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Tiago Cruz

Publications and source records attributed to Tiago Cruz.

16 recordsLinked to original sources
arXiv2025

A new formula for the classical dominant dimension using bimodules

We show that a faithful projective-injective module over a finite-dimensional algebra $A$ has the double centraliser property if and only if $A$ as a bimodule is reflexive. More generally, we provide a new characterisation of the classical dominant dimension by showing that having dominant dimension at least $n$ is equivalent to the bimodule $A$ being $n$-torsion-free. This allows us to find new connections between the classical Tachikawa and Nakayama conjectures and Gorenstein homological algebra. Furthermore, we use our results to give new interpretations of Hochschild (co)homology of finite-dimensional algebras using higher Auslander-Reiten translates and the canonical bimodule in the sense of Fang, Kerner and Yamagata.

math.RT↗
arXiv2025

Homological dimensions of Schur algebras $S(p,2p)$ and an Auslander-type correspondence

We study the homological properties of Schur algebras $S(p, 2p)$ over a field $k$ of positive characteristic $p$, focusing on their interplay with the representation theory of quotients of group algebras of symmetric groups via Schur-Weyl duality. Schur-Weyl duality establishes that the centraliser algebra, $\Lambda(p, 2p)$, of the tensor space $(k^p)^{\otimes 2p}$ (as a module over $S(p, 2p)$) is a quotient of the group algebra of the symmetric group. In this paper, we prove that Schur-Weyl duality between $S(p, 2p)$ and $\Lambda(p, 2p)$ is an instance of an Auslander-type correspondence. We compute the global dimension of Schur algebras $S(p, 2p)$ and their relative dominant dimension with respect to the tensor space $(k^p)^{\otimes 2p}$. In particular, we show that the pair $(S(p, 2p), (k^p)^{\otimes 2p})$ forms a relative $4(p-1)$-Auslander pair in the sense of Cruz and Psaroudakis, thereby connecting Schur algebras with higher homological algebra. Moreover, we determine the Hemmer-Nakano dimension associated with the quasi-hereditary cover of $\Lambda(p, 2p)$ that arises from Schur-Weyl duality. As an application, we show that the direct sum of some Young modules over $\Lambda(p, 2p)$ is a full tilting module when $p>2$.

math.RT↗
arXiv2025

An Auslander-Buchsbaum formula for higher Auslander algebras and applications

We provide a new non-commutative generalisation of the Auslander-Buchsbaum formula for higher Auslander algebras and use this to show that the class of tilted Auslander algebras, studied recently by Zito, and QF-1 algebras of global dimension at most 2, studied by Ringel in the 1970s, coincide. We furthermore give an explicit classification of this class of algebras and present generalisations to higher homological dimensions with a new local characterisation of QF-1 higher Auslander algebras.

math.RT↗
arXiv2024

A higher dimensional Auslander-Iyama-Solberg correspondence

In this paper, we prove a higher dimensional version of Auslander-Iyama-Solberg correspondence. Iyama and Solberg have shown a bijection between $n$-minimal Auslander-Gorenstein algebras and $n$-precluster tilting modules. If $A$ is an $n$-minimal Auslander-Gorenstein algebra, then the pair $(A,P)$ is a relative $(n+1)$-Auslander-Gorenstein pair in the sense of the authors, where $P$ is the minimal faithful projective-injective left $A$-module. We establish a higher dimensional Auslander-Iyama-Solberg, where $P$ is replaced by any self-orthogonal module $Q$ having finite projective and injective dimension. This new correspondence provides a bijection between relative Auslander--Gorenstein pairs and a new class of objects that generalise precluster tilting modules. This way, we obtain a new correspondence coming from the modular representation theory of general linear groups.

