Homological Properties Research Articles

Nakayama Algebras of Small Homological Dimension and Pattern Avoiding Permutations

Nakayama Algebras of Small Homological Dimension and Pattern Avoiding Permutations

Unforeseen observations about the homological dimension of Hilbert spaces

Unforeseen observations about the homological dimension of Hilbert spaces

Rigidity dimensions of self-injective Nakayama algebras

Rigidity dimension is a new homological dimension which is intended to measure the quality of the best resolution of an algebra. In this paper, we determine the rigidity dimensions of self-injective Nakayama algebras An,m with n simple modules and the Loewy length m⩾n.

Characteristic modules and Gorenstein (co-)homological dimension of groups

In this paper, we examine the Gorenstein dimension of modules over the group algebra kG of a group G with coefficients in a commutative ring k. As a Gorenstein analogue of the classical case, we bound this dimension in terms of the Gorenstein dimension of the underlying k-module and the Gorenstein dimension of G over k. Our method is based on the notion of a characteristic module for G, introduced by the second author, and uses the stability properties of the Gorenstein categories. We also examine the class of hierarchically decomposable groups defined by Kropholler and use the module of bounded Z-valued functions on such a group G to characterize the Gorenstein flat ZG-modules, in terms of flat modules, and the Gorenstein injective ZG-modules, in terms of injective modules (by complete analogy with the characterization of Gorenstein projective ZG-modules, in terms of projective modules, obtained by Dembegioti and the second author). It follows that, for a group G in Kropholler's class, (a) any Gorenstein projective ZG-module is Gorenstein flat and (b) a ZG-module is Gorenstein flat if its Pontryagin dual module is Gorenstein injective.

The May filtration on THH and faithfully flat descent

In this article, we study descent properties of topological Hochschild homology and topological cyclic homology. In particular, we verify that both of these invariants satisfy faithfully flat descent and 1-connective descent for connective E2-ring spectra. This generalizes a result of Bhatt–Morrow–Scholze from [6] and a result of Dundas–Rognes from [11], respectively. Along the way, we develop some basic theory for cobar constructions and give an alternative presentation of the May filtration on topological Hochschild homology, originally due to Angelini-Knoll–Salch [3].

Homological Properties of Persistent Homology

Homological Properties of Persistent Homology

On cohomological dimension of homomorphisms

The (co)homological dimension of a homomorphism ϕ : G → H \phi :G\to H is the maximal number k k such that the induced homomorphism in k k -th (co)homology groups is nonzero for coefficients in some H H -module. It is known that for geometrically finite groups G , c d ( G ) = h d ( G ) G, cd(G)=hd(G) and c d ( G × G ) = 2 c d ( G ) cd(G\times G)=2cd(G) . We prove analogous theorems for homomorphisms of geometrically finite groups. The analogy stops working on the Eilenberg-Ganea equality c d ( G ) = g d ( G ) cd(G)=gd(G) where c d ( G ) > 2 cd(G)>2 and g d ( G ) gd(G) is the geometric dimension of G G . We show that for every k > 2 k>2 there is a group homomorphism ϕ k : π k → Z k \phi _k:\pi _k\to \mathbb {Z}^k with c d ( ϕ k ) > k cd(\phi _k)>k and g d ( ϕ k ) = k gd(\phi _k)=k where π k \pi _k is the fundamental group of a closed aspherical ( k + 1 ) (k+1) -dimensional manifold.

Homological Properties of Ideals Generated by Fold Products of Linear Forms

Homological Properties of Ideals Generated by Fold Products of Linear Forms

Certain homological properties of triangular matrix rings

In this paper, we study the (Gorenstein) flat dimension of modules over triangular matrix rings, and give estimations of the (Gorenstein) weak global dimension and the supremum of flat lengths of injective modules over triangular matrix rings. As applications, some new Ding–Chen rings are constructed.

