[Paper Review] Cosilting complexes and AIR-cotilting modules

Peiyu Zhang, Jiaqun Wei|arXiv (Cornell University)|Jan 7, 2016

Algebraic structures and combinatorial models18 references3 citations

TL;DR

This paper introduces and unifies three dual notions in module theory—cosilting complexes, cosilting modules, and AIR-cotilting modules—proving they are equivalent over arbitrary rings. It establishes a bijection between equivalent classes of these modules and 2-term cosilting complexes, torsion-free cover classes, and torsion-free special precover classes, extending classical duality results from tilting theory to the cotilting setting.

ABSTRACT

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

Motivation & Objective

• To introduce and systematically study the new concepts of cosilting complexes, cosilting modules, and AIR-cotilting modules as duals to silting and tilting notions.
• To establish the equivalence between AIR-cotilting modules, cosilting modules, and quasi-cotilting modules over arbitrary rings, contrasting with the known distinctions in the tilting case.
• To extend Auslander and Reiten's correspondence between contravariantly finite subcategories and cotilting modules to the derived and cosilting setting.
• To establish a bijection between equivalent classes of AIR-cotilting (or cosilting) modules and key classes of torsion-free precover systems.

Proposed method

• Define a cosilting complex $ T $ over a ring $ R $ as a bounded complex in injective modules satisfying three conditions: (1) $ T o ext{Inj}(R) $, (2) $ T $ is prod-semi-selforthogonal, and (3) $ ext{K}^b( ext{Inj}(R)) $ is the smallest triangulated subcategory containing $ ext{Adp}_{ ext{D}}(T) $.
• Use the derived category $ ext{D}( ext{Mod}R) $ to define isomorphism classes of cosilting complexes and relate them to modules via derived isomorphism.
• Introduce AIR-cotilting modules via two exact sequences involving projective presentations and surjectivity of Hom functors on direct products.
• Define $ ext{Cogen}M $ and $ ext{F}_ ho $ for injective copresentations $ ho $, and use these to characterize partial and full AIR-cotilting modules.
• Apply Lemma 4.15 and Lemma 4.17 to decompose injective maps and relate $ ext{F}_ ho $-classes to kernel conditions on Hom functors.
• Prove equivalence of the three module classes via a chain of implications: (1) ⇒ (2) ⇒ (3) ⇒ (1), using properties of precover maps and Ext-injectivity.

Experimental results

Research questions

• RQ1 Are cosilting complexes, cosilting modules, and AIR-cotilting modules equivalent concepts over arbitrary rings?
• RQ2 How do cosilting complexes relate to torsion-free cover classes and torsion-free special precover classes?
• RQ3 Can the classical Auslander–Reiten correspondence between contravariantly finite subcategories and cotilting modules be extended to the derived and cosilting setting?
• RQ4 What characterizations and structural properties do cosilting modules possess, particularly regarding purity and cofiniteness?
• RQ5 What is the role of minimal injective copresentations in characterizing partial and full AIR-cotilting modules?

Key findings

• Cosilting complexes, cosilting modules, and AIR-cotilting modules are equivalent concepts over arbitrary unital rings.
• Cosilting modules are always pure-injective and cofinendo, and every cosilting module is quasi-cotilting.
• There is a bijection between equivalent classes of AIR-cotilting (or cosilting, or quasi-cotilting) modules and equivalent classes of 2-term cosilting complexes.
• There is a bijection between equivalent classes of AIR-cotilting modules and torsion-free cover classes.
• There is a bijection between equivalent classes of AIR-cotilting modules and torsion-free special precover classes.
• The classical Auslander–Reiten correspondence between contravariantly finite subcategories and cotilting modules is extended to cosilting complexes over arbitrary rings.