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We introduce and study the new concepts of cosilting complexes, cosiltingmodules and AIR-cotilting modules. We prove that the three conceptsAIR-cotilting modules, cosilting modules and quasi-cotilting modules coincidewith each other, in contrast with the dual fact that AIR-tilting modules,silting modules and quasi-tilting modules are different. Further, we show thatthere are bijections between the following four classes (1) equivalent classesof AIR-cotilting (resp., cosilting, quasi-cotilting) modules, (2) equivalentclasses of 2-term cosilting complexes, (3) torsion-free cover classes and (4)torsion-free special precover classes. We also extend a classical result ofAuslander and Reiten on the correspondence between certain contravariantlyfinite subcategories and cotilting modules to the case of cosilting complexes.

Cosilting complexes and AIR-cotilting modules