arXiv subjects

Eduardo do Nascimento Marcos

Publications and source records attributed to Eduardo do Nascimento Marcos.

6 recordsLinked to original sources

arXiv2026

A relative homology criterion of smoothness

We investigate the relationship between smoothness and the relative global dimension of a ring extension. We prove that a smooth commutative algebra $A$ over $B$ has finite relative global dimension $\text{gdim}(A,B)$. Conversely, under a mild condition on $B$, the finiteness of $\text{gdim}(A,B)$ implies that the map $B \to A$ is smooth. We also relate the relative global dimension to the usual global dimension of the fibers of $B \to A$, and establish a formula for the relative global dimension of tensor products of extensions. Finally, we present examples and an alternative characterisation of smoothness in terms of relative Hochschild homology.

math.AC↗

arXiv2018

Homogeneous triples for homogeneous algebras with two relations

In our preceding paper we have introduced the notion of an $s$-homogeneous triple. In this paper we use this technique to study connected $s$-homogeneous algebras with two relations. For such algebras, we describe all possible pairs $(A,M)$, where $A$ is the $s$-Veronese ring and $M$ is the $(s,1)$-Veronese bimodule of the $s$-homogeneous dual algebra. For each such a pair we give an intrinsic characterization of algebras corresponding to it. Due to results of our previous work many pairs determine the algebra uniquely up to isomorphism. Using our partial classification, we show that, to check the $s$-Koszulity of a connected $s$-homogeneous algebras with two relations, it is enough to verify an equality for Hilbert series or to check the exactness of the generalized Koszul complex in the second term. For each pair $(A,M)$ not belonging to one specific series of pairs, we check if there exists an $s$-Koszulity algebra corresponding to it. Thus, we describe a class of possible ${\rm Ext}$-algebras of $s$-Koszul connected algebras with two relations and realize all of them except a finite number of specific algebras as ${\rm Ext}$-algebras. Another result that follows from our classification is that an $s$-homogeneous algebra with two dimensional $s$-th component cannot be $s$-Koszul for $s>2$.

math.RT↗

arXiv2017

$s$-homogeneous algebras via $s$-homogeneous triples

To study $s$-homogeneous algebras, we introduce the category of quivers with $s$-homogeneous corelations and the category of $s$-homogeneous triples. We show that both of these categories are equivalent to the category of $s$-homogeneous algebras. We prove some properties of the elements of $s$-homogeneous triples and give some consequences for $s$-Koszul algebras. Then we discuss the relations between the $s$-Koszulity and the Hilbert series of $s$-homogeneous triples. We give some application of the obtained results to $s$-homogeneous algebras with simple zero component. We describe all $s$-Koszul algebras with one relation recovering the result of Berger and all $s$-Koszul algebras with one dimensional $s$-th component. We show that if the $s$-th Veronese ring of an $s$-homogeneous algebra has two generators, then it has at least two relations. Finally, we classify all $s$-homogeneous algebras with $s$-th Veronese rings ${\bf k}\langle x,y\rangle/(xy,yx)$ and ${\bf k}\langle x,y\rangle/(x^2,y^2)$. In particular, we show that all of these algebras are not $s$-Koszul while their $s$-homogeneous duals are $s$-Koszul.

math.KT↗

arXiv2015

Stratifying systems over the hereditary path algebra with quiver $\mathbb{A}_{p,q}$

The authors have proved in [J. Algebra Appl. 14 (2015), no. 6] that the size of a stratifying system over a finite-dimensional hereditary path algebra $A$ is at most $n$, where $n$ is the number of isomorphism classes of simple $A$-modules. Moreover, if $A$ is of Euclidean type a stratifying system over $A$ has at most $n-2$ regular modules. In this work, we construct a family of stratifying systems of size $n$ with a maximal number of regular elements, over the hereditary path algebra with quiver $\widetilde{\mathbb {A}}_{p,q} $, canonically oriented.

math.RT↗

arXiv2015

Hochschild cohomology of a smash product in the nonsemisimple case

In this paper we explore the relations between the Hochschild cohomology of an algebra over some field and the Hochschild cohomology of its smash product with a finite group. Basically we are concentrated on the case where the group under consideration is an extension of a cyclic $p$-group by some $p'$-group, where $p$ is the characteristic of the ground field.

math.RT↗

arXiv2003

Hochschild Cohomology of skew group rings and invariants

Let $A$ be a $k$-algebra and $G$ be a group acting on $A$. We show that $G$ also acts on the Hochschild cohomology algebra $HH^{\bullet}(A)$ and that there is a monomorphism of rings $HH^{\bullet}(A)^G \hookrightarrow HH^{\bullet}(A[G])$. That allows us to show the existence of a monomorphism from $HH^{\bullet}(\widetilde{A})^G$ into $HH^{\bullet}(A)$, where $\widetilde{A}$ is a Galois covering with group $G$.

math.RT↗