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We introduce (dual) strongly relative CS-Rickart objects in abelian categories, as common generalizations of (dual) strongly relative Rickart objects and strongly extending (lifting) objects. We gi...

We study the transfer of D4 and dually C4 objects via functors between abelian categories. We also give applications to Grothendieck categories, comodule categories, (graded) module categories.

In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category $\mathcal {A}$ from the point of view of lattice theory. Motivated by τ-tilting reduction of Jasso, we mainly focus on intervals $[\mathcal {U},\mathcal {T}]$ in the lattice $\operatorname {\mathsf {tors}} \mathcal {A}$ of torsion classes in $\mathcal {A}$ such that $\mathcal {W}:=\mathcal {U}^{\perp } \cap \mathcal {T}$ is a wide subcategory of $\mathcal {A}$ ; we call these intervals wide intervals. We prove that a wide interval $[\mathcal {U},\mathcal {T}]$ is isomorphic to the lattice $\operatorname {\mathsf {tors}} \mathcal {W}$ of torsion classes in the abelian category $\mathcal {W}$ . We also characterize wide intervals in two ways: First, in purely lattice theoretic terms based on the brick labeling established by Demonet–Iyama–Reading–Reiten–Thomas; and second, in terms of the Ingalls–Thomas correspondences between torsion classes and wide subcategories, which were further developed by Marks–Sťovicek.

We introduce and study (dual) relative F-Baer objects as specializations of (dual) relative split objects with respect to a fully invariant short exact sequence in AB3* (AB3) abelian categories. We...

Let R be a commutative Noetherian ring, $${\mathfrak {a}}$$ an ideal of R, and M an R-module. We prove that the category of $${\mathfrak {a}}$$ -weakly cofinite modules is a Melkersson subcategory of R-modules whenever $$\dim R\le 1$$ and is an Abelian subcategory whenever $$\dim R\le 2$$ . We also prove that if $$(R,{\mathfrak {m}})$$ is a local ring with $$\dim R/{\mathfrak {a}}\le 2$$ and $${{\,\mathrm{Supp}\,}}_R(M)\subseteq {{\,\mathrm{V}\,}}({\mathfrak {a}})$$ , then M is $${\mathfrak {a}}$$ -weakly cofinite if (and only if) $${{\,\mathrm{Hom}\,}}_R(R/{\mathfrak {a}}, M)$$ , $${{\,\mathrm{Ext}\,}}_{R}^{1}(R/{\mathfrak {a}}, M)$$ and $${{\,\mathrm{Ext}\,}}_{R}^{2}(R/{\mathfrak {a}}, M)$$ are weakly Laskerian. In addition, we prove that if $$(R,{\mathfrak {m}})$$ is a local ring with $$\dim R/{\mathfrak {a}}\le 2$$ and $$n\in \mathbb {N}_0$$ , such that $${{{\,\mathrm{Ext}\,}}}^{i}_R(R/{\mathfrak {a}},M)$$ is weakly Laskerian for all i, then $${{\,\mathrm{H}\,}}_{{\mathfrak {a}}}^{i}(M)$$ is $${\mathfrak {a}}$$ -weakly cofinite for all i if (and only if) $${{\,\mathrm{Hom}\,}}_R(R/{\mathfrak {a}}, {{\,\mathrm{H}\,}}_{{\mathfrak {a}}}^{i}(M))$$ is weakly Laskerian for all i.

We study the transfer via functors between abelian categories of the (dual) relative splitness of objects with respect to a fully invariant short exact sequence. We mainly consider fully faithful f...

Consider the abelian category ${\mathcal C}$ of commutative group schemes of finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp. ${\rm Pro}({\mathcal F})$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group $\varpi_1 : {\rm Pro}({\mathcal C}) \to {\rm Pro}({\mathcal F})$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors $\varpi_i$ vanish for $i \geq 2$. Along the way, we describe the indecomposable projective objects of ${\rm Pro}({\mathcal C})$ over an arbitrary field $k$., Final version, accepted for publication at Annales Henri Lebesgue

For a triangulated category T, if C is a cluster-tilting subcategory of T, then the quotient category T\C is an abelian category. Under certain conditions, the converse also holds. This is an very important result of cluster-tilting theory, due to Koenig-Zhu and Beligiannis. Now let B be a suitable extriangulated category, which is a simultaneous generalization of triangulated categories and exact categories. We introduce the notion of pre-cluster tilting subcategory C of B, which is a generalization of cluster tilting subcategory. We show that C is cluster tilting if and only if B/C is abelian., 17 pages

This paper focuses on the Ki-groups of two types of extensions of abelian categories, which are the trivial extension and the gluing of abelian categories. We prove that, under some conditions, Ki-groups of a certian subcategory of the trivial extension category is isomorphic to Ki-groups of the similar subcategory of the original category. Moreover, under some conditions, we show that the Ki-groups of a left (right) gluing of two abelian categories are isomorphic to the direct sum of Ki-groups of two abelian categories. As their applications, we obtain some results of the Ki-groups of the trivial extension of a ring by a bimodule (i∈N).

