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On central stability
Author(s) -
Gan Wee Liang,
Li Liping
Publication year - 2017
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12044
Subject(s) - mathematics , injective function , categorical variable , stability (learning theory) , ideal (ethics) , class (philosophy) , pure mathematics , finite set , algebra over a field , discrete mathematics , mathematical analysis , statistics , computer science , epistemology , machine learning , artificial intelligence , philosophy
The notion of central stability was first formulated for sequences of representations of the symmetric groups by Putman. A categorical reformulation was subsequently given by Church, Ellenberg, Farb, and Nagpal using the notion of FI‐modules, where FI is the category of finite sets and injective maps. We extend the notion of central stability from FI to a wide class of categories, and prove that a module is presented in finite degrees if and only if it is centrally stable. We also introduce the notion of d ‐step central stability, and prove that if the ideal of relations of a category is generated in degrees at most d , then every module presented in finite degrees is d ‐step centrally stable.