We describe the stable module categories of the self-injective finite-dimensional algebras of finite representation type over an algebraically closed field which are Calabi-Yau (in the sense of Kontsevich).

The Cartan matrix of a finite dimensional algebra A is an important combinatorial invariant reflecting frequently structural properties of the algebra and its module category. For example, one of the important features of the modular representation theory of finite groups is the nonsingularity of Cartan matrices of the associated group algebras (Brauer’s theorem). Recently, the class of all tame selfinjective algebras having simply connected Galois coverings and the stable Auslander-Reiten quiver...

Let $R$ be a ring. A subclass $𝒯$ of left $R$-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let $𝒯$ be a weak torsion class of left $R$-modules and $n$ a positive integer. Then a left $R$-module $M$ is called $𝒯$-finitely generated if there exists a finitely generated submodule $N$ such that $M/N\in 𝒯$; a left $R$-module $A$ is called $(𝒯,n)$-presented if there exists an exact sequence of left $R$-modules $$0⟶K_{n-1}⟶F_{n-1}⟶\cdots ⟶F_1⟶F_0⟶M⟶0$$ such that $F_0,\cdots ,F_{n-1}$ are finitely generated free and $K_{n-1}$ is $𝒯$-finitely generated; a left $R$-module...

We give axiomatic conditions in order to calculate the local cohomology of some idempotent kernel functors. These results lie in some new dimension introduced by T. Levasseur for Auslander-Gorenstein rings. Under some hypothesis, we generalize previous results.

In this paper, we show the existence of copure injective preenvelopes over noetherian rings and copure flat preenvelopes over commutative artinian rings. We use this to characterize $n$-Gorenstein rings. As a consequence, if the full subcategory of strongly copure injective (respectively flat) modules over a left and right noetherian ring $R$ has cokernels (respectively kernels), then $R$ is $2$-Gorenstein.

The purpose of this paper is to further the study of countably thick modules via weak injectivity. Among others, for some classes $ℳ$ of modules in $\sigma \left[M\right]$ we study when direct sums of modules from $ℳ$ satisfies a property $\mathbb{P}$ in $\sigma \left[M\right]$. In particular, we get characterization of locally countably thick modules, a generalization of locally q.f.d. modules.