We define the notion of an infinitely generated tilting object of infinite homological dimension in an abelian category. A one-to-one correspondence between $\infty$-tilting objects in complete, cocomplete abelian categories with an injective cogenerator and $\infty$-cotilting objects in complete, cocomplete abelian categories with a projective generator is constructed. We also introduce $\infty$-tilting pairs, consisting of an $\infty$-tilting object and its $\infty$-tilting class, and obtain a bijective correspondence between $\infty$-tilting and $\infty$-cotilting pairs. Finally, we discuss the related derived equivalences and t-structures.

A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.

Let R be a commutative Noetherian ring and I be an ideal of R. In this article, we answer affirmatively a question raised by the present author. Also, as a consequence, it is shown that the categor...

Let $\C$ be an $(n+2)$-angulated category with shift functor $\Sigma$ and $\X$ be a cluster-tilting subcategory of $\C$. Then we show that the quotient category $\C/\X$ is an $n$-abelian category. If $\C$ has a Serre functor, then $\C/\X$ is equivalent to an $n$-cluster tilting subcategory of an abelian category $\textrm{mod}(\Sigma^{-1}\X)$. Moreover, we also prove that $\textrm{mod}(\Sigma^{-1}\X)$ is Gorenstein of Gorenstein dimension at most $n$. As an application, we generalize recent results of Jacobsen-J{\o}rgensen and Koenig-Zhu., Comment: 14 pages

The Frobenius-Perron dimension for an abelian category was recently introduced. We apply this theory to the category of representations of the finite-dimensional radical squared zero algebras associated to certain modified ADE graphs. In particular, we take an ADE quiver with arrows in a certain orientation and an arbitrary number of loops at each vertex. We show that the Frobenius-Perron dimension of this category is equal to the maximum number of loops at a vertex. Along the way, we introduce a result which can be applied in general to calculate the Frobenius-Perron dimension of a radical square zero bound quiver algebra. We use this result to introduce a family of abelian categories which produce arbitrarily large irrational Frobenius-Perron dimensions., Comment: typos corrected

Let $R\to U$ be an associative ring epimorphism such that $U$ is a flat left $R$-module. Assume that the related Gabriel topology $\mathbb G$ of right ideals in $R$ has a countable base. Then we show that the left $R$-module $U$ has projective dimension at most $1$. Furthermore, the abelian category of left contramodules over the completion of $R$ at $\mathbb G$ fully faithfully embeds into the Geigle-Lenzing right perpendicular subcategory to $U$ in the category of left $R$-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an assocative ring $R$, we consider the induced topology on every left $R$-module, and for a perfect Gabriel topology $\mathbb G$ compare the completion of a module with an appropriate Ext module. Finally, we characterize the $U$-strongly flat left $R$-modules by the two conditions of left positive-degree Ext-orthogonality to all left $U$-modules and all $\mathbb G$-separated $\mathbb G$-complete left $R$-modules., Comment: LaTeX 2e with pb-diagram and xy-pic, 64 pages, 6 commutative diagrams + Corrigenda, LaTeX 2e with ulem.sty, 10 pages; v.6: corrigenda added (two mistakes, one in Remark 3.3 and the other one in Section 5); v.7: third section added to corrigenda (confusion in Remark 11.3); v.8: fourth section added to corrigenda (about an unjustified assertion in the preliminaries), main results unaffected

Let $${\mathcal {A}}$$ be an abelian category, and $${\mathcal {X}}$$ and $${\mathcal {Y}}$$ subcategories. We derive in this paper an additive functor $$T(-, -)$$ using proper $${\mathcal {X}}$$ -resolutions (respectively, proper $${\mathcal {Y}}$$ -coresolutions). Under certain conditions, we establish balance results for such relative homology (respectively, cohomology) over $${\mathcal {A}}$$ . Our main theorem simultaneously recovers theorems of Sather-Wagstaff et al. (J Math Kyoto Univ 48(3):571–596, 2008), Enochs et al. (Relative homological algebra. De Gruyter, Berlin 2000) and Di et al. (Bull Korean Math Soc 52(1):137–147, 2015) as special cases.

