Jan Šťovíček1
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, [4]. 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, [8]. Next,
is proved to be of finite type, generalizing results in [6], which already implies the definability of
.
In the cotilting case, the problem was reduced to the following conjecture in the paper [2]:
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 [7] by analyzing the cokernel of
, where
is a module and
is a pure-injective hull of
.