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?