We introduce and study the notion of Gorenstein silting complexes, which is a generalization of Gorenstein tilting modules in Gorenstein-derived categories. We give the equivalent characterization of Gorenstein silting complexes. We give a sufficient condition for a partial Gorenstein silting complex to have a complement.

Let Λ be an artin algebra and $0=I_{0}\subseteq I_{1} \subseteq I_{2}\subseteq\cdots \subseteq I_{n}$ a chain of ideals of Λ such that $(I_{i+1}/I_{i})\rad(\Lambda/I_{i})=0$ for any $0\leq i\leq n-1$ and $\Lambda/I_{n}$ is semisimple. If either none or the direct sum of exactly two consecutive ideals has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. As a consequence, we have that if either none or the direct sum of exactly two consecutive terms in the radical series of Λ has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. Some known results are obtained as corollaries.

Let $R$ be a ring and $T$ be a good Wakamatsu-tilting module with $S=\text{End}(T_{R})^{op}$. We prove that $T$ induces an equivalence between stable repetitive categories of $R$ and $S$ (i.e., stable module categories of repetitive algebras $\hat{R}$ and ${\hat{S}}$). This shows that good Wakamatsu-tilting modules seem to behave in Morita theory of stable repetitive categories as that tilting modules of finite projective dimension behave in Morita theory of derived categories.

In order to better unify the tilting theory and the Auslander–Reiten theory, Xi introduced a general transpose called the relative transpose. Originating from this, we introduce and study the cotranspose of modules with respect to a left A-module T called n-T-cotorsion-free modules. Also, we give many properties and characteristics of n-T-cotorsion-free modules under the help of semi-Wakamatsu-tilting modules AT.

The goal of this article is to study the coaugmented curved A∞-coalgebra structure of the Koszul codual of a filtered dg algebra over a field k. More precisely, we first extend one result of B. Keller that allowed to compute the A∞-coalgebra structure of the Koszul codual of a nonnegatively graded connected algebra to the case of any unitary dg algebra provided with a nonnegative increasing filtration whose zeroth term is k. We then show how to compute the coaugmented curved A∞-coalgebra structure of the Koszul codual of a Poincaré-Birkhoff-Witt (PBW) deformation of an N-Koszul algebra.

Khovanov–Lauda–Rouquier (KLR) algebras of finite Lie type come with families of standard modules, which under the Khovanov–Lauda–Rouquier categorification correspond to PBW bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of a standard module are injective. We present applications to the extensions between standard modules and modular representation theory of KLR algebras.

We consider the notion of a free resolution. In general, a free resolution can be of any length depending on the group ring under investigation. The metacyclic groups $G(pq)$ however admit periodic resolutions. In the particular case of $G(21)$ it is possible to achieve a fully diagonalized resolution. In order to achieve a diagonal resolution, we obtain a complete list of indecomposable modules over $\unicode[STIX]{x1D6EC}$. Such a list aids the decomposition of the augmentation ideal (the first syzygy) into a direct sum of indecomposable modules. Therefore, we are able to achieve a diagonalized map here. From this point it is possible to decompose all of the remaining syzygies in terms of indecomposable modules, leaving a diagonal resolution.

Let A be an artin algebra with representation dimension not more than 3. Assuming that AV is an Auslander generator and M ∈ addAV, we show that both findim(EndAM) and findim(EndAM)op are finite, and consequently the Gorenstein symmetry conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for EndAM.

Sather-Wagstaff et al. proved in [8] (S. Sather-Wagsta, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc.(2), 77(2) (2008), 481–502) that iterating the process used to define Gorenstein projective modules exactly leads to the Gorenstein projective modules. Also, they established in [9] (S. Sather-Wagsta, T. Sharif and D. White, AB-contexts and stability for Goren-stein at modules with respect to semi-dualizing modules, Algebra Represent. Theory14(3) (2011), 403–428) a stability of the subcategory of Gorenstein flat modules under a procedure to build R-modules from complete resolutions. In this paper we are concerned with another kind of stability of the class of Gorenstein flat modules via-à-vis the very Gorenstein process used to define Gorenstein flat modules. We settle in affirmative the following natural question in the setting of a left GF-closed ring R: Given an exact sequence of Gorenstein flat R-modules G = ⋅⋅⋅ G2G1G0G−1G−2 ⋅⋅⋅ such that the complex H ⊗RG is exact for each Gorenstein injective right R-module H, is the module M:= Im(G0 → G−1) a Gorenstein flat module?

Let $A$ be a graded algebra finitely generated in degree 1 over a field $k$. Point modules over $A$ introduced by Artin, Tate and Van den Bergh play an important role in studying $A$ in noncommutative algebraic geometry. In this paper, we define a dual notion of point module in terms of Koszul duality, which we call a co-point module. Using co-point modules, we will construct counter-examples to the following condition due to Auslander: for every finitely generated right module $\pi$ over a ring $R$, there is a natural number $n_M\in {\mathbb N}$ such that, for any finitely generated right module $N$ over $R$, ${\rm Ext}^i_R(M, N)=0$ for all $i\gg 0$ implies ${\rm Ext}^i_R(M, N)=0$ for all $i>n_M$.

The main result of this paper is that, over a non-commutative Koszul algebra, high truncations of finitely generated graded modules have linear free resolutions.