Eduardo Marcos

University of São Paulo | USP · Department of Mathematics (IME) (São Paulo)

In this paper we study the lower triangular matrix K-algebra Λ:=[T0MU], where U and T are basic K-algebras with enough idempotents and M is an U-T-bimodule where K acts centrally. Moreover, we characterise in terms of U, T and M when, on one hand, the lower triangular matrix K-algebra Λ is standardly stratified in the sense of [17]; and on the othe...

Let B⊂A be a left or right bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that B satisfies Han's conjecture if and only if A does, regardless if the extension splits or not. We provide conditions ensuring that an extension by arrows and relations is left or right bounded. Finally we gi...

Let G be a group acting on a small category C over a field k, that is C is a G-k-category. We first obtain an unexpected result: C is resolvable by a category which is G-k-equivalent to it, on which G acts freely on objects. This resolving category enables to show that if the coinvariants and the invariants functors are exact, then the coinvariants...

Corrigendum: Jacobi–Zariski long nearly exact sequences for associative algebras Claude Cibils, Marcelo Lanzilotta, Eduardo N. Marcos, Andrea Solotar Volume 53, Issue 6, Bulletin of the London Mathematical Society, pages: 1651-1652. First Published online: September 24, 2021

In this paper we study the lower triangular matrix $\mathbb{K}$-algebra $\Lambda:=\left[\begin{smallmatrix} T & 0 \\ M & U \end{smallmatrix}\right],$ where $U$ and $T$ are basic $\mathbb{K}$-algebras with enough idempotents and $M$ is an $U$-$T$-bimodule where $\mathbb{K}$ acts centrally. Moreover, we characterise in terms of $U,$ $T$ and $M$ when,...

Let $B\subset A$ be a bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that $B$ satisfies Han's conjecture if and only if $A$ does, regardless if the extension splits or not. We provide conditions ensuring that an extension by arrows and relations is bounded. Examples of non split bounde...

Let $G$ be a group acting on a small category $\mathcal C$ over a field $k$, that is $\mathcal C$ is a $G$-$k$-category. We first obtain that $\mathcal C$ is resolvable by a category which is $G$-$k$-equivalent to it, on which $G$ acts freely on objects. This resolvent category enables to show that if the coinvariants and the invariants functors ar...

For an extension of associative algebras $B\subset A$ over a field and an $A$-bimodule $X$, we obtain a Jacobi-Zariski long nearly exact sequence relating the Hochschild homologies of $A$ and $B$, and the relative Hochschild homology, all of them with coefficients in $X$. This long sequence is exact twice in three. There is a spectral sequence whic...

We provide a formula for the change of the dimension of the first Hochschild cohomology vector space of bound quiver algebras when adding new arrows. For this purpose we show that there exists a short exact sequence which relates the first cohomology vector spaces of the algebras to the first relative cohomology. Moreover, we show that the first Ho...

A main purpose of this paper is to prove that the class of finite dimensional algebras which verify Han's conjecture is closed under split bounded extensions.

We provide a formula for the change of the dimension of the first Hoch\-schild cohomology vector space of bound quiver algebras when adding new arrows. For this purpose we show that there exists a short exact sequence which relates the first cohomology vector spaces of the algebras to the first relative cohomology. Moreover, we show that the first...

We describe how the Hochschild (co)homology of a bound quiver algebra changes when adding or deleting arrows to the quiver. The main tools are relative Hochschild (co)homology, the Jacobi-Zariski long exact sequence obtained by A. Kaygun and a one step relative projective resolution of a tensor algebra.

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 alge...

Let $\mathcal C$ be category over a commutative ring $k$, its Hochschild-Mitchell homology and cohomology are denoted respectively $HH_*(\mathcal C)$ and $HH^*(\mathcal C).$ Let $G$ be a group acting on $\mathcal C$, and $\mathcal C[G]$ be the skew category. We provide decompositions of the (co)homology of $\mathcal C[G]$ along the conjugacy classe...

We study the cokernel of the application given by the Car-tan Matrix C Λ of a finite dimensional k-algebra Λ. This produces a finitely generated abelian group, the Cartan group G Λ , which is invariant under derived equivalences. We are interested in the case when G Λ is finite. For a standardly stratified algebra, it is shown that this group is al...