For a given hereditary abelian category satisfying some finiteness conditions, in certain twisted cases it is shown that the modified Ringel-Hall algebra is isomorphic to the naive lattice algebra and there exists an epimorphism from the modified Ringel-Hall algebra to the lattice algebra. Furthermore, the kernel of this epimorphism is described explicitly. Finally, we show that the naive lattice algebra is invariant under the derived equivalences of hereditary abelian categories.

In this paper we shall investigate the concepts of cofiniteness of local cohomology modules and Abelian categories of cofinite modules over arbitrary Noetherian rings. Then we shall improve some of the results given in the literature., Comment: 24 pages

We study the transfer of (dual) relative CS-Rickart properties via functors between abelian categories. We consider fully faithful functors as well as adjoint pairs of functors. We give several applications to Grothendieck categories and, in particular, to (graded) module and comodule categories., 14 pages

We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor $\Sigma$ on a finite abelian category $\mathcal{M}$, we introduce the notion of a $\Sigma$-twisted trace on the class $\mathrm{Proj}(\mathcal{M})$ of projective objects of $\mathcal{M}$. In our framework, there is a one-to-one correspondence between the set of $\Sigma$-twisted traces on $\mathrm{Proj}(\mathcal{M})$ and the set of natural transformations from $\Sigma$ to the Nakayama functor of $\mathcal{M}$. Non-degeneracy and compatibility with the module structure (when $\mathcal{M}$ is a module category over a finite tensor category) of a $\Sigma$-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular., Comment: 39 pages; to appear in Algebras and Representation Theory

In the last few years, Lopez-Permouth and several collaborators have introduced a new approach in the study of the classical projectivity, injectivity and flatness of modules. This way, they introduced subprojectivity domains of modules as a tool to measure, somehow, the projectivity level of such a module (so not just to determine whether or not the module is projective). In this paper we develop a new treatment of the subprojectivity in any abelian category which shed more light on some of its various important aspects. Namely, in terms of subprojectivity, some classical results are unified and some classical rings are characterized. It is also shown that, in some categories, the subprojectivity measures notions other than the projectivity. Furthermore, this new approach allows, in addition to establishing nice generalizations of known results, to construct various new examples such as the subprojectivity domain of the class of Gorenstein projective objects, the class of semi-projective complexes and particular types of representations of a finite linear quiver. The paper ends with a study showing that the fact that a subprojectivity domain of a class coincides with its first right Ext-orthogonal class can be characterized in terms of the existence of preenvelopes and precovers.

For a locally presentable abelian category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived category $\mathsf D(\mathsf B)$ is generated, as a triangulated category with coproducts, by the projective generator of $\mathsf B$. For a Grothendieck abelian category $\mathsf A$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's principle, we prove that the derived category $\mathsf D(\mathsf A)$ is generated, as a triangulated category with products, by the injective cogenerator of $\mathsf A$. More generally, we define the notion of an exact category with an object size function and prove that the derived category of any such exact category with exact $\kappa$-directed colimits of chains of admissible monomorphisms has Hom sets. In particular, the derived category of any locally presentable abelian category has Hom sets., Comment: LaTeX 2e with xy-pic; 50 pages, 5 commutative diagrams; v.2: Remarks 6.4 and 9.2 inserted, Introduction expanded, many references added; v.3: several misprints corrected

This is the first chapter which develops the basic language of homological algebra. It introduces categories, abelian categories, and the homology theory in an abelian category. Further, it also introduces the bi-functor \(EXT (-, -)\).

An endomorphism $f$ of an Abelian group $A$ is said to be inessential (in the category of Abelian groups) if it can be extended to an endomorphism of any Abelian group which contains $A$ as a subgroup. In this paper we show that $f$ is as above if and only if $(f-v\operatorname{id}_A)(A)$ is contained in the maximal divisible subgroup of $A$ for some $v \in \mathbb{Z}$.

In the paper "Deformation theory of abelian categories", the last two authors proved that an abelian category with enough injectives can be reconstructed as the category of finitely presented modules over the category of its injective objects. We show a generalization of this to pretriangulated dg-categories with a left bounded non-degenerate t-structure with enough derived injectives, the latter being derived enhancements of the injective objects in the heart of the t-structure. Such dg-categories (with an additional hypothesis of closure under suitable products) can be completely described in terms of left bounded twisted complexes of their derived injectives., 51 pages; postprint version

We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find intrinsic axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting subcategories, and cluster tilting subcategories of abelian categories. As a consequence we prove that any $d$-abelian category is equivalent to a $d$-cluster tilting subcategory of an abelian category, without any assumption on the categories being projectively generated., Comment: 29 pages. Accepted for publication in Journal of Pure and Applied Algebra

We introduce and study (strongly) [Formula: see text]-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) [Formula: see text]-Rickart. We derive consequences for module and comodule categories.

