Log in

Relevant bibliographies by topics / Category theory; homological algebra / Journal articles

To see the other types of publications on this topic, follow the link: Category theory; homological algebra.

Journal articles on the topic 'Category theory; homological algebra'

Author: Grafiati

Published: 4 June 2021

Last updated: 3 March 2023

Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles

Select a source type:

Consult the top 50 journal articles for your research on the topic 'Category theory; homological algebra.'

Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.

You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.

Browse journal articles on a wide variety of disciplines and organise your bibliography correctly.

1

CHRISTENSEN, J. DANIEL, and MARK HOVEY. "Quillen model structures for relative homological algebra." Mathematical Proceedings of the Cambridge Philosophical Society 133, no. 2 (September 2002): 261–93. http://dx.doi.org/10.1017/s0305004102006126.

Full text

Abstract:

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.

APA, Harvard, Vancouver, ISO, and other styles

2

Antosz, Jakub, and Stanislaw Betley. "Homological Algebra in the Category of Γ-Modules." Communications in Algebra 33, no. 6 (May 2005): 1913–36. http://dx.doi.org/10.1081/agb-200063342.

Full text

APA, Harvard, Vancouver, ISO, and other styles

3

Rennemo, Jørgen Vold. "The homological projective dual of." Compositio Mathematica 156, no. 3 (January 17, 2020): 476–525. http://dx.doi.org/10.1112/s0010437x19007772.

Full text

Abstract:

We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

APA, Harvard, Vancouver, ISO, and other styles

4

SANTIAGO, VALENTE. "STRATIFYING SYSTEMS FOR EXACT CATEGORIES." Glasgow Mathematical Journal 61, no. 03 (July 23, 2018): 501–21. http://dx.doi.org/10.1017/s0017089518000320.

Full text

Abstract:

AbstractIn this paper, we develop the theory of stratifying systems in the context of exact categories as a generalisation of the notion of stratifying systems in module categories, which have been studied by different authors. We prove that attached to a stratifying system in an exact category $(\mathcal{A},\mathcal{E})$ there is an standardly stratified algebra B such that the category $\mathscr{F}$F(Θ), of F-filtered objects in the exact category $(\mathcal{A},\mathcal{E})$ is equivalent to the category $\mathscr{F}$(Δ) of Δ-good modules associated to B. The theory we develop in exact categories, give us a way to produce standardly stratified algebras from module categories by just changing the exact structure on it. In this way, we can construct exact categories whose bounded derived category is equivalent to the bounded derived category of an standardly stratified algebra. Finally, applying the relative homological algebra developed by Auslander–Solberg, we can construct examples of stratifying systems that are not a stratifying system in the classical sense, so our approach really produces new stratifying systems.

APA, Harvard, Vancouver, ISO, and other styles

5

Assem, Ibrahim, and Flávio Ulhoa Coelho. "Complete slices and homological properties of tilted algebras." Glasgow Mathematical Journal 36, no. 3 (September 1994): 347–54. http://dx.doi.org/10.1017/s0017089500030950.

Full text

Abstract:

It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).

APA, Harvard, Vancouver, ISO, and other styles

6

Cristian Iovanov, Miodrag. "Generalized Frobenius Algebras and Hopf Algebras." Canadian Journal of Mathematics 66, no. 1 (February 2014): 205–40. http://dx.doi.org/10.4153/cjm-2012-060-7.

Full text

Abstract:

Abstract“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

APA, Harvard, Vancouver, ISO, and other styles

7

Kaushal Rana. "Homological Algebra and Its Application: A Descriptive Study." Integrated Journal for Research in Arts and Humanities 2, no. 1 (January 31, 2022): 29–35. http://dx.doi.org/10.55544/ijrah.2.1.47.

