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We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler 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.

Let A be a finite dimensional algebra and D^b(A) be the bounded derived category of finitely generated left A-modules. In this paper we consider lengths of compact exceptional objects in D^b(A), proving a sufficient condition such that these lengths are bounded by the number of isomorphism classes of simple A-modules. Moreover, we show that algebras satisfying this condition is bounded derived simple.

In this paper we use a homological approach to obtain upper bounds for a few homological invariants of FI_G-modules V. These upper bounds are expressed in terms of the generating degree and torsion degree, which measure the top and socle of V under actions of non-invertible morphisms in the category respectively.

We show that the FI-homology of an FI-module can be computed via a Koszul complex. As an application, we prove that the Castelnuovo-Mumford regularity of a finitely generated torsion FI-module is equal to its degree.

Let k be a commutative Noetherian ring. In this paper we consider filtered modules of the category FI firstly introduced by Nagpal. We show that a finitely generated FI-module V is filtered if and only if its higher homologies all vanish, and if and only if a certain homology vanishes. Using this homological characterization, we characterize finitely generated FI-modules V whose projective dimension is finite, and describe an upper bound for it. Furthermore, we give a new proof for the fact that V induces a finite complex of filtered modules, and use it as well as a result of Church and Ellenberg to obtain another upper bound for homological degrees of V.

In this paper we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras, and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.

In this paper we consider several homological dimensions of crossed products $A _{\alpha} ^{\sigma} G$, where $A$ is a left Noetherian ring and $G$ is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of $A ^{\sigma} _{\alpha} G$ are classified: global dimension of $A ^{\sigma} _{\alpha} G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^{\sigma} _{\alpha} G$ coincides with that of $A$. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that $A$ is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow $p$-subgroup $S \leqslant G$, we show that $A$ and $A _{\alpha} ^{\sigma} G$ share the same homological dimensions under extra assumptions.

Let k be a commutative Noetherian ring and C−− be a locally finite k-linear category equipped with a self-embedding functor of degree 1. We show under a moderate condition that finitely generated torsion representations of C−− are super finitely presented (that is, they have projective resolutions each term of which is finitely generated). In the situation that these self-embedding functors are genetic functors, we give upper bounds for homological degrees of finitely generated torsion modules. These results apply to quite a few categories recently appearing in representation stability theory. In particular, when k is a field of characteristic 0, we obtain another upper bound for homological degrees of finitely generated FI-modules.

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.

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