Closure properties of tilting and cotilting classes

Jan Šťovíček¹

Charles University, Prague; Heinrich Heine Universität Düsseldorf

Let $R$ be any (associative, unital) ring and $n \ge 0$ a natural number. Then any $n$-tilting as well as any $n$-cotilting class within the class of all (right $R$-) 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 $\mathcal C$ is of countable (finite) type if there is a set of countably (finitely) presented modules $\mathcal S$ such that $\mathcal C = \mathcal S^{\perp_1} = {\mathrm{Ker}}{\mathrm{Ext}}^1_R(\mathcal S, -)$. The proof involves two steps. First, any $n$-tilting class $\mathcal T$ is proved to be of countable type using set-theoretic methods, [8]. Next, $\mathcal T$ is proved to be of finite type, generalizing results in [6], which already implies the definability of $\mathcal T$.

In the cotilting case, the problem was reduced to the following conjecture in the paper [2]:

If $U$ is a module and ${^{\perp_1} U} = {\mathrm{Ker}}{\mathrm{Ext}}^1_R(-,U)$ is closed under direct products and pure submodules, then ${^{\perp_1} U}$ is closed under direct limits.

The proof of the latter assertion presented here is given in [7] by analyzing the cokernel of $M \hookrightarrow PE(M)$, where $M$ is a module and $PE(M)$ is a pure-injective hull of $M$.

Footnotes

...ček¹

joint work with Silvana Bazzoni and Jan Trlifaj