Showing papers on "Abelian category published in 2010"
TL;DR: In this article, the authors introduce the notion of an Auslander-Reiten sequence in a Krull-Schmidt category and describe the shapes of its semi-stable components.
Abstract: We rst introduce the notion of an Auslander-Reiten sequence in a Krull-Schmidt category. This uni es the notion of an almost split sequence in an abelian category and that of an Auslander-Reiten triangle in a triangulated category. We then de ne the Auslander-Reiten quiver of a Krull-Schmidt category and describe the shapes of its semi-stable components. The main result generalizes those for an artin algebra and specializes to an arbitrary triangulated categories, in particular to the derived category of bounded complexes of nitely generated modules over an artin algebra of nite global dimension.
71 citations
TL;DR: In this article, the notion of balanced pair of additive subcategories in an abelian category was introduced and sufficient conditions under which a balanced pair gave rise to a triangle-equivalence between two homotopy categories of complexes.
70 citations
TL;DR: In this article, the Popescu-Gabriel theorem is replaced by a triangulated category which is well generated (in the sense of Neeman) and algebraic (in a sense of Keller).
44 citations
TL;DR: The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms as discussed by the authors.
Abstract: The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called `algebraic' because they originate from abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the source categories are usually very `non-additive' before passing to homotopy classes of morphisms. Because of their origin I refer to these examples as `topological triangulated categories'.
In these extended talk notes I explain some systematic differences between these two kinds of triangulated categories. There are certain properties -- defined entirely in terms of the triangulated structure -- which hold in all algebraic examples, but which fail in some topological ones. These differences are all torsion phenomena, and rationally there is no difference between algebraic and topological triangulated categories.
39 citations
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TL;DR: In this paper, a notion of local homology and cosupport for triangulated categories is developed, building on earlier work of the authors on local cohomology and support.
Abstract: The Hom closed colocalizing subcategories of the stable module category of a finite group are classified. Along the way, the colocalizing subcategories of the homotopy category of injectives over an exterior algebra, and the derived category of a formal commutative differential graded algebra, are classified. To this end, and with an eye towards future applications, a notion of local homology and cosupport for triangulated categories is developed, building on earlier work of the authors on local cohomology and support.
37 citations
TL;DR: In this article, the authors consider a path algebra over k of the linearly oriented quiver and determine for which pairs (n, r) the algebra Λ(n,r) is piecewise hereditary.
Abstract: Let k be an algebraically closed field. Let Λ be the path algebra over k of the linearly oriented quiver \(\mathbb A_n\) for n ≥ 3. For r ≥ 2 and n > r we consider the finite dimensional k −algebra Λ(n,r) which is defined as the quotient algebra of Λ by the two sided ideal generated by all paths of length r. We will determine for which pairs (n,r) the algebra Λ(n,r) is piecewise hereditary, so the bounded derived category Db(Λ(n,r)) is equivalent to the bounded derived category of a hereditary abelian category \(\mathcal H\) as triangulated category.
29 citations
TL;DR: In this paper, the image of the free abelian category in Mod-R is described and related to special bases of the Ziegler and rep-Zariski spectra restricted to the set of indecomposable injectives.
26 citations
TL;DR: In this paper, the authors propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences.
26 citations
TL;DR: A general existence theorem for flat covers in locally finitely presented categories is obtained from an additive Ramsey type theorem in this article, where applications to categories of separated presheaves or sheaves, localizations of Bousfield type, torsion-free classes of finite type, and categories of filtered objects or complexes are given.
22 citations
TL;DR: For a wide class of abelian categories relevant in representation theory and algebraic geometry, the authors showed that the bounded derived categories have no non-trivial strongly finitely generated thick subcategories containing all perfect complexes.
Abstract: We show, for a wide class of abelian categories relevant in representation theory and algebraic geometry, that the bounded derived categories have no non-trivial strongly finitely generated thick subcategories containing all perfect complexes. In order to do so we prove a strong converse of the Ghost Lemma for bounded derived categories.
20 citations
TL;DR: In this article, all k-linear abelian 1-Calabi-Yau categories over an algebraically closed field k are derived equivalent to either the category of coherent sheaves on an elliptic curve, or to the finite dimensional representations of k.
Abstract: In this paper, we show all k-linear abelian 1-Calabi-Yau categories over an algebraically closed field k are derived equivalent to either the category of coherent sheaves on an elliptic curve, or to the finite dimensional representations of k[[t]]. Since all abelian categories derived equivalent with these two are known, we obtain a classification of all k-linear abelian 1-Calabi-Yau categories up to equivalence.
