Homological Properties Research Articles

An elementary approach to the homological properties of constant-rank operators

An elementary approach to the homological properties of constant-rank operators

How is a graph not like a manifold?

For an equivariantly formal action of a compact torus $T$ on a smooth manifold $X$ with isolated fixed points we investigate the global homological properties of the graded poset $S(X)$ of face submanifolds. We prove that the condition of the $j$-independency of tangent weights at each fixed point implies the $(j+1)$-acyclicity of the skeleta $S(X)_r$ for $r>j+1$. This result provides a necessary topological condition for a GKM graph to be a GKM graph of some GKM manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold of dimension $2n$ with an $(n-1)$-independent action of the $(n-1)$-dimensional torus, under certain colourability assumptions on its GKM graph. This description relates the equivariant cohomology algebra to the face algebra of a simplicial poset. This observation underlines a certain similarity between actions of complexity $1$ and torus manifolds. Bibliography: 27 titles.

A new construction of CAT(0) cube complexes

We introduce the notion of coupled link cube complex (CLCC) as a means of constructing interesting cocompactly cubulated groups. CLCCs are defined locally, making them a useful tool when precise control over the links is required. In this paper we study some general properties of CLCCs, such as their (co)homological dimension and criteria for hyperbolicity. Some examples of fundamental groups of CLCCs are RAAGs, RACGs, surface groups and some manifold groups. As immediate applications of our criteria we produce a number of cubulated 3 and 4 manifolds with hyperbolic fundamental group.

On homological properties of strict polynomial functors of degree p

We study the homological algebra in the category Pp of strict polynomial functors of degree p over a field of positive characteristic p. We determine the decomposition matrix of our category and we calculate the Ext-groups between functors important from the point of view of representation theory. Our results include computations of the Ext-algebras of simple functors and Schur functors. We observe that the category Pp has a Kazhdan-Lusztig theory and we show that the DG algebras computing the Ext-algebras for simple functors and Schur functors are formal. These last results allow one to describe the bounded derived category of Pp as derived categories of certain explicitly described graded algebras. We also generalize our results to all blocks of p-weight 1 in Pe for e>p.

Module structure of the homology of right-angled Artin kernels

In this paper, we study the module structure of the homology of Artin kernels, i.e., kernels of non-resonant characters from right-angled Artin groups onto the integer numbers, the module structure being with respect to the ring $\mathbb{K}[t^{\pm 1}]$, where $\mathbb{K}$ is a field of characteristic zero. Papadima and Suciu determined some part of this structure by means of the flag complex of the graph of the Artin group. In this work, we provide more properties of the torsion part of this module, e.g., the dimension of each primary part and the maximal size of Jordan forms (if we interpret the torsion structure in terms of a linear map). These properties are stated in terms of homology properties of suitable filtrations of the flag complex and suitable double covers of an associated toric complex.

Homogenized skew PBW extensions

In this paper, we provide a new and more general filtration to the family of noncommutative rings known as skew PBW extensions. We introduce the notion of sigma -filtered skew PBW extension and study some homological properties of these algebras. We show that the homogenization of a sigma -filtered skew PBW extension A over a ring R is a graded skew PBW extension over the homogenization of R. Using this fact, we prove that if the homogenization of R is Auslander-regular, then the homogenization of A is a domain Noetherian, Artin–Schelter regular, and A is Noetherian, Zariski and (ungraded) skew Calabi–Yau.

Gorenstein FP∞-Injective Modules and w-Noetherian Rings

We study some homological properties of Gorenstein [Formula: see text]-injective modules, and we prove (1) a ring [Formula: see text]is not necessarily coherent if every Gorenstein [Formula: see text]-injective [Formula: see text]-module is injective, and (2) a ring [Formula: see text] is not necessarily coherent if every Gorenstein injective [Formula: see text]-module is injective. In addition, we characterize [Formula: see text]-Noetherian rings in terms of Gorenstein [Formula: see text]-injective modules, and we prove that a ring [Formula: see text] is [Formula: see text]-Noetherian if and only if every GV-torsion-free FP-injective [Formula: see text]-module is Gorenstein [Formula: see text]-injective, if and only if any direct sum of GV-torsion-free FP-injective [Formula: see text]-modules is Gorenstein [Formula: see text]-injective.