Full text

Abstract:

Algebra has been used to define and answer issues in almost every field of mathematics, science, and engineering. Homological algebra depends largely on computable algebraic invariants to categorise diverse mathematical structures, such as topological, geometrical, arithmetical, and algebraic (up to certain equivalences). String theory and quantum theory, in particular, have shown it to be of crucial importance in addressing difficult physics questions. Geometric, topological and algebraic algebraic techniques to the study of homology are to be introduced in this research. Homology theory in abelian categories and a category theory are covered. the n-fold extension functors EXTn (-,-) , the torsion functors TORn (-,-), Algebraic geometry, derived functor theory, simplicial and singular homology theory, group co-homology theory, the sheaf theory, the sheaf co-homology, and the l-adic co-homology, as well as a demonstration of its applicability in representation theory.

APA, Harvard, Vancouver, ISO, and other styles

8

Lekili, Yankı, and Alexander Polishchuk. "Homological mirror symmetry for higher-dimensional pairs of pants." Compositio Mathematica 156, no. 7 (June 18, 2020): 1310–47. http://dx.doi.org/10.1112/s0010437x20007150.

Full text

Abstract:

Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$, for $k\geqslant n$, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ($n$-dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$. By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$-dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$. We also prove similar equivalences for finite abelian covers of the $n$-dimensional pair of pants.

APA, Harvard, Vancouver, ISO, and other styles

9

BARAKAT, MOHAMED, and MARKUS LANGE-HEGERMANN. "AN AXIOMATIC SETUP FOR ALGORITHMIC HOMOLOGICAL ALGEBRA AND AN ALTERNATIVE APPROACH TO LOCALIZATION." Journal of Algebra and Its Applications 10, no. 02 (April 2011): 269–93. http://dx.doi.org/10.1142/s0219498811004562.

Full text

Abstract:

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 R𝔪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R𝔪. As a corollary, we obtain the computability of the category of finitely presented R𝔪-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.

APA, Harvard, Vancouver, ISO, and other styles

10

WEI, REN, and ZHONGKUI LIU. "A QUILLEN MODEL STRUCTURE APPROACH TO HOMOLOGICAL DIMENSIONS OF COMPLEXES." Journal of Algebra and Its Applications 13, no. 03 (October 31, 2013): 1350106. http://dx.doi.org/10.1142/s0219498813501065.

Full text

Abstract:

In this paper, we first give an alternative characterization of the derived functor Ext via the Quillen model structure on the category of complexes induced by a given cotorsion pair [Formula: see text] in the category of modules, then based on this, we consider homological dimensions of complexes related to [Formula: see text]. As applications, we extend Gorenstein projective dimension of homologically bounded below complexes (in the sense of Christensen and coauthors) to unbounded complexes whenever R is Gorenstein. Moreover, we extend Stenström's FP-injective dimension from modules to complexes, define FP-projective dimension for complexes, and characterize Noetherian and von Neumann regular rings by these dimensions.

APA, Harvard, Vancouver, ISO, and other styles

11

Betley, Stanisław. "Stable derived functors, the Steenrod algebra and homological algebra in the category of functors." Fundamenta Mathematicae 168, no. 3 (2001): 279–93. http://dx.doi.org/10.4064/fm168-3-4.

Full text

APA, Harvard, Vancouver, ISO, and other styles

12

Zhang, Chao. "On the representation type of subcategories of derived categories." Journal of Algebra and Its Applications 18, no. 05 (May 2019): 2050032. http://dx.doi.org/10.1142/s0219498820500322.

Full text

Abstract:

Let [Formula: see text] be a finite-dimensional [Formula: see text]-algebra. In this paper, we mainly study the representation type of subcategories of the bounded derived category [Formula: see text]. First, we define the representation type and some homological invariants including cohomological length, width, range for subcategories. In this framework, we provide a characterization for derived discrete algebras. Moreover, for a finite-dimensional algebra [Formula: see text], we establish the first Brauer–Thrall type theorem of certain contravariantly finite subcategories [Formula: see text] of [Formula: see text], that is, [Formula: see text] is of finite type if and only if its cohomological range is finite.

APA, Harvard, Vancouver, ISO, and other styles

13

Gillespie, James, and Mark Hovey. "Gorenstein model structures and generalized derived categories." Proceedings of the Edinburgh Mathematical Society 53, no. 3 (August 5, 2010): 675–96. http://dx.doi.org/10.1017/s0013091508000709.