Width and Local Homology Dimension for Triangulated Categories

Let T be a compactly generated triangulated category. In this paper, the width and local homology dimension of an object X with respect to a homogeneous ideal a, widthR(a,X) and hd(a,X), respectively, are introduced. The local nature and some basic properties of widthR(a,X) and hd(a,X) are provided. In addition, we give an upper bound and lower bound of widthR(a,X). What is more, we give the relationship between the local homology dimension hd(a,X) and the arithmetic rank of a and dimR.

On Noetherian algebras, Schur functors and Hemmer–Nakano dimensions

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander’s correspondence, classical Schur–Weyl duality and Soergel’s Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B B , and module categories of endomorphism algebras of generator-relative cogenerators over B B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n n so that the endomorphism algebra of a generator-cogenerator provides an n n -faithful cover, in the sense of Rouquier, of B B . The degree n n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B B is bounded above by the number of non-isomorphic simple modules of B B . We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category O \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category O \mathcal {O} establishing an integral version of Soergel’s Struktursatz. We show that deformations of the combinatorial Soergel’s functor have better homological properties than the classical one.

Comparative Analysis of Hydrosol Volatile Components of Citrus × Aurantium 'Daidai' and Citrus × Aurantium L. Dried Buds with Different Extraction Processes Using Headspace-Solid-Phase Microextraction with Gas Chromatography-Mass Spectrometry.

This work used headspace solid-phase microextraction with gas chromatography-mass spectrometry (HS-SPME-GC-MS) to analyze the volatile components of hydrosols of Citrus × aurantium 'Daidai' and Citrus × aurantium L. dried buds (CAVAs and CADBs) by immersion and ultrasound-microwave synergistic-assisted steam distillation. The results show that a total of 106 volatiles were detected in hydrosols, mainly alcohols, alkenes, and esters, and the high content components of hydrosols were linalool, α-terpineol, and trans-geraniol. In terms of variety, the total and unique components of CAVA hydrosols were much higher than those of CADB hydrosols; the relative contents of 13 components of CAVA hydrosols were greater than those of CADB hydrosols, with geranyl acetate up to 15-fold; all hydrosols had a citrus, floral, and woody aroma. From the pretreatment, more volatile components were retained in the immersion; the relative contents of linalool and α-terpineol were increased by the ultrasound-microwave procedure; and the ultrasound-microwave procedure was favorable for the stimulation of the aroma of CAVA hydrosols, but it diminished the aroma of the CADB hydrosols. This study provides theoretical support for in-depth exploration based on the medicine food homology properties of CAVA and for improving the utilization rate of waste resources.

Categories of quiver representations and relative cotorsion pairs

We study the category Rep(Q,C) of representations of a quiver Q with values in an abelian category C. For this purpose we introduce the mesh and the cone-shape cardinal numbers associated to the quiver Q and we use them to impose conditions on C that allow us to prove interesting homological properties of Rep(Q,C) that can be constructed from C. For example, we compute the global dimension of Rep(Q,C) in terms of the global one of C. We also review a result of H. Holm and P. Jørgensen which states that (under certain conditions on C) every hereditary complete cotorsion pair (A,B) in C induces the hereditary complete cotorsion pairs (Rep(Q,A),Rep(Q,A)⊥1) and (Ψ⊥1(B),Ψ(B)) in Rep(Q,C), and then we obtain a strengthened version of this and other related results. Finally, we will apply the above developed theory to study the following full abelian subcategories of Rep(Q,C), finite-support, finite-bottom-support and finite-top-support representations. We show that the above mentioned cotorsion pairs in Rep(Q,C) can be restricted nicely on the aforementioned subcategories and under mild conditions we also get hereditary complete cotorsion pairs.

