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

12 pages