math.RT↗
arXiv2024

Characteristic tilting modules and Ringel duality in the Noetherian world

The foundations of Ringel duality for split quasi-hereditary algebras over commutative Noetherian rings are strengthened. Several descriptions and properties of the smallest resolving subcategory containing all standard modules over split quasi-hereditary algebras over commutative Noetherian rings are provided. In particular, given two split quasi-hereditary algebras $A$ and $B$, we prove that any exact equivalence between the smallest resolving subcategory containing all standard modules over $A$ and the smallest resolving subcategory containing all standard modules over $B$ lifts to a Morita equivalence between $A$ and $B$ which preserves the quasi-hereditary structure.

math.RT↗
arXiv2023

Higher Morita-Tachikawa correspondence

Important correspondences in representation theory can be regarded as restrictions of the Morita--Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo-symmetric algebras. Explicitly, the Morita--Tachikawa correspondence describes that endomorphism algebras of generators-cogenerators over finite-dimensional algebras are exactly the finite-dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi-generators and quasi-cogenerators which generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita--Tachikawa correspondence that takes into account relative dominant dimension with respect to a self-orthogonal module with arbitrary projective and injective dimension. These new versions also hold over Noetherian algebras which are finitely generated and projective over a commutative Noetherian ring.

math.RT↗
arXiv2023

Relative Auslander--Gorenstein Pairs

In this paper, we introduce and study relative Auslander--Gorenstein pairs. This consists of a finite-dimensional Gorenstein algebra together with a self-orthogonal module that provides a further homological feature of the algebra in terms of relative dominant dimension. These pairs will be called relative Auslander pairs whenever the algebra in question has finite global dimension. We characterize relative Auslander pairs by the existence and uniqueness of tilting-cotilting modules having higher values relative dominant and codominant dimension with respect to the self-orthogonal module. The same characterisation remains valid for relative Auslander--Gorenstein pairs if the self-orthogonal module has injective or projective dimension at most one. Our relative approach generalises and unifies the known results from the literature, for instance, the characterization of minimal Auslander--Gorenstein algebras. As an application of our methods, we prove that for any relative Auslander pair pieces of the module category of the endomorphism algebra of the self-orthogonal module can be identified with pieces of the module category of the endomorphism algebra of the unique tilting-cotilting module associated with the relative Auslander pair. We provide explicit examples of relative Auslander pairs.

math.RT↗
arXiv2022

Quasi-hereditary covers of Temperley-Lieb algebras and relative dominant dimension

Many connections and dualities in representation theory can be explained using quasi-hereditary covers in the sense of Rouquier. The concepts of relative dominant and codominant dimension with respect to a module, introduced recently by the first-named author, are important tools to evaluate and classify quasi-hereditary covers. In this paper, we prove that the relative dominant dimension of the regular module of a quasi-hereditary algebra with a simple preserving duality with respect to a summand $Q$ of a characteristic tilting module equals twice the relative dominant dimension of a characteristic tilting module with respect to $Q$. To resolve the Temperley-Lieb algebras of infinite global dimension, we apply this result to the class of Schur algebras $S(2, d)$ and $Q=V^{\otimes d}$ the $d$-tensor power of the 2-dimensional module and we completely determine the relative dominant dimension of the Schur algebra $S(2, d)$ with respect to $V^{\otimes d}$. The $q$-analogues of these results are also obtained. As a byproduct, we obtain a Hemmer-Nakano type result connecting the Ringel duals of $q$-Schur algebras and Temperley-Lieb algebras. From the point of view of Temperley-Lieb algebras, we obtain the first complete classification of their connection to their quasi-hereditary covers formed by Ringel duals of $q$-Schur algebras. These results are compatible with the integral setup, and we use them to deduce that the Ringel dual of a $q$-Schur algebra over the ring of Laurent polynomials over the integers together with some projective module is the best quasi-hereditary cover of the integral Temperley-Lieb algebra.