TL;DR: For a Hopf algebra in a braided monoidal abelian category, the stable anti-Yetter-Drinfeld module was introduced in this article, where the authors associate a para-cocyclic and a cocyclic object to the Hopf cyclic cohomology with coecients.
Abstract: We extend the formalism of Hopf cyclic cohomology to the context of braided categories. For a Hopf algebra in a braided monoidal abelian category we introduce the notion of stable anti-Yetter-Drinfeld module. We associate a para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution in the sense of Connes and Moscovici. When the braiding is symmetric the full formalism of Hopf cyclic cohomology with coecients can be extended to our categorical setting.
TL;DR: The distributive property can be studied through bilinear maps and various morphisms between these maps as discussed by the authors, which is a complete abelian category with projectives and admits a duality.
Abstract: The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the adjoint category is not a module category but nevertheless it is suitably familiar. The universal properties have geometric perspectives. For example, products are orthogonal sums. The bilinear division maps are the simple bimaps with respect to nondegenerate adjoint-morphisms. That formalizes the understanding that the atoms of linear geometries are algebraic objects with no zero-divisors. Adjoint-isomorphism coincides with principal isotopism; hence, nonassociative division rings can be studied within this framework.
This also corrects an error in an earlier pre-print; see Remark 2.11.
01 Jan 2010
TL;DR: The notion of semidirect product was introduced by Bourn and Janelidze in this paper, where they considered the case where the category C is pointed, has coequalizers of reexive pairs and binary coproducts.
Abstract: The categorical notion of semidirect product was introduced by Bourn and Janelidze in (1). A category C with split pullbacks is said to be a category with semidirect products if, for every morphism p:E! B in C, the pullback functor p : Pt(B)! Pt(E) has a left adjoint and is monadic. In this note we consider the case where the category C is pointed, has coequalizers of reexive pairs and binary coproducts. Forming the category of internal actions (as in (2)) we have by denition that C has semidirect products if the category Pt(C) of points is equivalent to the category Act(C) of internal actions. It is well known that: (a) a variety of universal algebras has semidirect products if and only if it is protomodular; (b) every semiabelian category has semidirect products; (c) not every homological category has semidirect products. This talk is divided in two parts. In the rst part we analyze the monadicity of p and give some necessary and sucient conditions for a category C to have semidirect products. In the second part we introduce the notion of strict action and show that C has semidirect products if and only if it is protomodular and every internal action is strict.
TL;DR: In this paper, it was shown that for an Abelian variety A defined over a Hilbertian field K every extension L of K in K (A tor ) is Hilbertian.
Abstract: A theorem of Kuyk says that every Abelian extension of a Hilbertian field is Hilbertian. We conjecture that for an Abelian variety A defined over a Hilbertian field K every extension L of K in K ( A tor ) is Hilbertian. We prove our conjecture when K is a number field. The proof applies a result of Serre about l -torsion of Abelian varieties, information about l -adic analytic groups, and Haran's diamond theorem.
TL;DR: It is shown that every cartesian ' quasi-MV algebra is embeddable into an interval in a particular Abelian l-group with operators and a purely group-theoretical equivalence is obtained between the mentioned category of l-groups with operator and the category of AbelianL-groups (both with strong order unit).
Abstract: In previous investigations into the subject [Giuntini et al. (2007, Studia Logica, 87, 99–128), Paoli et al. (2008, Reports on Mathematical Logic, 44, 53–85), Bou et al. (2008, Soft Computing, 12, 341–352)], ' quasi-MV algebras have been mainly viewed as preordered structures w.r.t. the induced preorder relation of their quasi-MV term reducts. In this article, we shall focus on a different relation which partially orders cartesian ' quasi-MV algebras. We shall prove that: (i) every cartesian ' quasi-MV algebra is embeddable into an interval in a particular Abelian l-group with operators; (ii) the category of cartesian ' quasi-MV algebras isomorphic with the pair algebras over their own polynomial MV subreducts is equivalent both to the category of such l-groups (with strong order unit), and to the category of MV algebras. As a by-product of these results we obtain a purely group-theoretical equivalence, namely between the mentioned category of l-groups with operators and the category of Abelian l-groups (both with strong order unit).