We study the cokernel of the application given by the Cartan Matrix $C_\Lambda$ of a finite dimensional $k$-algebra $\Lambda.$ This produces a finitely generated abelian group, the Cartan group $G_\Lambda,$ which is invariant under derived equivalences. We are interested in the case when $G_\Lambda$ is finite. For a standardly stratified algebra, i...

Let $K\Delta$ be the incidence algebra associated with a finite poset $(\Delta,\preceq)$ over the algebraically closed field $K$. We present a study of incidence algebras $K\Delta$ that are piecewise hereditary, which we denominate PHI algebras. We investigate the strong global dimension, the simply conectedeness and the one-point extension algebra...

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 $s$...

We provide a framework connecting several well-known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras. Finally, we provide a tool to evaluate the possible degrees of a module appearing in a graded projective reso...

In this paper, we study partial actions of groups on $R$-algebras, where $R$ is a commutative ring. We describe the partial actions of groups on the indecomposable algebras with enveloping actions. Then we work on algebras that can be decomposed as product of indecomposable algebras and we give a description of the partial actions of groups on thes...

The main objective of this paper is to present a theory for computing the Hochschild cohomology of algebras built on a specific data, namely multi-extension algebras. The computation relies on cohomological functors evaluated on the data, and on the combinatorics of an ad hoc quiver. One-point extensions are occurrences of this theory, and Happel's...

Let T be a tilting object in a triangulated category which is equivalent to the bounded derived category of a finite-dimensional hereditary algebra. The text investigages the strong global dimension, in the sense of Ringel, of the opposite algebra A of the endomorphism algebra of T. This invariant is expressed in terms of the lengths of the sequenc...

We explore some properties of wide subcategories of the category (Formula presented.) of finitely generated left (Formula presented.)-modules, for some artin algebra (Formula presented.) In particular we look at wide finitely generated subcategories and give a connection with the class of standard modules and standardly stratified algebras. Further...

We present a study on the description of incidence algebras that are piecewise hereditary, which we denominate Phia algebras. We describe the quiver with relations of the Phia algebras of Dynkin type and introduce a new family of Phia algebras of extended Dynkin type, which we call ANS family, in reference to Assem, Neh\-ring, and Skowro\'nski. In...

Let $\cQ$ be a quiver and $K$ a field. We study the interrelationship of homological properties of algebras associated to convex subquivers of $\cQ$ and quotients of the path algebra $K\cQ$. We introduce the homological heart of $\cQ$ which is a particularly nice convex subquiver of $\cQ$. For any algebra of the form $K\cQ/I$, the algebra associate...

We provide a framework connecting several well known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras. Finally, we provide a tool to evaluate the possible degrees of a module appearing in a graded projective reso...

We explore some properties of coherent subcategories of the category $\mathrm{mod}\,(\Lambda)$ of finitely generated left $\Lambda$-modules, for some artin algebra $\Lambda.$ In particular we look at coherently finitely generated subcategories and give a connection with the class of standard modules and standardly stratified algebras.

In [99. Giraldo , H. , Merklen , H. ( 2009 ). Irreducible morphisms of categories of complexes . Journal of Algebra 321 : 2716 – 2736 . View all references] Giraldo and Merklen studied irreducible morphisms in the categories 𝒞(𝒜) and D−(Λ), where 𝒞(𝒜) is the category of complexes over an abelian Krull–Schmidt category 𝒜 and D−(Λ) is the derived c...

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 thi...

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...

We consider an homogeneous action of a finite group on a free linear category over a field in order to prove that the subcategory of invariants is still free. Moreover we show that the representation type is preserved when considering invariants.

We consider an homogeneous action of a finite group on a free linear category over a field in order to prove that the subcategory of invariants is still free. Moreover we show that the representation type is preserved when considering invariants.

We study the behaviour of the Igusa-Todorov functions for radical square zero algebras. We show that the left and the right $\phi$-dimensions coincide, in this case. Some general results are given, but we concentrate more in the radical square zero algebras. Our study is based on two notions of hearth and member of a quiver $Q$. We give some bounds...

