where pr$(M)$ is the full subcategory of $\mathcal{T}$ of objects finitely presented by $M$, $A$ is the endomorphism algebra of $M$ and $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ is the homotopy category of complexes of finitely projective $A$-modules concentrated in degrees $-1$ and $0$. This functor is shown to be full and dense and its kernel is described. It detects isomorphisms, indecomposability and extriangles. In the Hom-finite case, it induces a bijection from the set of isomorphism classes of basic relative cluster-tilting objects of pr$(M)$ to that of basic silting complexs of $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$, which commutes with mutations. These results are applied to cluster categories of self-injective quivers with potential to recover a theorem of Mizuno on the endomorphism algebras of certain 2-term silting complexes. As an interesting consequence of the main results, if $\mathcal{T}$ is a 2-Calabi--Yau triangulated category and $M$ is a cluster-tilting object such that $A$ is self-injective, then $\mathbb{P}$ is an equivalence, in particular, $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ admits a triangle structure. In the appendix by Iyama it is shown that for a finite-dimensional algebra $A$, if $\mathcal{H}^{[-1,0]}({\rm proj}\, A)$ admits a triangle structure, then $A$ is necessarily self-injective.