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TL;DR: In this article, the Eilenberg-Mac Lane cohomology of groups was studied in the semi-abelian category Gp of groups, and conditions on the existence of isomorphisms with coefficients in the slice category D/Y were given.
Abstract: We make explicit some conditions on a semi-abelian category D such that, for any abelian group A in D and any object Y in D, the cohomology group homomorphisms with coefficients in A, induced by the inclusion of the abelian objects of D at the level of the slice category D/Y, are actually isomorphisms. These conditions hold in particular when D is the category Gp of groups, and this allows us to give a new insight on the Eilenberg-Mac Lane cohomology of groups. They hold also when D is the category K-Lie of Lie-algebras.
TL;DR: For the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representation for the dual fail in this article, and a localizing subcategory for which the inclusion into K(ab) does not have a right adjoint is presented.
Abstract: We show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory of K(Ab) for which the inclusion into K(Ab) does not have a right adjoint.
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TL;DR: The concept of entropy function h of an abelian category is introduced, and the Pinsker radical with respect to h is defined, so that the class of all objects with trivial P Insker radical is the torsion-free class of a torsionsion theory.
Abstract: The Pinsker subgroup of an abelian group with respect to an endomorphism was introduced in the context of algebraic entropy. Motivated by the nice properties and characterizations of the Pinsker subgroup, we generalize its construction in two directions. We introduce the concept of entropy function h of an abelian category and define the Pinsker radical with respect to h, so that the class of all objects with trivial Pinsker radical is the torsion class of a torsion theory.
TL;DR: The category of additive graded functors from the category of associated graded categories of modΛ to graded vector spaces decomposes into subcategories corresponding to the components of the Auslander-Reiten quiver as discussed by the authors.
TL;DR: In this article, the classical Hodge conjecture for the middle cohomology of an abelian variety is shown to be equivalent to the general Hodge conjectures for a smooth ample divisor in the abelians.
Abstract: We show how the classical Hodge conjecture for the middle cohomology of an abelian variety is equivalent to the general Hodge conjecture for the middle cohomology of a smooth ample divisor in the abelian variety. This is best suited to abelian varieties with actions of imaginary quadratic fields.
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TL;DR: In this article, the string number of self-maps is defined as a combinatorial entropy function and its global version for abelian groups, providing several examples involving also Hopfian abelians.
Abstract: The string number of self-maps arose in the context of algebraic entropy and it can be viewed as a kind of combinatorial entropy function Later on its values for endomorphisms of abelian groups were calculated in full generality We study its global version for abelian groups, providing several examples involving also Hopfian abelian groups Moreover, we characterize the class of all abelian groups with string number zero in many cases and discuss its stability properties
TL;DR: In this article, a new approach to the description of quotient divisible Abelian groups is proposed, which allows us to give an explicit and natural proof of the duality of the categories of torsion-free Abelian group of finite rank.
Abstract: A new approach to the description of quotient divisible Abelian groups is proposed. This approach allowed us to give an explicit and natural proof of the duality of the categories of torsion-free Abelian groups of finite rank and quotient divisible Abelian groups with distinguished basic subgroups. Bibliography: 3 titles.
TL;DR: The symmetric Auslander category A^s(R) as mentioned in this paper consists of complexes of projective modules whose left-and right-tails are equal to the left and right tails of totally acyclic complexes of projects.
Abstract: We define the symmetric Auslander category A^s(R) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right tails of totally acyclic complexes of projective modules.
The symmetric Auslander category contains A(R), the ordinary Auslander category. It is well known that A(R) is intimately related to Gorenstein projective modules, and our main result is that A^s(R) is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories:
\underline{GMor}(R) --> A^s(R) / K^b(Prj R).
Here \underline{GMor}(R) is the stable category of Gorenstein projective objects in the abelian category Mor(R) of homomorphisms of R-modules, and K^b(Prj R) is the homotopy category of bounded complexes of projective R-modules.
This result is set in the wider context of a theory for A^s(R) and B^s(R), the symmetric Bass category which is defined dually.
TL;DR: The notion of a pseudo-morphism is introduced and the equivalence of categories is proved: PsCat(A)~PsMor (A) between pseudo-categories and pseudo- morphisms in an additive 2-category, A, with kernels, extending thus the well known equivalence Cat(Ab)~Mor(Ab).