math.RT↗
arXiv2022

On split quasi-hereditary covers and Ringel duality

In this paper, we develop two new homological invariants called relative dominant dimension with respect to a module and relative codominant dimension with respect to a module. These are used to establish precise connections between Ringel duality, split quasi-hereditary covers and double centraliser properties. These homological invariants are studied over Noetherian algebras which are finitely generated and projective as a module over the ground ring and they are shown to behave nicely under change of rings techniques. It turns out that relative codominant dimension with respect to a summand of a characteristic tilting module is a useful tool to construct quasi-hereditary covers of Noetherian algebras and measure their quality. In particular, this homological invariant is used to construct split quasi-hereditary covers of quotients of Iwahori-Hecke algebras using Ringel duality of $q$-Schur algebras. Combining techniques of cover theory with relative dominant dimension theory we obtain a new proof for Ringel self-duality of the blocks of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$.

math.RT↗
arXiv2022

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 Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, $B$, and module categories of endomorphism algebras of generators-relative cogenerators over $B$ which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree $n$ so that the endomorphism algebra of a generator-cogenerator provides an $n$-faithful cover, in the sense of Rouquier, of $B$. The degree $n$ is known as Hemmer-Nakano dimension of the standard modules. We prove that 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 $B$ is bounded above by the number of non-isomorphic simple modules of $B$. We establish methods how to reduce computations of Hemmer-Nakano dimensions in the integral setup to computations of Hemmer-Nakano dimensions over finite-dimensional algebras. This theory allows us to derive results for Schur algebras and the BGG category O in the integral setup from the finite-dimensional case using relative dominant dimension. We exhibit several structural properties of deformations of the blocks of the BGG category 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.

math.RT↗
arXiv2021

Cellular Noetherian algebras with finite global dimension are split quasi-hereditary

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers.

math.RT↗
arXiv2021

An ELEGANT dataset with Denial of Service and Man in The Middle attacks

This document describes a dataset with diverse types of Denial of Service (DoS) attacks and Man-in-the-Middle (MiTM) attacks. The data is available at the following DOI 10.21227/mewp-g646. This document describes the data collection process and provides useful information on how data can be employed to devise models for cybersecurity in critical infrastructures using Programmable Logic Controllers (PLCs)

cs.CR↗
arXiv2021

On properly stratified Gorenstein algebras

We show that a properly stratified algebra is Gorenstein if and only if the characteristic tilting module coincides with the characteristic cotilting module. We further show that properly stratified Gorenstein algebras $A$ enjoy strong homological properties such as all Gorenstein projective modules being properly stratified and all endomorphism rings $\operatorname{End}_A(\Delta(i))$ being Frobenius algebras. We apply our results to the study of properly stratified algebras that are minimal Auslander-Gorenstein algebras in the sense of Iyama-Solberg and calculate under suitable conditions their Ringel duals. This applies in particular to all centraliser algebras of nilpotent matrices.

math.RT↗
arXiv2020

A characterisation of Morita algebras in terms of covers

A pair $(A, P)$ is called a cover of $\operatorname{End}_A(P)^{op}$ if the Schur functor $\operatorname{Hom}_A(P, -)$ is fully faithful on the full subcategory of projective $A$-modules, for a given projective $A$-module $P$. By definition, Morita algebras are the covers of self-injective algebras and then $P$ is a faithful projective-injective module. Conversely, we show that $A$ is a Morita algebra and $\operatorname{End}_A(P)^{op}$ is self-injective whenever $(A, P)$ is a cover of $\operatorname{End}_A(P)^{op}$ for a faithful projective-injective module $P$.

math.RT↗
arXiv2020

Schur--Weyl duality over commutative rings

The classical case of Schur--Weyl duality states that the actions of the group algebras of $GL_n$ and $S_d$ on the $d^{th}$-tensor power of a free module of finite rank centralize each other. We show that Schur--Weyl duality holds for commutative rings where enough scalars can be chosen whose non-zero differences are invertible. This implies all the known cases of Schur--Weyl duality so far. We also show that Schur--Weyl duality fails for $\mathbb{Z}$ and for any finite field when $d$ is sufficiently large.

math.GR↗