Full text

Abstract:

AbstractIn a paper from 2002, Hovey introduced the Gorenstein projective and Gorenstein injective model structures on R-Mod, the category of R-modules, where R is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce: the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable module category of R. If such a ring R has finite global dimension, the graded ring R[x]/(x2) is Gorenstein and the three associated Gorenstein model structures on R[x]/(x2)-Mod, the category of graded R[x]/(x2)-modules, are nothing more than the usual projective, injective and flat model structures on Ch(R), the category of chain complexes of R-modules. Although these correspondences only recover these model structures on Ch(R) when R has finite global dimension, we can set R = ℤ and use general techniques from model category theory to lift the projective model structure from Ch(ℤ) to Ch(R) for an arbitrary ring R. This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when ℤ[x]/(x2) is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring R and the derived category of R and we give some examples of such generalizations.

APA, Harvard, Vancouver, ISO, and other styles

14

EICK, BETTINA, and DAVID J. GREEN. "COCHAIN SEQUENCES AND THE QUILLEN CATEGORY OF A COCLASS FAMILY." Journal of the Australian Mathematical Society 102, no. 2 (May 12, 2016): 185–204. http://dx.doi.org/10.1017/s1446788716000185.

Full text

Abstract:

We introduce the concept of infinite cochain sequences and initiate a theory of homological algebra for them. We show how these sequences simplify and improve the construction of infinite coclass families (as introduced by Eick and Leedham-Green) and also how they can be applied to prove that almost all groups in such a family have equivalent Quillen categories. We also include some examples of infinite families of$p$-groups from different coclass families that have equivalent Quillen categories.

APA, Harvard, Vancouver, ISO, and other styles

15

Castillo, Jesús. "The hitchhiker guide to Categorical Banach space theory. Part II." Extracta Mathematicae 37, no. 1 (June 1, 2022): 1–56. http://dx.doi.org/10.17398/2605-5686.37.1.1.

Full text

Abstract:

What has category theory to offer to Banach spacers? In this second part survey-like paper we will focus on very much needed advanced categorical and homological elements, such as Kan extensions, derived category and derived functor or Abelian hearts of Banach spaces.

APA, Harvard, Vancouver, ISO, and other styles

16

Thomas, Weichert. "A homological characterization of the category of socle projective modules." Communications in Algebra 18, no. 10 (January 1990): 3547–63. http://dx.doi.org/10.1080/00927879008824090.

Full text

APA, Harvard, Vancouver, ISO, and other styles

17

Pascaleff, James, and Nicolò Sibilla. "Topological Fukaya category and mirror symmetry for punctured surfaces." Compositio Mathematica 155, no. 3 (March 2019): 599–644. http://dx.doi.org/10.1112/s0010437x19007073.

Full text

Abstract:

In this paper we establish a version of homological mirror symmetry for punctured Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured surface$\unicode[STIX]{x1D6F4}$via the topological Fukaya category. We prove that the topological Fukaya category of$\unicode[STIX]{x1D6F4}$is equivalent to the category of matrix factorizations of a certain mirror LG model$(X,W)$. Along the way we establish new gluing results for the topological Fukaya category of punctured surfaces which are of independent interest.

APA, Harvard, Vancouver, ISO, and other styles

18

Sheridan, Nick. "Formulae in noncommutative Hodge theory." Journal of Homotopy and Related Structures 15, no. 1 (November 21, 2019): 249–99. http://dx.doi.org/10.1007/s40062-019-00251-2.

Full text

Abstract:

AbstractWe prove that the cyclic homology of a saturated $$A_\infty $$A∞ category admits the structure of a ‘polarized variation of Hodge structures’, building heavily on the work of many authors: the main point of the paper is to present complete proofs, and also explicit formulae for all of the relevant structures. This forms part of a project of Ganatra, Perutz and the author, to prove that homological mirror symmetry implies enumerative mirror symmetry.

APA, Harvard, Vancouver, ISO, and other styles

19

Song, Weiling, Tiwei Zhao, and Zhaoyong Huang. "Homological Dimensions Relative to Special Subcategories." Algebra Colloquium 28, no. 01 (January 20, 2021): 131–42. http://dx.doi.org/10.1142/s1005386721000122.