Let $\mathfrak{a}$ be an ideal of a commutative noetherian (not necessarily local) ring $R$. In the case $\cd(\mathfrak{a},R)\leq 1$, we show that the subcategory of $\mathfrak{a}$-cofinite $R$-modules is abelian. Using this and the technique of way-out functors, we show that if $\cd(\mathfrak{a},R)\leq 1$, or $\dim(R/\mathfrak{a}) \leq 1$, or $\dim(R) \leq 2$, then the local cohomology module $H^{i}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$-cofinite for every $R$-complex $X$ with finitely generated homology modules and every $i \in \mathbb{Z}$. We further answer Question 1.3 in the three aforementioned cases, and reveal a correlation between Questions 1.1, 1.2, and 1.3., Comment: It will appear in Kyoto Journal of Mathematics

Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of $n$-exact categories in the sense of Jasso and $(n+2)$-angulated in the sense of Geiss-Keller-Oppermann. Let $\mathscr C$ be an $n$-exangulated category with enough projectives and enough injectives, and $\mathscr X$ a cluster tilting subcategory of $\mathscr C$. In this article, we show that the quotient category $\mathscr C/\mathscr X$ is an $n$-abelian category. This extends a result of Zhou-Zhu for $(n+2)$-angulated categories. Moreover, it highlights new phenomena when it is applied to $n$-exact categories., 18 pages. arXiv admin note: text overlap with arXiv:1909.13284 and arXiv:1807.06733

For a length abelian category, we show that all torsion-free classes can be classified by using only the information on bricks, including non functorially-finite ones. The idea is to consider the set of simple objects in a torsion-free class, which has the following property: it is a set of bricks where every non-zero map between them is an injection. We call such a set a monobrick. In this paper, we provide a uniform method to study torsion-free classes and wide subcategories via monobricks. We show that monobricks are in bijection with left Schur subcategories, which contains all subcategories closed under extensions, kernels and images, thus unifies torsion-free classes and wide subcategories. Then we show that torsion-free classes bijectively correspond to cofinally closed monobricks. Using monobricks, we deduce several known results on torsion(-free) classes and wide subcategories (e.g. finiteness result and bijections) in length abelian categories, without using $\tau$-tilting theory. For Nakayama algebras, left Schur subcategories are the same as subcategories closed under extensions, kernels and images, and we show that its number is related to the large Schr\"oder number., Comment: 28 pages, final version. a minor correction. to appear in Adv. Math

Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of absolutely pure group valued functors over $\mathcal{M}$, denote by $\mathcal{L}_2(\mathcal{M},\mathcal{G})$, is an abelian category and $\mathcal{M}$ is equivalent to a full subcategory of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ in such a way that $n$-kernels and $n$-cokernels are precisely exact sequences of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ with terms in $\mathcal{M}$. This gives a higher-dimensional version of the Freyd-Mitchell embedding theorem for $n$-abelian categories.

We give a classification of torsion classes (or nullity classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda’s clas...

Given two complete hereditary cotorsion pairs ( Q , R ) and ( Q ′ , R ′ ) in a bicomplete abelian category G such that Q ′ ⊆ Q and Q ∩ R = Q ′ ∩ R ′ , Becker showed that there exists a hereditary abelian model structure M = ( Q , W , R ′ ) on G , where W is a thick subcategory of G . We prove that the homotopy category Ho( M ) of M is triangulated equivalent to the triangulated quotient category D b ( G ) [ Q , R ′ ] ˆ / K b ( Q ′ ∩ R ′ ) , where D b ( G ) [ Q , R ′ ] ˆ is the subcategory of D b ( G ) consisting of all homology bounded complexes with both finite Q dimension and R ′ dimension and K b ( Q ′ ∩ R ′ ) is the bounded homotopy category of Q ′ ∩ R ′ (core) objects. Applications are given in the category of modules. It is shown that the homotopy category of the Gorenstein flat (resp., Ding projective and Gorenstein AC-projective) model structure on the category of modules established by Gillespie and his coauthors can be realized as a certain triangulated quotient category.

We introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories. Our theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories., 17 pages. arXiv admin note: text overlap with arXiv:1709.05872, arXiv:1803.02683

Alexander L. Rosenberg has constructed a spectrum for abelian categories which is able to reconstruct a quasi-separated scheme from its category of quasi-coherent sheaves. In this note we present a detailed proof of this result which is due to Ofer Gabber. Moreover, we determine the automorphism class group of the category of quasi-coherent sheaves.