If $f$ is an idempotent in a ring $\Lambda$, then we find sufficient \linebreak conditions which imply that the cohomology rings $\oplus_{n\ge 0}Ext^n_{\Lambda}(\Lambda/{\br},\Lambda/{\br})$ and \linebreak $\oplus_{n\ge 0}Ext^n_{f\Lambda f}(f\Lambda f/f{\br} f,f\Lambda f/f{\br} f)$ are eventually isomorphic. This result allows us to compare finite...

In this paper, we study the derived categories of a Koszul algebra and its Yoneda algebra to determine when those categories are triangularly equivalent. We prove that the simply connected Koszul algebras are derived equivalent to their Yoneda algebras. We have considered discrete Koszul algebras and we gave necessary and sufficient conditions for...

This paper deals with stratifying systems over hereditary algebras. In the case of tame hereditary algebras we obtain a bound for the size of the stratifying systems composed only by regular modules and we conclude that stratifying systems can not be complete. For wild hereditary algebras with more than 2 vertices we show that there exists a comple...

In this article, we study subrings of the Ext-algebra of a graded module over a graded ring R. We show these subrings can be defined by equivalence relations on exact sequences over the ring. In particular, the shriek ring, R , and the even part of an Ext-algebra of a d-Koszul algebra can be defined by equivalence relations on exact sequences.

In this paper we introduce the definition of partial action on small $k$-categories generalizing the similar well known notion of partial actions on algebras. The point of view of partial action which we use in this paper is the one which was introduced by Exel in his work on $C^*$-algebras, see \cite{E}. Various generalizations were done afterward...

Let A be a finite dimensional k-algebra, (Θ, ≤) be a stratifying system in mod(A) and ℱ(Θ) be the class of Θ-filtered A-modules. In this article, we give the definition and also study some of the properties of the relative socle in ℱ(Θ). We approach the relative socle in three ways. Namely, we view it as (1) a Θ-semisimple subobject of M having the...

The relationship between an algebra and its associated monomial algebra is investigated when at least one of the algebras is d-Koszul. It is shown that an algebra which has a reduced Gröbnerbasis that is composed of homogeneous elements of degree d is d-Koszul if and only if its associated monomial algebra is d-Koszul. The class of 2-d-determined a...

Using the Luthar--Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Suzuki sporadic simple group Suz. As a consequence, for this group we confirm the Kimmerle's conjecture on prime graphs.

In this note, we show that, if A congruent to kQ(A)/I-A is a schurian strongly simply connected algebra given by its normed presentation, and Sigma is the unique poset whose Hasse quiver coincides with Q(A), then A congruent to k Sigma if and only if I-A has a generating set consisting of exactly chi(Q(A)) elements, where chi(Q(A)) is the Euler cha...

The aim of this work is to characterize the algebras which are standardly stratified with respect to any order of the simple modules. We show that such algebras are exactly the algebras with all idempotent ideals projective. We also deduce as a corollary a characterization of hereditary algebras, originally due to Dlab and Ringel

In this work we extend, to the path algebras context, some results obtained in the commutative context, [E. L. Green, J. Symb. Comput. 29, No. 4-5, 601-623 (2000; Zbl 1002.16043)]. The main result is that one can extend the Gröbner bases of an ungraded ideal to one possible definition of homogenization for the non commutative case.

Let R be an algebra, and let (θ,≼) be a stratifying system of R-modules. If the category F(θ) is θ-directing, then we prove that indF(θ) is finite. In order to do that, we introduce a quadratic form qθ which depends on θ. Moreover, we also give sufficient conditions to get the correspondence X↦dim̲θX from indF(θ) to the set of positive roots of qθ.

Given an Ext-injective stratifying system of A-modules (theta, (Y) over bar, less than or similar to) satisfying that the projective dimension of Y is finite, we prove that the finitistic dimension of the algebra A is equal to the finitistic dimension of the category I(theta) = {X is an element of mod Lambda : Ext(Lambda)(1) (-, X)vertical bar(F(th...

We study the stratifications of algebras with radical square zero. We show that an algebra with radical square zero is standardly stratified in some order if and only if it is quasi-triangular. We also show that for such algebras there is an order such that F(Δ)=Proj and also an order such F(Δ)=P <∞ .