Full text

Abstract:

Let [Formula: see text] be an abelian category, [Formula: see text] an additive, full and self-orthogonal subcategory of [Formula: see text] closed under direct summands, [Formula: see text] the right Gorenstein subcategory of [Formula: see text] relative to [Formula: see text], and [Formula: see text] the left orthogonal class of [Formula: see text]. For an object [Formula: see text] in [Formula: see text], we prove that if [Formula: see text] is in the right 1-orthogonal class of [Formula: see text], then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical; if the [Formula: see text]-projective dimension of [Formula: see text] is finite, then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical. We also prove that the supremum of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension and that of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension coincide. Then we apply these results to the category of modules.

APA, Harvard, Vancouver, ISO, and other styles

20

Huebschmann, J. "Minimal Free Multi-Models for Chain Algebras." gmj 11, no. 4 (December 2004): 733–52. http://dx.doi.org/10.1515/gmj.2004.733.

Full text

Abstract:

Abstract Let 𝑅 be a local ring and 𝐴 a connected differential graded algebra over 𝑅 which is free as a graded 𝑅-module. Using homological perturbation theory techniques, we construct a minimal free multi-model for 𝐴 having properties similar to those of an ordinary minimal model over a field; in particular the model is unique up to isomorphism of multialgebras. The attribute ‘multi’ refers to the category of multicomplexes.

APA, Harvard, Vancouver, ISO, and other styles

21

ALB, ALINA, and MIHAIL URSUL. "A FEW HOMOLOGICAL CHARACTERIZATIONS OF COMPACT SEMISIMPLE RINGS." Journal of Algebra and Its Applications 04, no. 05 (October 2005): 539–49. http://dx.doi.org/10.1142/s0219498805001411.

Full text

Abstract:

Fix any compact ring R with identity. We associate to R the following categories of topological R-modules: (i) R𝔇 (𝔇R) the category of all discrete topological left (right) R-modules; (ii) Rℭ (ℭR) the category of all compact left (right) R-modules. We have introduced the following notions (analogous with classical notions of module theory): (i) the tensor product [Formula: see text] of A ∈ ℭR and B ∈Rℭ ([Formula: see text] has a structure of a compact Abelian group); (ii) a topologically semisimple module; (iii) a compact topologically flat module. We give a characterization of compact semisimple rings by using of flat modules.

APA, Harvard, Vancouver, ISO, and other styles

22

Tu, Junwu. "Matrix factorizations via Koszul duality." Compositio Mathematica 150, no. 9 (July 17, 2014): 1549–78. http://dx.doi.org/10.1112/s0010437x14007295.

Full text

Abstract:

AbstractIn this paper we prove a version of curved Koszul duality for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}$-graded curved coalgebras and their cobar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations $\mathsf{MF}(R,W)$. We show how Dyckerhoff’s generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel–Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category $\mathsf{MF}(R,W)$. Similar results are also obtained in the orbifold case and in the graded case.

APA, Harvard, Vancouver, ISO, and other styles

23

NEHANIV, CHRYSTOPHER LEV. "ALGEBRAIC CONNECTIVITY." International Journal of Algebra and Computation 01, no. 04 (December 1991): 445–71. http://dx.doi.org/10.1142/s0218196791000316.

Full text

Abstract:

