MathDB

Wakamatsu Tilting Conjecture

At least 15 years old · documented by

Let AA be a basic finite dimensional kk-algebra and let TT be a finitely generated AA-module. Call TT a Wakamatsu tilting module if Ext⁡Ai(T,T)=0\operatorname{Ext}_A^i(T,T)=0 for all i>0i>0 and there is a coresolution

0→A→T0→T1→T2→…0 \rightarrow A \rightarrow T_0 \rightarrow T_1 \rightarrow T_2 \rightarrow \ldots

with each TiT_i in add⁡T\operatorname{add} T and the relevant first extension groups vanishing. Wakamatsu Tilting Conjecture. If TT is a Wakamatsu tilting module with finite projective dimension, then TT is a tilting module. Tilting modules are known to be Wakamatsu tilting, but the converse under finite projective dimension is an important unresolved homological question.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Wakamatsu tilting conjecture

    Let AA be a finite-dimensional algebra over a field. Let T∈A-mod⁡T\in A\operatorname{-mod}. Suppose that TT is self-orthogonal, has finite projective dimension, and satisfies T-⁡domdim⁡AA=+∞T\operatorname{-}\operatorname{domdim}_A A=+\infty. Wakamatsu tilting conjecture. Then TT is a tilting module. This is presented as a related conjecture to the self-orthogonal quasi-generator conjecture; the source does not provide evidence of resolution.

    source: Tiago Cruz, “Higher Morita-Tachikawa correspondence”, arXiv:2304.01370 (2023).

References

Primary source

Dag Madsen, “On a common generalization of Koszul duality and tilting equivalence”, arXiv:1007.3282 (2010).

homological-algebrarepresentation-theorytilting-theory
Posted

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.