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Wakamatsu Tilting Conjecture

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 1

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).

Sources & referencesView supporting material

Primary source

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

homological-algebrarepresentation-theorytilting-theory

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