Let [Formula: see text] be a type of algebra in the sense of universal algebra. By defining singular simplices in algebras and emulating singular [co] homology, we introduce for each variety, pseudo-variety, and divisional class V of type [Formula: see text], a homology and cohomology theory which measure the V-connectivity of type-[Formula: see text] algebras. Intuitively, if we were to think of an algebra as a space and subalgebras which lie in V as simplices, then V-connectivity describes the failure of subalgebras to lie in V, i.e., it describes the "holes" in this space. These [co]homologies are functorial on the class of type-[Formula: see text] algebras and are characterized by a natural topological interpretation. All these notions extend to subsets of algebras. One obtains for this algebraic connectivity, the long exact sequences, relative [co]homologies, and the analogues of the usual [co]homological notions of the algebraic topologists. In fact, we show that the [co]homologies are actually the same as the simplicial [co]homology of simplicial complexes that depend functorially on the algebras. Thus the connectivities in question have a natural geometric meaning. This allows the wholesale import into algebra of the concepts, results, and techniques of algebraic topology. In particular, functoriality implies that the [co]homology of a pair of algebras A ⊆ B is an invariant of the position of A in B. When one V contains another, we obtain relationships between the [co] homology theories in the form of long exact sequences. Furthermore for finite algebras, V-[co]homology is effectively computable if membership in V is. We obtain an analogue of the Poincaré lemma (stating that subsets of an algebra in V are V-homologically trivial), extremely general guarantees of the existence of subsets with non-trivial V-homology for algebras not in V, long exact V-homotopy sequences, as well as analogues of the powerful Eilenberg-Zilber theorems and Kunneth theorems in the setting of V-connectivity for V a variety or pseudo-variety. Also in the more general case of any divisionally closed V, we construct the long exact Mayer-Vietoris sequences for V-homology. Results for homomorphisms include an algebraic version of contiguity for homomorphisms (which implies they are V-homotopic) and a proof that V-surmorphisms are V-homotopy equivalences. If we allow the divisional classes to vary, then algebraic connectivity may be viewed as a functor from the category of pairs W ⊆ V of divisional classes of [Formula: see text]-algebras with inclusions as morphisms' to the category of functors from pairs of [Formula: see text]-algebras to pairs of simplicial complexes. Examples show the non-triviality of this theory (e.g. "associativity tori"), and two preliminary applications to semigroups are given: 1) a proof that the group connectivity of a torsion semigroup S is homotopy equivalent to a space whose points are the maximal subgroups of S, and 2) an aperiodic connectivity analogue of the fundamental lemma of complexity.

APA, Harvard, Vancouver, ISO, and other styles

24

KIM, BUMSIG, and HWAYOUNG LEE. "WALL-CROSSINGS FOR TWISTED QUIVER BUNDLES." International Journal of Mathematics 24, no. 05 (May 2013): 1350038. http://dx.doi.org/10.1142/s0129167x13500389.

Full text

Abstract:

Given a double quiver, we study homological algebra of twisted quiver sheaves with the moment map relation using the short exact sequence of Crawley-Boevey, Holland, Gothen, and King. Then in a certain one-parameter space of the stability conditions, we obtain a wall-crossing formula for the generalized Donaldson–Thomas invariants of the abelian category of framed twisted quiver sheaves on a smooth projective curve. To do so, we closely follow the approach of Chuang, Diaconescu and Pan in the ADHM quiver case, which makes use of the theory of Joyce and Song. The invariants virtually count framed twisted quiver sheaves with the moment map relation and directly generalize the ADHM invariants of Diaconescu.

APA, Harvard, Vancouver, ISO, and other styles

25

Martsinkovsky, Alex, and Jeremy Russell. "Injective stabilization of additive functors, III. Asymptotic stabilization of the tensor product." Algebra and Discrete Mathematics 31, no. 1 (2021): 120–51. http://dx.doi.org/10.12958/adm1728.

Full text

Abstract:

The injective stabilization of the tensor product is subjected to an iterative procedure that utilizes its bifunctor property. The limit of this procedure, called the asymptotic stabilization of the tensor product, provides a homological counterpart of Buchweitz's asymptotic construction of stable cohomology. The resulting connected sequence of functors is isomorphic to Triulzi's J-completion of the Tor functor. A comparison map from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be always epic. The category of finitely presented functors is shown to be complete and cocomplete. As a consequence, the inert injective stabilization of the tensor product with fixed variable a finitely generated module over an artin algebra is shown to be finitely presented. Its defect and consequently all right-derived functors are determined. New notions of asymptotic torsion and cotorsion are introduced and are related to each other.

APA, Harvard, Vancouver, ISO, and other styles

26

Chuang, Joseph, and Andrey Lazarev. "Rank functions on triangulated categories." Journal für die reine und angewandte Mathematik (Crelles Journal) 2021, no. 781 (October 16, 2021): 127–64. http://dx.doi.org/10.1515/crelle-2021-0052.

