# Closure properties of tilting and cotilting classes

Jan Šťovíček1

Charles University, Prague; Heinrich Heine Universität Düsseldorf

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 .

#### Footnotes

...ček1
joint work with Silvana Bazzoni and Jan Trlifaj

2005-05-23