Free Factors and Profinite Completions

Abstract Can one detect free products of groups via their profinite completions? We answer positively among virtually free groups. More precisely, we prove that a subgroup of a finitely generated virtually free group $G$ is a free factor if and only if its closure in the profinite completion of $G$ is a profinite free factor. This generalises results by Parzanchevski and Puder for free groups that were later also proved by Wilton. Our methods are entirely different to theirs, combining homological properties of profinite groups and the decomposition theory of Dicks and Dunwoody.

Quasi-Gorenstein projective and quasi-Gorenstein injective modules

In this paper, we introduce quasi-Gorenstein projective and quasi-Gorenstein injective modules, study some of their properties and investigate their behavior with respect to short exact sequences. Also, we give homological descriptions of quasi-Gorenstein homological dimensions of modules when they are finite and in this case, we prove that quasi-Gorenstein projective precover and quasi-Gorenstein injective preenvelope exist.

Some homological properties of Borel type ideals

ABSTRACT We study ideals of Borel type, including k-Borel ideals and t-spread Veronese ideals. We determine their free resolutions and their homological shift ideals. The multiplicity and the analytic spread of equigenerated squarefree principal Borel ideals are computed. For the multiplicity, the result is given under an additional assumption which is always satisfied for squarefree principal Borel ideals. These results are used to analyze the behavior of height, multiplicity and analytic spread of the homological shift ideals as functions of j, when I is an equigenerated squarefree Borel ideal.

Homological properties of trivial ring extensions

Let [Formula: see text] be a trivial extension of a ring [Formula: see text] by an [Formula: see text]-[Formula: see text]-bimodule [Formula: see text]. We first study the transfer of some rings such as left [Formula: see text]-rings, left Kasch rings, left PS rings between [Formula: see text] and [Formula: see text]. Then we investigate how to construct [Formula: see text]-tilting modules and Gorenstein modules in [Formula: see text]-Mod from [Formula: see text]-Mod. Finally we exhibit the connections between preenvelopes (respectively, precovers) in [Formula: see text]-Mod and preenvelopes (respectively, precovers) in [Formula: see text]-Mod.

Homological properties of pinched Veronese rings

Pinched Veronese rings are formed by removing an algebra generator from a Veronese subring of a polynomial ring. We study the homological properties of such rings, including the Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco and Martino classified Cohen-Macaulayness of pinched Veronese rings by the maximum entry of the exponent vector of the pinched monomial; we re-prove their results with semigroup methods and correct an omission of a small class of examples of Cohen-Macaulay pinched Veronese rings. When the underlying field is of prime characteristic, we show that pinched Veronese rings exhibit a variety of F-singularities, including F-regular, F-injective, and F-nilpotent. We also compute upper bounds on the Frobenius test exponents of pinched Veronese rings, a computational invariant which controls the Frobenius closure of all parameter ideals simultaneously.

Rigidity dimensions of Hochschild extensions of hereditary algebras of type D

Rigidity dimension of algebras is a new homological dimension which measures the quality of resolutions of algebras by algebras of finite global dimension and large dominant dimension. In this paper, we calculate the rigidity dimension of the Hochschild extension of a hereditary algebra H of Dynkin type D n ( n ≥ 4 ) by the standard duality H -bimodule. It turns out that this dimension is independent of n and the orientation of D n .

Categorical Torelli theorems: results and open problems

We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible subcategories of the bounded derived category of coherent sheaves of such a variety. The focus is on Enriques surfaces, prime Fano threefolds and cubic fourfolds.

Homological Properties of Some Types of Generalized Matrix Banach Algebras

Homological Properties of Some Types of Generalized Matrix Banach Algebras

Some interactions between Hopf Galois extensions and noncommutative rings

In this paper, our objects of interest are Hopf Galois extensions (e.g., Hopf algebras, Galois field extensions, strongly graded algebras, crossed products, principal bundles, etc.) and families of noncommutative rings (e.g., skew polynomial rings, PBW extensions and skew PBW extensions, etc.) We collect and systematize questions, problems, properties and recent advances in both theories by explicitly developing examples and doing calculations that are usually omitted in the literature. In particular, for Hopf Galois extensions we consider approaches from the point of view of quantum torsors (also known as quantum heaps) and Hopf Galois systems, while for some families of noncommutative rings we present advances in the characterization of ring-theoretic and homological properties. Every developed topic is exemplified with abundant references to classic and current works, so this paper serves as a survey for those interested in either of the two theories. Throughout, interactions between both are presented.

