Mackey Functors with Chain Conditions

✍ Scribed by Florian Luca

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 336 KB
• Volume: 205
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1997.7417

✦ Synopsis

Let A be a Green functor for a finite group G and let M be a left A-module. M Ž . is called noetherian artinian if the lattice of all submodules of M satisfies ACC Ž . Ž . respectively DCC . M is called totally noetherian totally artinian if M is X Ž . noetherian respectively artinian for every finite G-set X. In this paper it is shown Ž . that the totally noetherian totally artinian left A-modules behave in the same Ž . Ž way as the classical noetherian artinian modules. The left noetherian left . artinian Green functors are less well behaved but some properties from classical algebra still hold. It is shown that Cohen's theorem holds in this setting. Our main result shows that if a commutative Green functor A is artinian, then the rings Ž . Ž Ž .. A H rJac A H are semisimple artinian for all subgroups H of G. ᮊ 1998 Academic Press Contents. 1. Introduction. 2. Definitions, notations, and terminology. 3. Characteristic Mackey functors and cocharacteristic subfunctors.

4. Mackey functors with chain conditions. 5. Commutati¨e artinian Green functors.