Narrow Results

Frobenius algebra is formulated within the Abelian monoidal category of operad of graphs. A not necessarily associative algebra $\mathsf{Y}$ is said to be a Frobenius algebra if there exists a $\mathsf{Y}$-module isomorphism. A new concept of a solvable Frobenius algebra is introduced: an algebra $\mathsf{Y}$ is said to be a solvable Frobenius algebra if it possesses a nonzero one-sided $\mathsf{Y}$-module morphism with nontrivial radical. In the category of operad of graphs, we can express the necessary and sufficient conditions for an algebra to be a solvable Frobenius algebra. The notion of a solvable Frobenius algebra makes it possible to find all commutative nonassociative Frobenius algebras (Conjecture 10.1), and to find all Frobenius structures for commutative associative Frobenius algebras. Frobenius algebra allows $S_2$-permuted opposite algebra to be extended to $S_3$-permuted algebras.

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e. a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal 𝔪 in a commutative ring R, we show how solving (in)homogeneous linear systems over R_𝔪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R_𝔪. As a corollary, we obtain the computability of the category of finitely presented R_𝔪-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a byproduct, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings, we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal $𝔞$ of a Noetherian ring $R$. It is shown that an $R$-module $M$ is cofinite with respect to $𝔞$, if and only if, ${Ext}_R^i(R/𝔞,M)$ is finitely generated for all $i\le \mathrm{\text{cd}}(𝔞,M)+1$, whenever $dimR/𝔞=1$. In addition, we show that if $M$ is finitely generated and $H_{𝔞}^i(M)$ are weakly Laskerian for all $i\le t-1$, then $H_{𝔞}^i(M)$ are $𝔞$-cofinite for all $i\le t-1$ and for any minimax submodule $K$ of $H_{𝔞}^t(M)$, the $R$-modules ${Hom}_R(R/𝔞,H_{𝔞}^t(M)/K)$ and ${Ext}_R^1(R/𝔞,H_{𝔞}^t(M)/K)$ are finitely generated, where $t$ is a non-negative integer. Finally, we explore a criterion for weakly cofiniteness of modules with respect to an ideal of dimension one. Namely, for such ideals it suffices that the two first $Ext$-modules in the definition for weakly cofiniteness are weakly Laskerian. As an application of this result, we deduce that the category of all $𝔞$-weakly cofinite modules over $R$ forms a full Abelian subcategory of the category of modules.

Let $R$ be a commutative noetherian ring, and ${𝒵}$ a stable under specialization subset of $Spec(R)$. We introduce a notion of ${𝒵}$-cofiniteness and study its main properties. In the case $dim({𝒵})\le 1$, or $dim(R)\le 2$, or $R$ is semilocal with $cd({𝒵},R)\le 1$, we show that the category of ${𝒵}$-cofinite $R$-modules is abelian. Also, in each of these cases, we prove that the local cohomology module $H_{{𝒵}}^i(X)$ is ${𝒵}$-cofinite for every homologically left-bounded $R$-complex $X$ whose homology modules are finitely generated and every $i\in \mathbb{Z}$.

We establish some properties involving regular morphisms in abelian categories. We show a decomposition theorem on the image of a regular sum of morphisms, a characterization of regular morphisms in terms of consecutive pairs of morphisms, and a description of certain equivalent morphisms. We also generalize Ehrlich’s Theorem on one-sided unit regular morphisms by showing that if $N$ is an $M$-regular object, then a morphism $f:M\to N$ is left (right) unit regular if and only if there exists a split monomorphism (epimorphism) $\mathrm{\text{Ker}}(f)\to \mathrm{\text{Coker}}(f)$. We also study regular morphisms determined by generalized inverses in additive categories.

We introduce and study (strongly) $\pi$-Rickart objects and their duals in abelian categories, which generalize (strongly) self-Rickart objects and their duals. We establish general properties of such objects, we analyze their behavior with respect to coproducts, and we study classes all of whose objects are (strongly) $\pi$-Rickart. We derive consequences for module and comodule categories.

Let $(R,𝔪)$ be a commutative Noetherian complete local ring and $I$ be a proper ideal of $R$. Suppose that $M$ is a nonzero $I$-cofinite $R$-module of Krull dimension $n$. In this paper, it shown that $dimR/(I+{Ann}_R{H}_{𝔪}^n(M))=n.$ As an application of this result, it is shown that $dimR/(I+𝔭)=n$, for each $𝔭\in {Att}_R{H}_{𝔪}^n(M).$ Also it shown that for each $j\ge 0$ the submodule ${\mathrm{\Sigma }}_j(M):=\cup \{K:K\le M$ and $dimK\le j\}$ of $M$ is $I$-cofinite, $dimR/(I+{Ann}_R{H}_{𝔪}^j(M/{\mathrm{\Sigma }}_j(M)))<j$ and $dimR/(I+{Ann}_R{H}_{𝔪}^j(M))\le j,$ whenever the category of all $I$-cofinite $R$-modules is an Abelian subcategory of the category of all $R$-modules. Also some applications of these results will be included.