Full text

Abstract:

Abstract We introduce the notion of a rank function on a triangulated category 𝒞 {\mathcal{C}} which generalizes the Sylvester rank function in the case when 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} is the perfect derived category of a ring A. We show that rank functions are closely related to functors into simple triangulated categories and classify Verdier quotients into simple triangulated categories in terms of particular rank functions called localizing. If 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} as above, localizing rank functions also classify finite homological epimorphisms from A into differential graded skew-fields or, more generally, differential graded Artinian rings. To establish these results, we develop the theory of derived localization of differential graded algebras at thick subcategories of their perfect derived categories. This is a far-reaching generalization of Cohn’s matrix localization of rings and has independent interest.

APA, Harvard, Vancouver, ISO, and other styles

27

Divaani-Aazar, Kamran, and Majid Rahro Zargar. "The derived category analogues of Faltings Local-global Principle and Annihilator Theorems." Journal of Algebra and Its Applications 18, no. 07 (July 2019): 1950140. http://dx.doi.org/10.1142/s0219498819501408.

Full text

Abstract:

Let [Formula: see text] be a specialization closed subset of Spec R and X a homologically left-bounded complex with finitely generated homologies. We establish Faltings’ Local-global Principle and Annihilator Theorems for the local cohomology modules [Formula: see text] Our versions contain variations of results already known on these theorems.

APA, Harvard, Vancouver, ISO, and other styles

28

MASHHAD, FATEMEH MOHAMMADI AGHJEH, and KAMRAN DIVAANI-AAZAR. "LOCAL HOMOLOGY AND GORENSTEIN FLAT MODULES." Journal of Algebra and Its Applications 11, no. 02 (April 2012): 1250022. http://dx.doi.org/10.1142/s0219498811005750.

Full text

Abstract:

Let R be a commutative Noetherian ring, 𝔞 be an ideal of R and [Formula: see text] denote the derived category of R-modules. We investigate the theory of local homology in conjunction with Gorenstein flat modules. Let X be a homologically bounded to the right complex and Q be a bounded to the right complex of Gorenstein flat R-modules such that Q and X are isomorphic in [Formula: see text]. We establish a natural isomorphism LΛ𝔞(X) ≃ Λ𝔞(Q) in [Formula: see text] which immediately asserts that sup LΛ𝔞(X) ≤ Gfd RX. This isomorphism yields several conseQuences. For instance, in the case R possesses a dualizing complex, we show that Gfd RLΛ𝔞(X) ≤ Gfd RX. Also, we establish a criterion for regularity of Gorenstein local rings.

APA, Harvard, Vancouver, ISO, and other styles

29

Qi, You. "Hopfological algebra." Compositio Mathematica 150, no. 1 (October 7, 2013): 1–45. http://dx.doi.org/10.1112/s0010437x13007380.

Full text

Abstract:

AbstractWe develop some basic homological theory of hopfological algebra as defined by Khovanov [Hopfological algebra and categorification at a root of unity: the first steps, Preprint (2006), arXiv:math/0509083v2]. Several properties in hopfological algebra analogous to those of usual homological theory of DG algebras are obtained.

APA, Harvard, Vancouver, ISO, and other styles

30

Henneaux, Marc. "Homological algebra and Yang-Mills theory." Journal of Pure and Applied Algebra 100, no. 1-3 (May 1995): 3–17. http://dx.doi.org/10.1016/0022-4049(95)00016-p.

Full text

APA, Harvard, Vancouver, ISO, and other styles

31

Hinich, Vladimir. "Homological algebra of homotopy algebras." Communications in Algebra 25, no. 10 (January 1997): 3291–323. http://dx.doi.org/10.1080/00927879708826055.

Full text

APA, Harvard, Vancouver, ISO, and other styles

32

Kleiner, Mark, and Aboubakr Lbekkouri. "Splitters and relative homological algebra." Journal of Algebra 153, no. 2 (December 1992): 476–93. http://dx.doi.org/10.1016/0021-8693(92)90168-l.

