An inductive machinery for representations of categories with shift functors

We describe an inductive machinery to prove various properties of representations of a category equipped with a generic shift functor. Specifically, we show that if a property (P) of representations of the category behaves well under the generic shift functor, then all finitely generated representat...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2019-12, Vol.371 (12), p.8513-8534
Main Authors: Gan, Wee Liang, Li, Liping
Format: Article
Language:English
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Summary:We describe an inductive machinery to prove various properties of representations of a category equipped with a generic shift functor. Specifically, we show that if a property (P) of representations of the category behaves well under the generic shift functor, then all finitely generated representations of the category have the property (P). In this way, we obtain simple criteria for properties such as Noetherianity, finiteness of Castelnuovo-Mumford regularity, and polynomial growth of dimension to hold. This gives a systemetic and uniform proof of such properties for representations of the categories \mathscr {FI}_G and \mathscr {OI}_G which appear in representation stability theory.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/7554