Let ${𝒵}$ be a specialization closed subset of $\mathrm{\text{Spec}}R$ and ${𝒮}$ be a Serre subcategory of $ModR$. As a generalization of the notion of cofiniteness, we introduce the concept of ${𝒵}$-coserreness with respect to ${𝒮}$ (see Definition 4.1). First, as a main result, for some special Serre subcategories ${𝒮}$, we show that an $R$-module $M$ with ${dim}_{R}M\le 1$ is ${𝒵}$-coserre with respect to ${𝒮}$ if and only if ${Ext}_R^{0,1}(R/𝔞,M)\in {𝒮}$ for all ideals $𝔞\in F({𝒵})$. Indeed, this result provides a partial answer to a question that was recently raised in [K. Bahmanpour, R. Naghipour and M. Sedghi, Modules cofinite and weakly cofinite with respect to an ideal, J. Algebra Appl. 16 (2018) 1–17]. As an application of this result, we show that the category of ${𝒵}$-coserre $R$-modules $M$ with ${dim}_{R}M\le 1$ is a full Abelian subcategory of $ModR$. Also, for every homologically bounded $R$-complex $X$ whose homology modules belong to ${𝒮}$ we show that the local cohomology modules ${\mathrm{\text{H}}}_{{𝒵}}^i(X)$ for all $i$, are ${𝒵}$-coserre in all the cases where ${dim}_R{𝒵}\le 1$, ${dim}_{R}X\le 2-supX$ and $cd({𝒵},X)\le 1-supX$.

We investigate relative CS-Baer objects in abelian categories in relationship with other relevant classes of objects such as relative Baer objects, extending objects, objects having certain summand intersection properties and relative CS-Rickart objects. Dual results are automatically obtained by applying the duality principle in abelian categories. We also study direct sums of relative CS-Baer objects, and we determine the complete structure of dual self-CS-Baer modules over Dedekind domains. Further applications are given to module categories.

Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$ such that the $R$-modules $H_I^i(M)$ are $I$-cofinite, for all finitely generated $R$-modules $M$ and all $i\in {\mathbb{N}}_0$. In this paper, we investigate a question of Hartshorne concerning the Abelianness of the category of all $I$-cofinite modules.

Let ${𝒜}$ be an abelian category and $({𝒳},{𝒴})$ a pair of classes of objects in ${𝒜}$. Inspired by Bouchiba’s work on generalized Gorenstein projective modules, we give a new way of measuring $({𝒳},{𝒴})$-Gorenstein projective dimension by defining a complete $n$-$({𝒳},{𝒴})$ resolution. Then we relate the relative global Gorenstein homological dimension to the invariants silp(${𝒜}$) and spli(${𝒜}$) under some conditions. Furthermore, we prove that, in the setting of a left and right coherent ring $R$, the supremum of Ding projective dimensions of all finitely presented (left or right) $R$-modules and the (left or right) Gorenstein weak global dimension are identical, generalizing a theorem of Ding, Li and Mao.

Let 𝒜 be an abelian category and $M \in \mathcal{A}$. Then M is called a $\left(D_4\right)$-object if, whenever A and B are subobjects of M with $M = A \oplus B$ and $f:A\to B$ is an epimorphism, $\mathrm{Ker}f$ is a direct summand of A. In this paper we give several equivalent conditions of $(D_4)$-objects in an abelian category. Among other results, we prove that any object M in an abelian category 𝒜 is $\left(D_4\right)$ if and only if for every subobject K of M such that K is the intersection $K_1\cap K_2$ of perspective direct summands $K_1$ and $K_2$ of M with $M = K_1 + K_2$, every morphismr $\phi : M \to M/K$ can be lifted to an endomorphism $\theta :M\to M$ in $En{d}_{\mathcal{A}}\left(M\right)$.

In this paper, we prove a reduction result on wide subcategories of abelian categories which is similar to the Calabi-Yau reduction, silting reduction and $\tau$-tilting reduction. More precisely, if an abelian category $\mathcal{A}$ admits a recollement relative to abelian categories $\mathcal{A}\prime$ and $\mathcal{A}\prime \prime$, which is denoted by $(\mathcal{A}\prime ,\mathcal{A},\mathcal{A}\prime \prime ,i^{\ast },i_{\ast },i^{!},j_{!},j^{\ast },j_{\ast })$, then the assignment $\mathcal{C}\mapsto j^{\ast }(\mathcal{C})$ defines a bijection between wide subcategories in $\mathcal{A}$ containing $i_{\ast }(\mathcal{A}\prime )$ and wide subcategories in $\mathcal{A}\prime \prime$. Moreover, a wide subcategory $\mathcal{C}$ of $\mathcal{A}$ containing $i_{\ast }(\mathcal{A}\prime )$ admits a new recollement relative to $\mathcal{A}\prime$ and $j^{\ast }(\mathcal{C})$ which is induced from the original recollement.

We consider stable representations of non-Dynkin quivers with respect to a central charge. These attract a lot of interest in mathematics and physics since they can be identified with so-called BPS states. Another motivation is the work of Dimitrov et al. on the phases of stable representations of the generalized Kronecker quiver. One aim is to explain for general Euclidean and wild quivers the behavior of phases of stable representations well known in some examples. In addition, we study especially the behavior of preinjective, postprojective and regular indecomposable modules. We show that the existence of a stable representation with self-extensions implies the existence of infinitely many stables without self-extensions for rigid central charges. In this case the phases of the stable representations approach one or two limit points. In particular, the phases are not dense in two arcs. The category of representations of acyclic quivers is a special case of rigid Abelian categories which show this behavior for rigid central charges.

It is well known that the Baer criterion for injectivity of $R$-modules, for a ring $R$ with unit, is not true in a general category, even in a general abelian category. In this paper, we prove some results analogous to the Baer criterion for injectivity in abelian categories and Grothendieck categories. In particular, we generalize the known fact that $G$-injectivity is the same as injectivity, if $G$ is a generator in a Grothendieck category. Furthermore, some Baer type theorems for general abelian categories are proved. Finally, equivalent conditions to satisfying a classical kind of Baer criterion are found in (locally presentable) abelian categories.