Homological Algebra Research Articles

About Supports of Local Cohomology Modules

The article provides the relation between the theory of local cohomology modules, and vanishing results, and also about the theory of support of such modules. Here, we put results about the theory, and also we provide a relation of local cohomology in the theory of commutative algebra and homological algebra.

Simplicial homology of differential graded algebras of Lie algebras

Simplicial homology of differential graded algebras of Lie algebras

Diagonal tensor algebra and naïve liftings

The notion of naïve lifting of differential graded (DG) modules along certain DG algebra extensions was introduced by the authors for the purpose of studying problems in homological commutative algebra that involve self-vanishing of Ext. Our goal in this paper is to deeply study naïve lifting property using the diagonal tensor algebra. Our main result provides several characterizations of naïve liftability of DG modules under certain Ext vanishing conditions. As an application, we affirmatively answer a question posed by the authors regarding a specific characterization of naïve liftability.

Higher-dimensional Heegaard Floer homology and Hecke algebras

Given a closed oriented surface \Sigma of genus greater than 0, we construct a map \mathcal{F} from the wrapped higher-dimensional Heegaard Floer homology of the cotangent fibers of T^{*}\Sigma to the Hecke algebra associated to \Sigma and show that \mathcal{F} is an isomorphism of algebras. We also establish analogous results for punctured surfaces.

The module structure of a group action on a ring

Consider a finite group G acting on a graded Noetherian k-algebra S, for some field k of characteristic p; for example S might be a polynomial ring. Regard S as a kG-module and consider the multiplicity of a particular indecomposable module as a summand in each degree. We show how this can be described in terms of homological algebra and how it is linked to the geometry of the group action on the spectrum of S.

Idempotents and homology of diagram algebras

AbstractThis paper provides a systematization of some recent results in homology of algebras. Our main theorem gives criteria under which the homology of a diagram algebra is isomorphic to the homology of the subalgebra on diagrams having the maximum number of left-to-right connections. From this theorem, we deduce the ‘invertible-parameter’ cases of the Temperley–Lieb and Brauer results of Boyd–Hepworth and Boyd–Hepworth–Patzt. We are also able to give a new proof of Sroka’s theorem that the homology of an odd-strand Temperley–Lieb algebra vanishes, as well as an analogous result for Brauer algebras and an interpretation of both results in the even-strand case. Our proofs are relatively elementary: in particular, no auxiliary chain complexes or spectral sequences are required. We briefly discuss the relationship to cellular algebras in the sense of Graham–Lehrer.

The definable content of homological invariants I: Ext$\mathrm{Ext}$ and lim1$\mathrm{lim}^1$

AbstractThis is the first installment in a series of papers illustrating how classical invariants of homological algebra and algebraic topology may be enriched with additional descriptive set theoretic information. To effect this enrichment, we show that many of these invariants may be naturally regarded as functors to the category, introduced herein, of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification. In the present work we focus on the first derived functors of and . The resulting definable for pairs of countable abelian groups and definable for towers of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable is a fully faithful contravariant functor from the category of finite‐rank torsion‐free abelian groups with no free summands; this contrasts with the fact that there are uncountably many non‐isomorphic such groups with isomorphic classical invariants . To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover; within this framework we prove several rigidity results for non‐Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of ‐adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem of classifying all group extensions of by up to base‐free isomorphism, when for prime numbers and .

Quantum van Est isomorphism

Motivated by the fact that the Hopf-cyclic (co)homologies of function algebras over Lie groups and universal enveloping algebras over Lie algebras capture the Lie group and Lie algebra (co)homologies, we hereby upgrade the classical van Est isomorphism to ones between the Hopf-cyclic (co)homologies of quantized algebras of functions and quantized universal enveloping algebras, both in h -adic and q -deformation frameworks.

