Let be any (associative, unital) ring and a natural number. Then any -tilting as well as any -cotilting class within the class of all (right -) modules is definable, that is, closed under direct products, direct limits and pure submodules.
In the tilting case, the notions of a class of finite and countable type are used, . A class is of countable (finite) type if there is a set of countably (finitely) presented modules such that . The proof involves two steps. First, any -tilting class is proved to be of countable type using set-theoretic methods, . Next, is proved to be of finite type, generalizing results in , which already implies the definability of .
In the cotilting case, the problem was reduced to the following conjecture in the paper :
If is a module and is closed under direct products and pure submodules, then is closed under direct limits.
The proof of the latter assertion presented here is given in  by analyzing the cokernel of , where is a module and is a pure-injective hull of .