In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category \mathcal{A} to an appropriate q-polynomial algebra. In the case that \mathcal{A} is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math. 55(10) 2045–2066, 2012). Moreover, if the underlying graph of Q associated with \mathcal{A} is bipartite and the matrix B associated to the quiver Q is of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.

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.

We develop a local cohomology theory for FI^m-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI^m-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.

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

For a Calabi-Yau triangulated category $\mathcal{C}$ of Calabi-Yau dimension $d$ with a $d-$cluster tilting subcategory $\mathcal{T}$, the decomposition of $\mathcal{C}$ is determined by the decomposition of $\mathcal{T}$ satisfying "vanishing condition" of negative extension groups, namely, $\mathcal{C}=\oplus_{i\in I}\mathcal{C}_i$, where $\mathcal{C}_i, i\in I$ are triangulated subcategories, if and only if $\mathcal{T}=\oplus_{i\in I}\mathcal{T}_i,$ where $\mathcal{T}_i, i\in I$ are subcategories with $\mbox{Hom} _{\mathcal{C}}(\mathcal{T} _i[t],\mathcal{T} _j)=0, \forall 1\leq t\leq d-2$ and $i\not= j.$ This induces that for any two cluster tilting objects $T, T'$ in a $2-$Calabi-Yau triangulated category $\mathcal{C}$, the Gabriel quivers of endomorphism algebra End$_{\mathcal{C}}T$ of $T$ is connected if and only if so is End$_{\mathcal{C}}T'$. As an application, we prove that indecomposable $2-$Calabi-Yau triangulated categories with cluster tilting objects have no non-trivial t-structures and no non-trivial co-t-structures. This allows us to give a classification of torsion pairs in those triangulated categories, and to determine further the hearts of torsion pairs in the sense of Nakaoka, which are equivalent to the module categories over the endomorphism algebras of the cores of the torsion pairs.