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In this paper we introduce an inductive method to study OI -modules presented in finite degrees, where OI is a skeleton of the category of totally ordered finite sets and order-preserving injective maps. As an application, we obtain an explicit upper bound for the Castelnuovo-Mumford regularity of OI -modules. [ABSTRACT FROM AUTHOR]

We show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite Castelnuovo-Mumford regularity if and only if they are presented in finite degrees, and hence the category consisting of them is abelian. These results apply to examples such as the categories FI m and FI G m. [ABSTRACT FROM AUTHOR]

In this paper we describe inductive machinery to investigate asymptotic behavior of homology groups and related invariants of representations of certain graded combinatorial categories over a commutative Noetherian ring k , via introducing inductive functors which generalize important properties of shift functors of FI-modules. In particular, a sufficient criterion for finiteness of Castelnuovo–Mumford regularity of finitely generated representations of these categories is obtained. As applications, we show that a few important infinite combinatorial categories appearing in representation stability theory (for example FI d , OI d , FI d , OI d ) are equipped with inductive functors, and hence the finiteness of Castelnuovo–Mumford regularity of their finitely generated representations is guaranteed. We also prove that truncated representations of these categories have linear minimal resolutions by relative projective modules, which are precisely linear minimal projective resolutions when k is a field of characteristic 0. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

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 ˜ − 1 V . [ABSTRACT FROM AUTHOR]

We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective. [ABSTRACT FROM AUTHOR]

Abstract: A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with . We study its representation theory, and in particular, its category . [Copyright &y& Elsevier]

Abstract: We determine the PBW deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine Dynkin quiver. In particular, we show that there is precisely one parameter which does not come from deformation of the preprojective algebra. We prove that the PBW deformation is Morita equivalent to a corresponding symplectic reflection algebra for wreath product. [Copyright &y& Elsevier]

We give a direct proof of (a slight generalization of) the recent result of Premet (1999) related to generalized Gelfand-Graev representations and of an equivalence due to Skryabin (2000). [ABSTRACT FROM AUTHOR]

We answer positively a question of Church, Miller, Nagpal and Reinhold on existence of a linear bound on the presentation degree of the homology of a complex of FI-modules. This implies a linear stable range for the homology of congruence subgroups of general linear groups. [ABSTRACT FROM AUTHOR]