Full text

APA, Harvard, Vancouver, ISO, and other styles

33

Salch, Andrew. "Relative homological algebra and Waldhausen $K$-theory." Homology, Homotopy and Applications 20, no. 1 (2018): 87–116. http://dx.doi.org/10.4310/hha.2018.v20.n1.a7.

Full text

APA, Harvard, Vancouver, ISO, and other styles

34

Gugenheim, V. K. A. M., L. A. Lambe, and J. D. Stasheff. "Perturbation theory in differential homological algebra II." Illinois Journal of Mathematics 35, no. 3 (September 1991): 357–73. http://dx.doi.org/10.1215/ijm/1255987784.

Full text

APA, Harvard, Vancouver, ISO, and other styles

35

Gugenheim, V. K. A. M., and L. A. Lambe. "Perturbation theory in differential homological algebra I." Illinois Journal of Mathematics 33, no. 4 (December 1989): 566–82. http://dx.doi.org/10.1215/ijm/1255988571.

Full text

APA, Harvard, Vancouver, ISO, and other styles

36

Bueso, J. L., F. Van Oystaeyen, and A. Verschoren. "Relating homological dimensions to relative homological dimensions." Communications in Algebra 15, no. 7 (January 1987): 1503–18. http://dx.doi.org/10.1080/00927878708823482.

Full text

APA, Harvard, Vancouver, ISO, and other styles

37

KONISHI, EIJI. "PLANAR HOMOLOGICAL MIRROR SYMMETRY." International Journal of Modern Physics A 22, no. 29 (November 20, 2007): 5351–68. http://dx.doi.org/10.1142/s0217751x07037202.

Full text

Abstract:

In this paper, we formulate a planar limited version of the B-side in homological mirror symmetry that formularizes Chern–Simons-type topological open string field theory using homotopy associative algebra (A∞ algebra). This formulation is based on the works by Dijkgraaf and Vafa. We show that our formularization includes gravity/gauge theory correspondence which originates in the AdS/CFT duality of Dijkgraaf–Vafa theory.

APA, Harvard, Vancouver, ISO, and other styles

38

Enochs, Edgar, and Sergio Estrada. "Relative homological algebra in the category of quasi-coherent sheaves." Advances in Mathematics 194, no. 2 (July 2005): 284–95. http://dx.doi.org/10.1016/j.aim.2004.06.007.

Full text

APA, Harvard, Vancouver, ISO, and other styles

39

Asadollahi, Javad, Azadeh Mehregan, and Somayeh Sadeghi. "Cotorsion classes in higher homological algebra." Journal of Pure and Applied Algebra 226, no. 2 (February 2022): 106839. http://dx.doi.org/10.1016/j.jpaa.2021.106839.

Full text

APA, Harvard, Vancouver, ISO, and other styles

40

Jekel, Solomon. "The Euler Class in homological algebra." Journal of Pure and Applied Algebra 215, no. 11 (November 2011): 2628–38. http://dx.doi.org/10.1016/j.jpaa.2011.03.005.

Full text

APA, Harvard, Vancouver, ISO, and other styles

41

Wei, Jiaqun. "Gorenstein homological theory for differential modules." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 3 (June 2015): 639–55. http://dx.doi.org/10.1017/s0308210513000541.

Full text

Abstract:

We show that a differential module is Gorenstein projective (injective, respectively) if and only if its underlying module is Gorenstein projective (injective, respectively). We then relate the Ringel–Zhang theorem on differential modules to the Avramov–Buchweitz–Iyengar notion of projective class of differential modules and prove that for a ring R there is a bijective correspondence between projectively stable objects of split differential modules of projective class not more than 1 and R-modules of projective dimension not more than 1, and this is given by the homology functor H and stable syzygy functor ΩD. The correspondence sends indecomposable objects to indecomposable objects. In particular, we obtain that for a hereditary ring R there is a bijective correspondence between objects of the projectively stable category of Gorenstein projective differential modules and the category of all R-modules given by the homology functor and the stable syzygy functor. This gives an extended version of the Ringel–Zhang theorem.

