# Equivalence of definitions of Tilting

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

2) $Ext^1(T,T)=0$

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$for i>0 3') There is an exact sequence (not necessarily short) like that of 3) above. On the other hand, Donkin gives us a definition for a (partial) tilting module over algebraic groups, if T admits both a standard and a costandard filtration. This is also the definition used for tilting modules over Lie algebras, etc. Property 2) follows from the filtrations, and something similar to 3) if we consider a full tilting module. My questions are the following Q: Under what conditions are the definitions of Ringel and Donkin the same? Reference? Q: Under what conditions does having a standard & constandard filtration imply the weaker 1')? Q:Suppose we have tilting modules, over some algebraic structure, in the sense of Donkin, which do not have finite projective dimension. Is there some kind of analogue of properties 1) or 1') to look for? ## 1 Answer Q1: Ringel defined, for any given quasi-hereditary algebra$\Lambda$, a particular tilting-cotilting module called the characteristic tilting module. (It is a tilting module in the generalized sense, the projective dimension can be larger than one.) This characteristic tilting module has both a standard and a costandard filtration. Due to a slight misunderstanding, Donkin used the name 'tilting module' for the summands of this particular module. It was a useful concept, and since then this has become the definition used in some parts of representation theory. The two definitions are almost never equivalent, since for any given algebra there are usually many different tilting modules in the original sense. For instance is$\Lambda\$ itself always a tilting module.

Q3: There is a concept called Wakamatsu tilting module (see Handbook of Tilting Theory, page 207) where you leave out condition 1), keep condition 2), and in condition 3) replace the sequence with a possibly infinite coresolution.