Products and powers of principal symmetric ideals

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

Universal distances for extended persistence

The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen–Steiner, Edelsbrunner, and Harer. We address the question of universality, which asks for the largest possible stable distance on extended persistence diagrams, showing that a more discriminative variant of the bottleneck distance is universal. Our result applies more generally to settings where persistence diagrams are considered only up to a certain degree. We achieve our results by establishing a functorial construction and several characteristic properties of relative interlevel set homology, which mirror the classical Eilenberg–Steenrod axioms. Finally, we contrast the bottleneck distance with the interleaving distance of sheaves on the real line by showing that the latter is not intrinsic, let alone universal. This particular result has the further implication that the interleaving distance of Reeb graphs is not intrinsic either.

Homological dimensions of the Jacobson radical

This work presents results on the finiteness, and on the symmetry properties, of various homological dimensions associated to the Jacobson radical and its higher syzygies, of a semiperfect ring.

Structural investigation of Trypanosoma cruzi Akt-like kinase as drug target against Chagas disease

According to the World Health Organization, Chagas disease (CD) is the most prevalent poverty-promoting neglected tropical disease. Alarmingly, climate change is accelerating the geographical spreading of CD causative parasite, Trypanosoma cruzi, which additionally increases infection rates. Still, CD treatment remains challenging due to a lack of safe and efficient drugs. In this work, we analyze the viability of T. cruzi Akt-like kinase (TcAkt) as drug target against CD including primary structural and functional information about a parasitic Akt protein. Nuclear Magnetic Resonance derived information in combination with Molecular Dynamics simulations offer detailed insights into structural properties of the pleckstrin homology (PH) domain of TcAkt and its binding to phosphatidylinositol phosphate ligands (PIP). Experimental data combined with Alpha Fold proposes a model for the mechanism of action of TcAkt involving a PIP-induced disruption of the intramolecular interface between the kinase and the PH domain resulting in an open conformation enabling TcAkt kinase activity. Further docking experiments reveal that TcAkt is recognized by human inhibitors PIT-1 and capivasertib, and TcAkt inhibition by UBMC-4 and UBMC-6 is achieved via binding to TcAkt kinase domain. Our in-depth structural analysis of TcAkt reveals potential sites for drug development against CD, located at activity essential regions.

Ring homomorphisms and local rings with quasi-decomposable maximal ideal

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper showed that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms.

Hochschild homology dimension and a class of Hochschild extension algebras of truncated quiver algebras

In this paper, we show that higher Hochschild homology groups do not vanish for a class of Hochschild extension algebras of truncated quiver algebras by the standard duality module. Consequently, this result extends the result of the trivial extension algebra by the standard duality module for truncated quiver algebras. Moreover, as an application, we show that for any Hochschild extension algebra of selfinjective Nakayama algebras by the standard duality module, its Hochschild homology dimension is infinite.

(X,Y)-Gorenstein Categories, Associated (Global) Homological Dimensions and Applications to Relative Foxby Classes

Recently, Gorenstein dimensions relative to a semidualizing module have been the subject of numerous studies with interesting extensions of the classical homological dimensions. Although all these studies share the same direction, a common basis, and similar final goals, there is no common framework encompassing them as parts of a whole, progressing, on different fronts, towards the same end. We provide this general and global framework in the context of abelian categories, standardizing terminology and notation: we establish a general context by defining Gorenstein categories relative to two classes of objects ((X,Y)-Gorenstein categories, denoted G(X,Y)), and carry out a study of the homological dimensions associated with them. We prove, under some mild standard conditions, the corresponding version of the Comparison Lemma that ensures the consistency of a homological-dimension theory. We show that Ext functors can be used as tools to compute these G(X,Y)-dimensions, and we compare the dimensions obtained using the classes G(X) with those computed using G(X,Y). We also initiate a research of the global dimensions obtained with these classes G(X,Y) and find conditions for them to be finite. Finally, we show that these classes of Gorenstein objects are closely and interestingly related to the Foxby classes induced by a pair of functors. Namely, we prove that the Auslander and Bass classes are indeed G(X,Y) categories for some specific classes X and Y.