APA, Harvard, Vancouver, ISO, and other styles

42

Holm, Henrik. "Gorenstein homological dimensions." Journal of Pure and Applied Algebra 189, no. 1-3 (May 2004): 167–93. http://dx.doi.org/10.1016/j.jpaa.2003.11.007.

Full text

APA, Harvard, Vancouver, ISO, and other styles

43

Bennett-Tennenhaus, Raphael, and Amit Shah. "Transport of structure in higher homological algebra." Journal of Algebra 574 (May 2021): 514–49. http://dx.doi.org/10.1016/j.jalgebra.2021.01.019.

Full text

APA, Harvard, Vancouver, ISO, and other styles

44

Ciampella, A., and L. A. Lomonaco. "Homological computations in the universal Steenrod algebra." Fundamenta Mathematicae 183, no. 3 (2004): 245–52. http://dx.doi.org/10.4064/fm183-3-4.

Full text

APA, Harvard, Vancouver, ISO, and other styles

45

Asefa, Dadi. "Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras." Journal of Mathematics 2021 (November 19, 2021): 1–8. http://dx.doi.org/10.1155/2021/8127282.

Full text

Abstract:

Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A , how to construct all the Gorenstein-projective A -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ = A M B A 0 B . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ = A M B A 0 B .

APA, Harvard, Vancouver, ISO, and other styles

46

Ágoston, István, Vlastimil Dlab, and Erzsébet Lukács. "Homological Duality and Quasi-Heredity." Canadian Journal of Mathematics 48, no. 5 (October 1, 1996): 897–917. http://dx.doi.org/10.4153/cjm-1996-046-0.

Full text

Abstract:

AbstractThis paper represents a general study of the (Yoneda) Ext-algebra A* of a finite dimensional K-algebra A. Our motivation lies in the problem of establishing conditions under which (i) the species of A* coincides with the dual species of A and (ii) the quasi-heredity of A (or A*) yields the quasi-heredity of A* (or A, respectively). These questions are closely related to the Kazhdan—Lusztig Theory as presented by [CPS2]. The main results include introducing the concept of a solid algebra and the relevant Theorem 4.5 as well as a rather complete description of the situation in the case of monomial algebras in Section 5.

APA, Harvard, Vancouver, ISO, and other styles

47

Gurmeet, K., and I. B. S. Passi. "Centralizers and homological dimension." Communications in Algebra 22, no. 14 (January 1994): 5703–8. http://dx.doi.org/10.1080/00927879408825156.

Full text

APA, Harvard, Vancouver, ISO, and other styles

48

Borceux, Francis, Maria Manuel Clementino, Marino Gran, and Lurdes Sousa. "Protolocalisations of homological categories." Journal of Pure and Applied Algebra 212, no. 8 (August 2008): 1898–927. http://dx.doi.org/10.1016/j.jpaa.2007.11.005.

Full text

APA, Harvard, Vancouver, ISO, and other styles

49

Tanton, James Stuart. "A Homological Fibration forGl∞." Journal of Algebra 190, no. 2 (April 1997): 540–55. http://dx.doi.org/10.1006/jabr.1996.6915.

Full text

APA, Harvard, Vancouver, ISO, and other styles

50

Zhang, Yingying. "Excellent Extensions and Homological Conjectures." Algebra Colloquium 25, no. 04 (December 2018): 619–26. http://dx.doi.org/10.1142/s1005386718000433.

Full text

Abstract:

In this paper, we introduce the notion of excellent extensions of rings. Let Γ be an excellent extension of an Artin algebra Λ, we prove that Λ satisfies the Gorenstein symmetry conjecture (resp., finitistic dimension conjecture, Auslander–Gorenstein conjecture, Nakayama conjecture) if and only if so does Γ. As a special case of excellent extensions, when G is a finite group whose order is invertible in Λ acting on Λ and Λ is G-stable, we prove that if the skew group algebra ΛG satisfies strong Nakayama conjecture (resp., generalized Nakayama conjecture), then so does Λ.

APA, Harvard, Vancouver, ISO, and other styles

We offer discounts on all premium plans for authors whose works are included in thematic literature selections. Contact us to get a unique promo code!