The paper: The odd part of an N-Koszul algebra, Comm. in Algebra, 33:101–108, 2005, contains some missprints that make it difficult to read. We correct here these mistakes.

In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In the case of the smash product for an infinite group our constr...

In this paper we point out that the “Process of standardization”, given in Dlab and Ringel (19922. Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras . Repr. Theory and Related Topics, London Math. Soc. LNS 168 : 200 – 224 . View all references), and also the “Comparison method” given in Platzeck and...

The so called n-Koszul algebras have been studied by Berger (2001) and by Green et al. (2004). They are natural generalizations of Koszul algebras however Koszul duality is not well understood in this case, some partial results in this direction were obtained in Green et al. (2004), where the following results is proved: Given an n-Koszul algebra w...

In this paper, we define concepts of crowns and quasi-crowns, valid in an arbitrary schurian algebra, and which generalise the corresponding concepts in an incidence algebra. We show first that a triangular schurian algebra is strongly simply connected if and only if it is simply connected and contains no quasi-crown. We then prove that the absence...

In this paper we continue the study of stratifying systems, which were introduced by K. Erdmann and C. Sáenz in [Comm. Algebra 31 (7) (2003) 3429–3446]. We show that this new concept provides a categorical generalization of the Δ-modules for a standardly stratified algebra.

In this paper we study d-Koszul algebras which were introduced by Berger. We show that when d⩾3, these are classified by the Ext-algebra being generated in degrees 0, 1, and 2. We show the Ext-algebra, after regrading, is a Koszul algebra and present the structure of the Ext-algebra.

Without Abstract

In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In case of the smash product for an infinite group our constructi...

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}(...

We consider associative algebras L over a field provided with a direct sum decomposition of a two-sided ideal M and a sub-algebra A - examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of L. We study the connecting h...

Algebras for which every module is a Koszul module are classified. A necessary condition for the subcategory of graded modules with linear presentations to be equal to the subcategory of Koszul modules is given. This condition is also a sufficient condition when the algebra is radical cube zero. Finally, these subcategories are studied when the alg...

We give a description of the indecomposable objects in the derived category of a finite-dimensional skewed-gentle algebra.

In this paper we study N-koszul algebras which were introduced by R. Berger. We show that when n 3, these are classified by the Ext-algebra being generated in degrees 0, 1, and 2. We give a description of the Ext-algebra using the analogous of the Koszul complex and we also show that it is, is a Koszul algebra, after regrading. This notions can be...

Given a finite-dimensional algebra, we present sufficient conditions on the projective presentation of the algebra modulo its radical for a tilted algebra to be a Koszul algebra and for the endomorphism ring of a tilting module to be a quasi-Koszul algebra. One condition we impose is that the algebra has global dimension no greater than 2. One of t...

In the first part, we study algebras A such that A=R⨿I, where R is a subalgebra and I a two-sided nilpotent ideal. Under certain conditions on I, we show that A is standardly stratified if and only if R is standardly stratified. Next, for A=U0MV, we show that A is standardly stratified if and only if the algebra R=U×V is standardly stratified and V...

When an algebra is graded by a group, any additive character of the group induces a diagonalizable derivation of the ring. This construction is studied in detail for the case of a path algebra modulo relations and its fundamental group. We describe an injection of the character group into the rst cohomology group following Assem-de la Pena. Rather...

In this paper, the electric field distribution obtained from solving Maxwell's equations was coupled to the energy equation to predict the temperature distribution during microwave heating of solids. The effect of sample rotation was incorporated to the model. Simulation runs showed that rotation of the sample results in a more uniform temperature...

In this paper we study the category of finitely generated modules of finite projective dimension over a class of weakly triangular algebras, which includes the algebras whose idempotent ideals have finite projective dimension. In particular, we prove that the relations given by the (relative) almost split sequences generate the group of all relatio...

The purpose of this paper is to study the Hochschild cohomology ring H *()of algebras of the form = kZe /JN , where Ze is an oriented cycle with e vertices and J is the ideal generated by the arrows, N≥2. We provide a new description of the Yoneda product in H *(). and prove that this is a finitely generated infinite dimensional ring. In addition w...