Koszul Complexes and Relative Homological Algebra of Functors Over Posets

AbstractUnder certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

Complexity of Supersymmetric Systems and the Cohomology Problem

We consider the complexity of the local Hamiltonian problem in the context of fermionic Hamiltonians with N=2 supersymmetry and show that the problem remains QMA-complete. Our main motivation for studying this is the well-known fact that the ground state energy of a supersymmetric system is exactly zero if and only if a certain cohomology group is nontrivial. This opens the door to bringing the tools of Hamiltonian complexity to study the computational complexity of a large number of algorithmic problems that arise in homological algebra, including problems in algebraic topology, algebraic geometry, and group theory. We take the first steps in this direction by introducing the k-local Cohomology problem and showing that it is QMA1-hard and, for a large class of instances, is contained in QMA. We then consider the complexity of estimating normalized Betti numbers and show that this problem is hard for the quantum complexity class DQC1, and for a large class of instances is contained in BQP. In light of these results, we argue that it is natural to frame many of these homological problems in terms of finding ground states of supersymmetric fermionic systems. As an illustration of this perspective we discuss in some detail the model of Fendley, Schoutens, and de Boer consisting of hard-core fermions on a graph, whose ground state structure encodes l-dimensional holes in the independence complex of the graph. This offers a new perspective on existing quantum algorithms for topological data analysis and suggests new ones.

On the second relative homology of Leibniz algebras

On the second relative homology of Leibniz algebras

Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings

Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings

Gorenstein Krull domains and their factor rings

The concept of Gorenstein Krull domains can be viewed as a Gorenstein analogue of Krull domains, which are not necessarily integrally closed. This allows us to study those domains from the point view of Gorenstein homological algebra. By the so-called w-operation, we prove that an integral domain is Gorenstein Krull if and only if for any nonzero nonunit, the Gorenstein global dimension of its w-factor ring is zero.

The completion of d-abelian categories

Let A be a finite-dimensional algebra, and M be a d-cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d-cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as an essentially small d-abelian category, known to be equivalent to a d-cluster tilting subcategory of an abelian category A. The completion of M, denoted by Ind(M), is defined as the universal completion of M with respect to filtered colimits. We explore Ind(M) and demonstrate its equivalence to the full subcategory Ld(M) of ModM, comprising left d-exact functors. Notably, Ind(M) as a subcategory of ModMEff(M) falls short of being a d-cluster tilting subcategory since it satisfies all properties of a d-cluster tilting subcategory except d-rigidity. For a d-cluster tilting subcategory M of mod A, M→ consists of all filtered colimits of objects from M, is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M→ is a d-rigid subcategory remains unanswered. However, if it is indeed d-rigid, it qualifies as a d-cluster tilting subcategory. In the case d=2, employing cotorsion theory, we establish that M→ is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether M→ is a d-cluster tilting subcategory of Mod A appears to be equivalent to Iyama's question about the finiteness of M. Furthermore, for general d, we address the problem and present several equivalent conditions for Iyama's question.

The Report of Homological Algebra

The Report of Homological Algebra

The homology of the partition algebras

The homology of the partition algebras

Representation Theory of Quivers and Finite-Dimensional Algebras

This workshop was about the representation theory of quivers and finite-dimensional (associative) algebras, and links to other areas of mathematics, including other areas of representation theory, homological algebra, cluster algebras, algebraic geometry and singularity theory. Particularly active topics included \tau -tilting theory, algebras arising from surface triangulations and the study of exact categories and their generalizations.

Computations Stiefel-Whitney classes of real representations for non-prime power groups

In the current study, the researchers conduct computation of the Stiefel- Whitney classes of real representations of non-prime power groups such as Mathieu groups, symmetric groups, alternating groups and Janko groups. Stiefel- Whitney classes are conducted by using polytope convex hull with vector in n-dimention our computation using the HAP (Homological Algebra Programming.

Simplicial approach to path homology of quivers, marked categories, groups and algebras

AbstractWe develop a generalisation of the path homology theory introduced by Grigor'yan, Lin, Muranov and Yau (GLMY theory) in a general simplicial setting. The new theory includes as particular cases the GLMY theory for path complexes and new homology theories: path homology of categories with a chosen set of morphisms (marked categories) groups with a chosen subset (marked groups) and path Hochschild homology of algebras with chosen vector subspaces (marked algebras). Using our general machinery, we also introduce a new homology theory for quivers that we call square‐commutative homology of quivers and compare it with the theory developed by Grigor'yan, Muranov, Vershinin and Yau.

On functors preserving projective resolutions

It is important for applications of Homological Algebra in Representation Theory to have control over the behavior of (minimal) projective resolutions under various functors. In this paper, we describe three broad families of functors that preserve such resolutions. We will use these results in our work on Representation Theory of Schur algebras.