There seem to be two types of definitions for what is a tilting module (as a reference, Handbook of Tilting Theory). I believe that the original definition of Ringel is
Def: T, a module over a finite dimensional associate algebra $\Lambda$, is tilting if
1) The projective dimension is at most 1
3) There is a short exact sequence $0 \rightarrow \Lambda \rightarrow T^1 \rightarrow T^2 \rightarrow 0$ and $T^i% are elements of add T
This definition is then generalized (ex. Reiten) for $\Lambda$ an Artin algebra 1') projective dimension is finite 2') $Ext^i(T,T)=0$ for i>0 3') There is an exact sequence (not necessarily short) like that of 3) above.
On the other hand, Donkin gives us a definition for a (partial) tilting module over algebraic groups, if T admits both a standard and a costandard filtration. This is also the definition used for tilting modules over Lie algebras, etc. Property 2) follows from the filtrations, and something similar to 3) if we consider a full tilting module. My questions are the following
Q: Under what conditions are the definitions of Ringel and Donkin the same? Reference?
Q: Under what conditions does having a standard & constandard filtration imply the weaker 1')?
Q:Suppose we have tilting modules, over some algebraic structure, in the sense of Donkin, which do not have finite projective dimension. Is there some kind of analogue of properties 1) or 1') to look for?