Infinite dimensional tilting theory

Abstract

Infinite dimensional tilting modules are abundant in representation theory. They occur when studying torsion pairs in module categories, when looking for complements to partial tilting modules, or in connection with the Homological Conjectures. They share many properties with classical tilting modules, but they also give rise to interesting new phenomena as they are intimately related with localization, both at the level of module categories and of derived categories. In these notes, we review the main features of infinite dimensional tilting modules. We discuss the relationship with approximation theory and with localization. Finally, we focus on some classification results and we give a geometric interpretation of tilting.

Cosilting Modules

“…Therefore, N d is not (partial) silting. By a similar argument it follows that the example constructed in [1,Section 3.4] can be used to see that over tame hereditary artin algebras of infinite representation type there exists a cotilting module W such that its dual W d is not (partial) silting.…”

“…Proof. (1) This follows from the inclusion Cogen(T ) ⊆ ⊥ T and [10, Proposition 1.4.2] by a standard proof. For reader's convenience, we present the details of this proof.…”

“…Conversely, if ( • T , B ζ ) is a torsion pair, then B ζ is closed under direct products and T ∈ B ζ . It follows that T is partial cosilting, and we apply (1) to obtain that ( • T , Cogen(T )) is a torsion pair. Then B ζ = Cogen(T ), and the proof is complete.…”

“…We mention that if we start with a cosilting module N then its dual N d is not necessarily a (partial) silting module. In order to see this we will use the examples constructed in [1,Section 3].…”

“…Proof. (1) This follows from the fact that every contravariant Hom functor sends direct sums into direct products.…”