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The following is an open problem:

Given a partial tilting module $T$ over a finite dimensional algebra $A$ (that is $Ext_A^i(T,T)=0$ for all $i \geq 1$ and $pd(T) < \infty$), then $T$ is a tilting module iff $|T|=n$ where $n$ denotes the number of simple $A$-modules and $|T|$ the number of non-isomorphic indecomposable summands of $T$.

Questions:

  • Who first conjectured this and where?
  • What is the status of the conjecture (for which classes of algebras it is known?) and its relation to the other homological conjectures? Maybe there is a survey on this and related open problems on tilting modules?
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2 Answers 2

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There is relevant information here, including a statement of the conjecture (as Conjecture 5.1).

http://www.math.uni-bonn.de/people/schroer/fd-problems-files/FD-RigidModulesConj.pdf

That preprint also gives a couple of references to papers of Happel. I have heard the conjecture attributed to him, but I cannot conveniently check whether it appears in the cited papers.

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I found the article https://link.springer.com/chapter/10.1007/978-94-009-2985-2_6 which is a compilation of questions and problems by several authors compiled by M.Schaps from 1988.

Question 1 in chapter 3 is due to D. Happel and asks the question of this thread. Question 1' is by J. Rickard and asks the analog question for tilting complexes. It is also noted that the truth of the question in the thread implies the Gorenstein symmetry conjecture.

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