Let $X_n$ be the set of Nakayama algebras with a linear connected quiver and n simple modules. Define a relation on $X_n$ by saying that two such algebras are equivalent if one is isomorphic to the endomorphism ring of a tilting module of the other algebra. Recall that here a tilting module is a module $T$ with $n$ indecomposable summands and $Ext^i(T,T)=0$ for all $i >0$.
- Are they equivalent iff they are derived equivalent? (if not, i hope this is still an equivalence relation) edit: It is not an equivalence relation, see the comment by Jeremy Rickard. Thus the answer to 1. is no.
- How many equivalence classes are there under derived equivalence?
- Is 1. true if we replace in the relation the tilted condition into "iterated tilted"?