Abstract: We describe the (tetra) category of pseudo-categories, pseudo-functors, natural transformations, pseudo-natural transformations, and modifications, as introduced in Martins-Ferreira (JHRS 1:47–78, 2006), internal to an additive 2-category with kernels, as formalized in Martins-Ferreira (Fields Inst Commun 43:387–410, 2004). In the context of a 2-Ab-category, we introduce the notion of a pseudo-morphism and prove the equivalence of categories: PsCat(A)~PsMor(A) between pseudo-categories and pseudo-morphisms in an additive 2-category, A, with kernels– extending thus the well known equivalence Cat(Ab)~Mor(Ab) between internal categories and morphisms of abelian groups. The leading example of an additive 2-category with kernels is Cat(Ab). In the case A=Cat(Ab) we obtain a description of the (tetra) category of internal pseudo-double categories in Ab, and particularize it to a description of the (tetra) category of internal bicategories in abelian groups. As expected, pseudo-natural transformations coincide with homotopies of 2-chain complexes (as in Bourn, J Pure Appl Algebra 66:229–249, 1990).
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TL;DR: In this article, a Tannakian category of mixed elliptic motives generated by an elliptic curve is defined, and the notion of a relative DGA over a reductive group is defined as the category of comodules over the relative bar construction of a certain DGA, which is constructed from cycle complexes.
Abstract: Bloch and Kriz construct an abelian category of mixed Tate motives as the category of comodules over a Hopf algebra obtained by the bar construction of the DGA of cycle complexes. In this paper we generalize their construction to give the definition of a category of mixed elliptic motives, i.e. a Tannakian category of mixed motives generated by an elliptic curve. We introduce the notion of a relative DGA over a reductive group. Then the category of mixed elliptic motives is defined as the category of comodules over the relative bar construction of a certain DGA $A_{EM}$ which is constructed from cycle complexes. The elliptic polylogarithm of Beilinson-Levin gives an interesting object in this category.
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TL;DR: In this paper, it was shown that the category of "L$-complete modules" is also idempotent under mild conditions on the left derived functor of the category.
Abstract: Let $\mathcal{C}$ be an abelian category and let $\Lambda : \mathcal{C}\rightarrow\mathcal{C}$ be an idempotent functor which is not right exact, so that the zeroth left derived functor $L_0\Lambda$ does not necessarily coincide with $\Lambda$. In this paper we show that, under mild conditions on $\Lambda$, $L_0\Lambda$ is also idempotent, and the category of $L_0\Lambda$-complete objects of $\mathcal{C}$ is the smallest exact subcategory of $\mathcal{C}$ containing the $\Lambda$-complete objects. In the main application, where $\Lambda$ is the $I$-adic completion functor on a category of modules, this gives us that the category of "$L$-complete modules," studied by Greenlees-May and Hovey-Strickland, is not an ad hoc construction but is in fact characterized by a universal property. Generalizations are also given to the case of relative derived functors, in the sense of relative homological algebra.
TL;DR: In this article, it was shown that if A is an abelian category, then the category of representations of a quiver in A is also abelians, and that twisted linear representations of quivers in A are equivalent to linear (untwisted) representations of dierent quivers.
Abstract: We consider representations of quivers in arbitrary categories and twisted representations of quivers in arbitrary ten- sor categories. We show that if A is an abelian category, then the category of representations of a quiver in A is also abelian, and that the category of twisted linear representations of a quiver is equivalent to the category of linear (untwisted) representations of a dierent quiver. We conclude by discussing how represen- tations of quivers arise naturally in certain important problems concerning monads ans sheaves on projective varieties.
TL;DR: In this article, it was shown that F is canonically isomorphic to the right derived DG functor RH0(F) and a similar result for bounded derived DG categories and a formula that expresses Hochschild cohomology of the categories Db dg(A), D+dg(B) as the Ext groups in the abelian category of left exact functors A → IndA.
Abstract: Assume that abelian categories A, B over a field admit countable direct limits and that these limits are exact. Let F : D dg(A)→ D + dg(B) be a DG quasi-functor such that the functor Ho(F) : D+(A)→ D+(B) carries D≥0(A) to D≥0(B) and such that, for every i > 0, the functor HiF : A → B is effaceable. We prove that F is canonically isomorphic to the right derived DG functor RH0(F). We also prove a similar result for bounded derived DG categories and a formula that expresses Hochschild cohomology of the categories Db dg(A), D + dg(A) as the Ext groups in the abelian category of left exact functors A → IndA . The proofs are based on a description of Drinfeld’s category of quasi-functors as the derived category of a category of sheaves.