This paper presents a simulation study of oxygen mass transfer in subsurface aeration systems using two models proposed in the literature. A modification in one of the models was introduced, resulting in significant differences in oxygen concentration in the gas phase for small airflow rate values. Simulation results show that the oxygen equilibriu...

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Let A be an artin algebra and let modA denote the category of finitely generated right A-modules. We denote by rad(modA) the ideal in modA generated by all non-invertible morphisms between in-decomposable modules in modA. The infinite radical rad ∞ (modA) of modA is the intersection of all powers rad i (modA), i ≥ 1, of rad(modA). It is known that...

General properties of finite dimensional Hopf algebras are investi-gated. In particular, we generalize some of the work on almost split sequences of Auslander and Carlson on group rings to (not necessarily cocommutative) Hopf algebras with involutive antipode. We also give a new proof of a theorem of Larson, [2], which says that for a finite dimens...

A local theory of graded modules is introduced in this paper. In the case when the graded ring is a quotient ring of a path algebra, it is shown how to retrieve the original algebra from knowledge of the local theory. Almost split sequences are investigated from a local point of view.

It is well known that an artin algebra A is of finite representation type if and only if red(modA)=0. In this note we deepen this result by showing that(rad(modA))=0 implies that A is of finite representation type.

Given a quotient singularity R = SG where S = C[[x1,., xn]]is the formal power series ring in n-variables over the complex numbers C, there is an epimorphism of Grothendieck groups yΨ: G0(S[G]) → G0(R), where S[G] is the skew group ring and Ψ is induced by the fixed point functor. The Grothendieck group of S[G] carries a natural structure of a ring...

Given a quotient singularity R = SG where S = C[[ x1, ..., xn]] is the formal power series ring in n-variables over the complex numbers C, there is an epimorphism of Grothendieck groups ?: G0(S[ G]) ? G0(R), where S[ G] is the skew group ring and ? is induced by the fixed point functor. The Grothendieck group of S[ G] carries a natural structure of...

Abstract In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In case of the smash product for an infinite group our c...

We show that if Λ=kQ/I is a monomial algebra then H 1 (Λ,Λ)=0 if and only if Q is a tree. Boundary maps are also presented to compute some examples of cohomology for monomial algebras and algebras of global dimension 2. In particular, we use the maps to provide a computational proof of D. Happel’s theorem [Topics in algebra, Banach Cent. Publ. 26,...

We study the first Hochschild cohomology group H 1 (Λ) of certain finite-dimensional algebras and how it relates to presentations of Λ. In particular, we consider this relationship for monomial, directed, and a generalization of Schurian algebras. The relationship between presentations and the fundamental group π 1 (Λ) is also studied.

Let A be an Artin algebra and let modA denote the category of finitely generated right A-modules. We denote by rad(modA) the ideal in modA generated by all non-invertible morphisms between indecomposable modules in modA. The infinite radical rad ∞ (modA) of modA is the intersection of all powers rad i (modA), i≥1, of rad(modA). It is known that A i...

Neste trabalho estudamos as álgebras estandarmente estratificadas, em geral e obtivemos os seguintes resultados: - uma caracterização das álgebras que são estandarmente estratificadas em todas as ordens dos simples, obtendo como corolário uma caracterização para as álgebras hereditárias; - estudamos a estratificação das álgebras com só dois simples...

Sejam 'lâmbda' uma k-álgebra de dimensão finita sobre o corpo k, 'tau' um 'lâmbda'-módulo inclinante e 'tau'= 'End IND.'lâmbda'(T), o anel de endomorfismo de 'tau' sobre 'lâmbda'. Através da carcterização dos morfismos, entre os somandos diretosde 'tau' estabelecemos um critério que permite decidir quando a álgebra inclinada graduada 'tau''APROXIMA...

Tese (Doutorado)--Instituto de Matemática e Estatística da Universidade de São Paulo, 12/07/2004.

A Teoria de Bases de Gröbner foi introduzida no contexto do anel de polinômios comutativo por Bruno Buchberger em sua tese de doutorado em 1965. Essa teoria encontra suas primeiras aplicações na área de Geometria Algébrica e possui um papel central em Álgebra Computacional. Grande parte de sua importância se deve ao fato de ela fornecer uma solução...