By analogy with the projective, injective and flat modules, in this paper we study some properties of $C$-Gorenstein projective, injective and flat modules and discuss some connections between $C$-Gorenstein injective and $C$-Gorenstein flat modules. We also investigate some connections between $C$-Gorenstein projective, injective and flat modules of change of rings.

Let G be a group, R a G-graded ring and X a right G-set. We study functors between categories of modules graded by G-sets, continuing the work of [M]. As an application we obtain generalizations of Cohen-Montgomery Duality Theorems by categorical methods. Then we study when some functors introduced in [M] (which generalize some functors ocurring in [D1], [D2] and [NRV]) are separable. Finally we obtain an application to the study of the weak dimension of a group graded ring.

We develop a diagrammatic categorification of the polynomial ring ℤ[x]. Our categorification satisfies a version of Bernstein-Gelfand-Gelfand reciprocity property with the indecomposable projective modules corresponding to xⁿ and standard modules to (x-1)ⁿ in the Grothendieck ring.

We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a $2\times 2$ matrix $A$ over a projective-free ring $R$ is strongly $J$-clean if and only if $A\in J\left(M_2\left(R\right)\right)$, or $I_2-A\in J\left(M_2\left(R\right)\right)$, or $A$ is similar to $\left(\begin{array}{cc}0& \lambda \\ 1& \mu \end{array}\right)$, where $\lambda \in J\left(R\right)$, $\mu \in 1+J\left(R\right)$, and the equation $x^2-x\mu -\lambda =0$ has a root in $J\left(R\right)$ and a root in $1+J\left(R\right)$. We further prove that $f\left(x\right)\in R\left[\right[x\left]\right]$ is strongly $J$-clean if $f\left(0\right)\in R$ be optimally $J$-clean.

A ring $R$ is (weakly) nil clean provided that every element in $R$ is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let $R$ be abelian, and let $n\in \mathbb{N}$. We prove that $M_n\left(R\right)$ is nil clean if and only if $R/J\left(R\right)$ is Boolean and $M_n\left(J\left(R\right)\right)$ is nil. Furthermore, we prove that $R$ is weakly nil clean if and only if $R$ is periodic; $R/J\left(R\right)$ is ${\mathbb{Z}}_3$, $B$ or ${\mathbb{Z}}_3\oplus B$ where $B$ is a Boolean ring, and that $M_n\left(R\right)$ is weakly nil clean if and only if $M_n\left(R\right)$ is nil clean for all $n\ge 2$.

A matrix $A\in M_n\left(R\right)$ is $e$-clean provided there exists an idempotent $E\in M_n\left(R\right)$ such that $A-E\in {\mathrm{GL}}_n\left(R\right)$ and $detE=e$. We get a general criterion of $e$-cleanness for the matrix $\left[[a_1,a_2,\cdots ,a_{n+1}]\right]$. Under the $n$-stable range condition, it is shown that $\left[[a_1,a_2,\cdots ,a_{n+1}]\right]$ is $0$-clean iff $(a_1,a_2,\cdots ,a_{n+1})=1$. As an application, we prove that the $0$-cleanness and unit-regularity for such $n\times n$ matrix over a Dedekind domain coincide for all $n\ge 3$. The analogous for $(s,2)$ property is also obtained.

Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\le 2$. We construct a triangulated category ${𝒞}_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$. When ${𝒞}_A$ is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also...

We define a graded twisted-coassociative coproduct on the tensor algebra the desuspension space of a graded vector space $V$. The coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences of multilinear maps on $V$ (resp. graded Loday structures on $V$, sequences that we call Loday infinity structures on $V$). We prove a minimal model theorem for Loday infinity algebras and observe that the ${\text{Lod}}_{\infty }$ category contains the ${\text{L}}_{\infty }$ category as...

Nous calculons la cohomologie de Hochschild directement sur les graphes de Kontsevich. Celle-ci est localisée sur les graphes totalement antisymétriques ayant autant de pieds que de pattes. La considération de cette cohomologie permet de réinterpréter l’équation de formalité pour l’espace ${ℝ}^d$.

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.