We introduce and investigate weak relative Rickart objects and dual weak relative Rickart objects in abelian categories. Several types of abelian categories are characterized in terms of (dual) wea...

Given a cotorsion pair ([Formula: see text], [Formula: see text]) in an abelian category [Formula: see text] , we define cotorsion pairs ([Formula: see text], dg[Formula: see text]) and (dg[Formula: see text], [Formula: see text]) in the category [Formula: see text]N([Formula: see text]) of N-complexes on [Formula: see text]. We prove that if the cotorsion pair ([Formula: see text], [Formula: see text]) is complete and hereditary in a bicomplete abelian category, then both of the induced cotorsion pairs are complete, compatible and hereditary. We also create complete cotorsion pairs (dw[Formula: see text], (dw[Formula: see text])⊥), (ex[Formula: see text], (ex[Formula: see text])⊥) and (⊥(dw[Formula: see text]), dw[Formula: see text]), (⊥(ex[Formula: see text]); ex[Formula: see text]) in a termwise manner by starting with a cotorsion pair ([Formula: see text], [Formula: see text]) that is cogenerated by a set. As applications of these results, we obtain more abelian model structures from the cotorsion pairs.

Let $${\mathcal {A}}$$ be an abelian category. In this paper we study monoform objects and atoms introduced by Kanda. We classify full subcategories of $${\mathcal {A}}$$ by means of subclasses of $${\mathrm{ASpec}}{\mathcal {A}}$$ , the atom spectrum of $${\mathcal {A}}$$ . We also study the atomical decomposition and localization theory in terms of atoms. As some applications of our results, we study the category Mod-A where A is a fully right bounded noetherian ring.

Let R be a commutative Noetherian ring, I ⊆ J be ideals of R and M be a finitely generated R-module. In this paper it is shown that q ( J , M ) ≤ q ( I , M ) + cd ( J , M / I M ) . Furthermore, it is shown that, for any ideal I of R and any finitely generated R-module M with q ( I , M ) ≤ 1 , the local cohomology modules H I i ( M ) are I-cofinite for all integers i ≥ 0 . As a consequence of this result it is shown that, if q ( I , R ) ≤ 1 , then for any finitely generated R-module M, the local cohomology modules H I i ( M ) are I-cofinite for all integers i ≥ 0 . Finally, it is shown that the category of all I-cofinite R-modules C ( R , I ) c o f is an Abelian subcategory of the category of all R-modules, whenever ( R , m ) is a complete Noetherian local ring and I is an ideal of R with q ( I , R ) ≤ 1 . These assertions answer affirmatively two questions raised by R. Hartshorne in [16] , in the some special cases.

Given a non-negative integer n and a complete hereditary cotorsion triple , the notion of subcategories in an abelian category is introduced. It is proved that a virtually Gorenstein ring R is n-Gorenstein if and only if the subcategory of Gorenstein injective R-modules is with respect to the cotorsion triple , where stands for the subcategory of Gorenstein projectives. In the case when a subcategory of is closed under direct summands such that each object in admits a right -approximation, a Bazzoni characterization is given for to be . Finally, an Auslander–Reiten correspondence is established between the class of subcategories and that of certain subcategories of which are -coresolving covariantly finite and closed under direct summands.

Generalizing our previous results on Walsh system, we consider the system of characters of zero-dimensional compact abelian group with the second axiom of countability and give for the sets of uniqueness with respect to this system a solution of category problem. We also obtain a generalization of a theorem on existence for this system a perfect M 0 -set whose Hausdorff h -measure equals zero.

We propose the first subquadratic-time algorithms to a number of natural problems inabelian pattern matching (also called jumbled pattern matching) for strings over a constant-sized alphabet.Two strings are considered equivalent in this model if the numbers of occurrences of respectivesymbols in both of them, specified by their so-called Parikh vectors, are the same. We consider theproblems of computing abelian squares, abelian periods, abelian runs, abelian covers, and abelian borders.This work can be viewed as a continuation of a recent very active line of research on subquadraticspace and time jumbled indexing for binary and constant-sized alphabets (e.g., Moosa and Rahman,2012). Our algorithms apply the so-called four Russian technique in a fancy way and work in linearspace. The purpose of this work is to show that breaking the O(n2) barrier is possible for the aforementionedproblems. In this paper we extend our previous work, published in a preliminary form atMACIS 2015 conference, with effcient computation of abelian runs.