Marcelo Lanzilotta - Academia.edu (original) (raw)

Papers by Marcelo Lanzilotta

Research paper thumbnail of Gerstenhaber structure on Hochschild cohomology of toupie algebras

arXiv (Cornell University), Mar 27, 2018

Research paper thumbnail of Split-by-nilpotent extensions algebras and stratifying systems

arXiv (Cornell University), Apr 18, 2013

Research paper thumbnail of Constructing nearly Frobenius algebras

arXiv (Cornell University), Jun 17, 2013

Research paper thumbnail of Strongly stratifying ideals, Morita contexts and Hochschild homology

arXiv (Cornell University), Mar 30, 2023

We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A... more We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context -or equivalently for a strongly stratifying idealwe show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture. If Han's conjecture is true for local algebras and an algebra Λ admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for Λ.

Research paper thumbnail of Erratum to the article Split bounded extension algebras and Han’s conjecture

Pacific Journal of Mathematics, Aug 1, 2022

Research paper thumbnail of Igusa-Todorov for radical square zero algebras

Research paper thumbnail of Strongly stratifying ideals, Morita contexts and Hochschild homology

Research paper thumbnail of Igusa-Todorov functions for Artin algebras

arXiv (Cornell University), May 31, 2016

In this paper we study the behaviour of the Igusa-Todorov functions for Artin algebras A with fin... more In this paper we study the behaviour of the Igusa-Todorov functions for Artin algebras A with finite injective dimension, and Gorenstein algebras as a particular case. We show that the φ-dimension and ψ-dimension are finite in both cases. Also we prove that monomial, gentle and cluster tilted algebras have finite φ-dimension and finite ψ-dimension.

Research paper thumbnail of The Φ-Dimension: A New Homological Measure

In [15], K. Igusa and G. Todorov introduced two functions φ and ψ, which are natural and importan... more In [15], K. Igusa and G. Todorov introduced two functions φ and ψ, which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become into a powerful tool to understand better the finitistic dimension conjecture. In this paper, for an artin R-algebra A and the Igusa-Todorov function φ, we characterise the φ-dimension of A in terms either of the bi-functors Ext i A (−, −) or Tor's bi-functors Tor A i (−, −). Furthermore, by using the first characterisation of the φ-dimension, we show that the finiteness of the φdimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz's result [3, Corollary 1] as follows: For an artin algebra A, a tilting A-module T and the endomorphism algebra B = End A (T) op , we have that φ dim (A) − pd T ≤ φ dim (B) ≤ φ dim (A) + pd T.

Research paper thumbnail of 1 Layer Lengths, Torsion Theories and the Finitistic Dimension

Let Λ be a left-artinian ring. Generalizing the Loewy length, we propose the layer length associa... more Let Λ be a left-artinian ring. Generalizing the Loewy length, we propose the layer length associated with a torsion theory, which is a new measure for finitely generated Λ-modules. As an application, we obtain a theorem having as corollaries the main results of [3] and [7]. 1. Layer lengths Throughout the paper, we fix the following notation. Λ will be a left-artinian ring and C := mod (Λ) the category of finitely generated left Λ-modules. We also denote by End Z (C) the category of all additive functors from C to C. Furthermore we let rad (resp. soc) denote the Jacobson's radical (resp. socle) lying in End Z (C). Note that the functors rad and soc are both subfunctors of the identity 1 C. Recall that if α and β belong to End Z (C) and α is a subfunctor of β, we have the quotient functor β/α ∈ End Z (C) which is defined as follows: (a) (β/α)(M) := β(M)/α(M) for M ∈ C, and (b) (β/α)(f) (x + α (M)) := β (f) (x) + α (N) for a morphism f : M → N in C. Furthermore, we set top := 1 C /rad ∈ End Z (C). Finally, we also recall that the functors rad and 1 C /soc preserve monomorphisms and epimorphisms in C. Given α ∈ End Z (C), we consider the α-radical functor F α := rad • α and the α-socle quotient functor G α := α/(soc • α) where • is the composition in End Z (C). Furthermore, we consider the classes F α = { M ∈ C : α(M) = 0 } and T α = { M ∈ C : α(M) = M }. Also we set min ∅ := ∞. Definition 1.1. For α and β in End Z (C), we define: (a) the (α, β)-layer length ℓℓ β α : C → N ∪ {∞} ℓℓ β α (M) := min {i ≥ 0 : α • β i (M) = 0 }; (b) the α-radical layer length ℓℓ α := ℓℓ Fα α and the α-socle layer length ℓℓ α := ℓℓ Gα α. Note that ℓℓ α (M) and ℓℓ α (M) are finite for all M in C. Example 1.2. The Loewy length is obtained by taking α = 1 C in 1.1 (b). This yields the usual radical layer length ℓℓ 1C and socle layer length ℓℓ 1C. In this case, it is well known that ℓℓ 1C = ℓℓ 1C .

