Let $U$ be a dg-$A$-module, $B$ the endomorphism dg-algebra of $U$. We know that if $U$ is a good silting object, then there exist a dg-algebra $C$ and a recollement among the derived categories $𝐃(C,d)$ of $C$, $𝐃(B,d)$ of $B$ and $𝐃(A,d)$ of $A$. We investigate the condition under which the induced dg-algebra $C$ is weak nonpositive. In order to deal with both silting and cosilting dg-modules consistently, the notion of weak silting dg-modules is introduced. Thus, similar results for good cosilting dg-modules are obtained....

Let $I$ be an ideal of a commutative Noetherian ring $R$. It is shown that the $R$-modules $H_I^j\left(M\right)$ are $I$-cofinite for all finitely generated $R$-modules $M$ and all $j\in {\mathbb{N}}_0$ if and only if the $R$-modules ${\mathrm{Ext}}_R^i(N,H_I^j\left(M\right))$ and ${\mathrm{Tor}}_i^R(N,H_I^j\left(M\right))$ are $I$-cofinite for all finitely generated $R$-modules $M$, $N$ and all integers $i,j\in {\mathbb{N}}_0$.

Let $p$ be a prime number. This paper introduces the Roquette category ${ℛ}_p$ of finite $p$-groups, which is an additive tensor category containing all finite $p$-groups among its objects. In ${ℛ}_p$, every finite $p$-group $P$ admits a canonical direct summand $\partial P$, called the edge of $P$. Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$-groups, and the tensor structure of ${ℛ}_p$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors...

Zhou and Zhu have shown that if $𝒞$ is an $(n+2)$-angulated category and $𝒳$ is a cluster tilting subcategory of $𝒞$, then the quotient category $𝒞/𝒳$ is an $n$-abelian category. We show that if $𝒞$ has Auslander-Reiten $(n+2)$-angles, then $𝒞/𝒳$ has Auslander-Reiten $n$-exact sequences.

In this paper we introduce the concept of $\tau$-extending modules by $\tau$-rational submodules and study some properties of such modules. It is shown that the set of all $\tau$-rational left ideals of ${}_{R}R$ is a Gabriel filter. An $R$-module $M$ is called $\tau$-extending if every submodule of $M$ is $\tau$-rational in a direct summand of $M$. It is proved that $M$ is $\tau$-extending if and only if $M=Re{j}_{M}E(R/\tau \left(R\right))\oplus N$, such that $N$ is a $\tau$-extending submodule of $M$. An example is given to show that the direct sum of $\tau$-extending modules need not...

Let $𝒳$ be a semibrick in an extriangulated category. If $𝒳$ is a $\tau$-semibrick, then the Auslander-Reiten quiver $\Gamma \left(ℱ\right(𝒳\left)\right)$ of the filtration subcategory $ℱ\left(𝒳\right)$ generated by $𝒳$ is $\mathbb{Z}{𝔸}_{\infty }$. If $𝒳={\left\{X_i\right\}}_{i=1}^t$ is a $\tau$-cycle semibrick, then $\Gamma \left(ℱ\right(𝒳\left)\right)$ is $\mathbb{Z}{𝔸}_{\infty }/{\tau }_{𝔸}^t$.

If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if ${𝒟}_X^{\left(\infty \right)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of ${𝒟}_X^{\left(\infty \right)}$ on an ${𝒪}_X$-module $ℰ$ is equivalent to giving an infinite sequence of ${𝒪}_X$-modules descending $ℰ$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f:X\to S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it...

We study equivalences for category ${𝒪}_p$ of the rational Cherednik algebras ${𝐇}_p$ of type $G_{\ell }\left(n\right)={\left({\mu }_{\ell }\right)}^n⋊{𝔖}_n$: a highest weight equivalence between ${𝒪}_p$ and ${𝒪}_{\sigma \left(p\right)}$ for $\sigma \in {𝔖}_{\ell }$ and an action of ${𝔖}_{\ell }$ on an explicit non-empty Zariski open set of parameters $p$; a derived equivalence between ${𝒪}_p$ and ${𝒪}_{p^{\text{'}}}$ whenever $p$ and $p^{\text{'}}$ have integral difference; a highest weight equivalence between ${𝒪}_p$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$. As a consequence, we confirm special cases...

We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $𝒯$ is of the form $\mathbb{Z}\Delta /G$ where $\Delta$ is a disjoint union of simply-laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $\mathbb{Z}\Delta$. Then we prove that for ‘most’ groups $G$, the category $𝒯$ is standard, ...

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$. Let $t\in {\mathbb{N}}_0$ be an integer and $M$ an $R$-module such that ${\mathrm{Ext}}_R^i(R/I,M)$ is minimax for all $i\le t+1$. We prove that if $H_I^i\left(M\right)$ is ${\mathrm{FD}}_{\le 1}$ (or weakly Laskerian) for all $i<t$, then the $R$-modules $H_I^i\left(M\right)$ are $I$-cominimax for all $i<t$ and ${\mathrm{Ext}}_R^i(R/I,H_I^t\left(M\right))$ is minimax for $i=0,1$. Let $N$ be a finitely generated $R$-module. We prove that ${\mathrm{Ext}}_R^j(N,H_I^i\left(M\right))$ and ${\mathrm{Tor}}_j^R(N,H_I^i\left(M\right))$ are $I$-cominimax for all $i$ and $j$ whenever $M$ is minimax and $H_I^i\left(M\right)$ is ${\mathrm{FD}}_{\le 1}$ (or weakly Laskerian) for all $i$.

Let $R$ be a Noetherian ring, and $I$ and $J$ be two ideals of $R$. Let $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$ and $M$ be a $ZD$-module. As a generalization of the $S$-$\mathrm{depth}(I,M)$ and $\mathrm{depth}(I,J,M)$, the $S$-$\mathrm{depth}$ of $(I,J)$ on $M$ is defined as $S$-$\mathrm{depth}(I,J,M)=inf\{S$-$\mathrm{depth}(𝔞,M):𝔞\in \tilde{\mathrm{W}}(I,J)\}$, and some properties of this concept are investigated. The relations between $S$-$\mathrm{depth}(I,J,M)$ and $H_{I,J}^i\left(M\right)$ are studied, and it is proved that $S$-$\mathrm{depth}(I,J,M)=inf\{i:H_{I,J}^i\left(M\right)\notin S\}$, where $S$ is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology...

For any positive integer $n$, let $A_n$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories ${ℳ}_{n+1}$ and ${ℱ}_n$, where ${ℳ}_{n+1}$ and ${ℱ}_n$ are the two extriangulated categories corresponding to the representation category of $A_{n+1}$ and the morphism category of projective representations of $A_n$, respectively. As a...

CONTENTSIntroduction................................................................................................................................................................................5§ 1. Notation and preliminaries.............................................................................................................................................6§ 2. Epimorphisms and monomorphisms.........................................................................................................................7§...

Let Λ be an artin algebra. We prove that for each sequence ${\left(h_i\right)}_{i\in \mathbb{Z}}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X{\in }^b\left(\Lambda \right)$, the bounded derived category of Λ, with $lengt{h}_{E\left(X\right)}{H}^i\left(X\right)=h_i$ for all i ∈ ℤ and E(X) the endomorphism ring of X in ${}^b\left(\Lambda \right)$ if and only if ${}^b\left(Mod\Lambda \right)$, the bounded derived category of the category $Mod\Lambda$ of all left Λ-modules, has no generic objects in the sense of [4].