On the negative-one shift functor for FI-modules

Abstract

We show that the negative-one shift functor S−1 on the category of FI-modules is a left adjoint of the shift functor S and a right adjoint of the derivative functor D. We show that for any FI-module V , the coinduction QV of V is an extension of V by S−1V .

FI$^m$-modules over Noetherian rings

“…Representation theory of the category FI and its plentiful applications in various fields such as algebra, algebraic topology, algebraic geometry, number theory, rooted from the work of Church, Ellenberg, Farb, and Nagpal in [4,5], have been actively studied. With their initial contributions and the followed explorations described in a series of papers [3,8,9,10,18,13,14,16,17,21,22,23,24], people now have a satisfactory understanding on representational and homological properties of FI-modules and their relationships with representation stability properties. Furthermore, quite a few combinatorial categories sharing similar structures as FI, including FI G , FI d , FI m , FIM have been introduced and studied with viewpoints from representation theory, commutative algebra, and combinatorics; see [6,7,24,26].…”