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Question whether the presentation functor preserves tilting objects

Let T\mathcal T be an idempotent complete algebraic triangulated category, let MM be a silting object, and put B=End⁡T(M).B=\operatorname{End}_{\mathcal T}(M). For the presentation functor PM ⁣:pr⁡(M)→K[−1,0](proj⁡B),\mathbb P_M\colon\operatorname{pr}(M)\to K^{[-1,0]}(\operatorname{proj}B), is PM(X)\mathbb P_M(X) tilting whenever X∈pr⁡(M)X\in\operatorname{pr}(M) is tilting?

References

Primary source

arXiv

Additional references

homological-algebrarepresentation-theorytriangulated-categories
Posted

Progress summary

Refreshed
Claimed solved

A new unrefereed preprint claims a concrete counterexample showing that the presentation functor does not always preserve tilting objects.

The question, attributed to Yu Zhou, asks whether a tilting object TT is sent by the presentation functor to a tilting complex. Earlier work described the general behavior on degree −1-1 extensions as unresolved.

Known results

  • The answer is positive when End⁡T(T)\operatorname{End}_{\mathcal T}(T) is hereditary (2025 preprint).

September 2026 negative counterexample

A preprint constructs, for a specified quiver algebra, T=ΣΛT=\Sigma\Lambda such that PM(T)P_M(T) is a two-term silting complex with a nonzero negative self-extension, so it is not tilting. This claims a negative resolution of the preservation question, but the result is unverified.

Current status (as of September 2026): A preprint claims the preservation question is settled negatively by an explicit finite-dimensional counterexample; independent verification remains outstanding.

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