A negative answer to a question on tilting objects and two-term complexes

Abstract

Let $M$ be a silting object in an idempotent complete algebraic triangulated category $\mathcal T$. Put $B={\rm End}_{\mathcal T}(M)$, and let $\mathbb{P}_M\colon {\rm pr}(M)\to K^{[-1,0]}({\rm proj}B)$ be the presentation functor associated with $M$. It was recently asked whether $\mathbb{P}_M(T)$ must be tilting whenever $T\in{\rm pr}(M)$ is a tilting object. We answer this question in the negative by giving an explicit finite-dimensional example. Namely, for $$ \Lambda=k(1\xrightarrow{\alpha}2\xrightarrow{\beta}3\xrightarrow{\gamma}4)/(\alpha\beta\gamma),$$ we construct a silting object $M\in K^b({\rm proj}\Lambda)$ and a tilting object $T=\Sigma\Lambda\in{\rm pr}(M)$ for which $$ {\rm dim}_k{\rm Hom}_{K^b({\rm proj}B)}\bigl(\mathbb{P}_M(T),\Sigma^{-1}\mathbb{P}_M(T)\bigr)=1.$$ Thus $\mathbb{P}_M(T)$ is a two-term silting complex but not a tilting complex.