Association study of leptin receptor polymorphisms in women with obesity and their impact on protein domains: a case-control study and in silico analyses

Leptin receptor (LEPR) is a member of the class I cytokine receptor family that receives and transmits leptin signals. It is primarily involved in the regulation of energy expenditure and food intake. This study aimed to evaluate the association of LEPR gene polymorphisms, Lys109Arg, Gln223Arg and Lys656Asn, with obesity in Moroccan women and to explore the structural and functional consequences of these SNPs. The variants were genotyped using the Sanger sequencing method. The three-dimensional structures of LEPR extracellular domains were determined using a template-based tertiary structure modeling web server and the protein variants were generated using in silico mutagenesis. The amino acids conservation analysis in the variants region was performed based on a protein’s evolutionary profile. The molecular dynamics simulations of the wild-types and variants N-terminal, cytokine receptor homology I and fibronectin type III domains of LEPR protein were performed to investigate their impact on the domain structures. We identified that only Lys656Asn polymorphism is associated with obesity in Moroccan women (P = 0.024). In silico analyses revealed that Lys109, Gln223 and Lys656 are exposed residues and their substitution leads to changes in protein structure through loss or gain of hydrogen bonds and hydrophobic interactions. Lys656Asn increases the stability and decreased flexibility of the fibronectin type III domain. Lys109Arg highly decreases the stability and increases flexibility and the overall dimension of N-terminal and cytokine receptor homology I domains. Gln223Arg increases the stability and the compaction level of these domains. These results provide insight into the involvement of LEPR variants in obesity development. Communicated by Ramaswamy H. Sarma

Frobenius functors and Gorenstein homological properties

We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it preserves and reflects Gorenstein projective objects. We give conditions on when a Frobenius functor preserves the stable categories of Gorenstein projective objects, the singularity categories and the Gorenstein defect categories, respectively. In the appendix, we give a direct proof of the following known result: for an abelian category with enough projectives and injectives, its global Gorenstein projective dimension coincides with its global Gorenstein injective dimension.

Some homological properties of the enveloping dual Banach algebra

Some homological properties of the enveloping dual Banach algebra

Impact of Ca2+-Induced PI(4,5)P2 Clusters on PH-YFP Organization and Protein-Protein Interactions.

Despite its low abundance, phosphatidylinositol 4,5-bisphosphate (PI(4,5)P2) is a key modulator of membrane-associated signaling events in eukaryotic cells. Temporal and spatial regulation of PI(4,5)P2 concentration can achieve localized increases in the levels of this lipid, which are crucial for the activation or recruitment of peripheral proteins to the plasma membrane. The recent observation of the dramatic impact of physiological divalent cation concentrations on PI(4,5)P2 clustering, suggests that protein anchoring to the plasma membrane through PI(4,5)P2 is likely not defined solely by a simple (monomeric PI(4,5)P2)/(protein bound PI(4,5)P2) equilibrium, but instead depends on complex protein interactions with PI(4,5)P2 clusters. The insertion of PI(4,5)P2-binding proteins within these clusters can putatively modulate protein–protein interactions in the membrane, but the relevance of such effects is largely unknown. In this work, we characterized the impact of Ca2+ on the organization and protein–protein interactions of PI(4,5)P2-binding proteins. We show that, in giant unilamellar vesicles presenting PI(4,5)P2, the membrane diffusion properties of pleckstrin homology (PH) domains tagged with a yellow fluorescent protein (YFP) are affected by the presence of Ca2+, suggesting direct interactions between the protein and PI(4,5)P2 clusters. Importantly, PH-YFP is found to dimerize in the membrane in the absence of Ca2+. This oligomerization is inhibited in the presence of physiological concentrations of the divalent cation. These results confirm that cation-dependent PI(4,5)P2 clustering promotes interactions between PI(4,5)P2-binding proteins and has the potential to dramatically influence the organization and downstream interactions of PI(4,5)P2-binding proteins in the plasma membrane.