Research paper thumbnail of Han's conjecture for bounded extensions

Journal of Algebra, 2022

Let B ⊂ A be a bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long n... more Let B ⊂ 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 bounded extensions are given and we obtain a structure result for the extensions of an algebra given by a quiver and admissible relations.

Research paper thumbnail of Hochschild cohomology of multi-extension zero algebras

The main objective of this paper is to present a theory for computing the Hochschild cohomology o... more 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 long exact sequence is a particular case of the long exact sequence of cohomology that we obtain via the study of trajectories of the quiver. We introduce cohomology along paths, and we compute it under suitable Tor vanishing hypotheses. The cup product on Hochschild cohomology enables us to describe the connecting homomorphism of the long exact sequence. Multi-extension algebras built on the round trip quiver provide square matrix algebras which have two algebras on the diagonal and two bimodules on the corners. If the bimodules are projective, we show that a five-term exact sequences arises. If the bimodules are free of rank one, we...

Research paper thumbnail of Adding or deleting arrows of a bound quiver algebra and Hochschild (co)homology

We describe how the Hochschild (co)homology of a bound quiver algebra changes when adding or dele... more 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.

Research paper thumbnail of Jacobi–Zariski long nearly exact sequences for associative algebras

Bulletin of the London Mathematical Society, 2021

For an extension of associative algebras B ⊂ A over a field and an Abimodule X, we obtain a Jacob... more For an extension of associative algebras B ⊂ A over a field and an Abimodule 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 which converges to the gap of exactness.

Research paper thumbnail of The Phi-dimension: A new homological measure

arXiv: Representation Theory, 2013

K. Igusa and G. Todorov introduced two functions phi\phiphi and psi,\psi,psi, which are natural and importa... more K. Igusa and G. Todorov introduced two functions phi\phiphi and psi,\psi,psi, which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become into a powerful tool to understand better the finitistic dimension conjecture. In this paper, for an artin RRR-algebra AAA and the Igusa-Todorov function phi,\phi,phi, we characterise the phi\phiphi-dimension of AAA in terms either of the bi-functors mathrmExtiA(−,−)\mathrm{Ext}^{i}_{A}(-, -)mathrmExtiA(−,−) or Tor's bi-functors mathrmTorAi(−,−).\mathrm{Tor}^{A}_{i}(-,-).mathrmTorAi(−,−). Furthermore, by using the first characterisation of the phi\phiphi-dimension, we show that the finiteness of the phi\phiphi-dimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz's result as follows: For an artin algebra A,A,A, a tilting AAA-module TTT and the endomorphism algebra B=mathrmEndA(T)op,B=\mathrm{End}_A(T)^{op},B=mathrmEndA(T)op, we have that $\mathrm{Fidim}\,(A)-\mathrm{pd}\,T\leq \mathrm{Fidim...

Research paper thumbnail of 12 Finitistic Dimension Through Infinite Projective Dimension

Abstract. We show that an artin algebra Λ having at most three radical layers of infinite project... more Abstract. We show that an artin algebra Λ having at most three radical layers of infinite projective dimension has finite finitistic dimension, generalizing the known result for algebras with vanishing radical cube. We also give an equivalence between the finiteness of fin.dim.Λ and the finiteness of a given class of Λ-modules of infinite projective dimension. 1. Introduction. Let Λ be an artin algebra, and consider mod Λ the class of finitely generated left Λ-modules. The finitistic dimension of Λ is then defined to be fin.dim. Λ = sup{pdM: M ∈ mod Λ and pdM < ∞}, where pd M denotes the projective dimension of M. It was conjectured by Bass in

Research paper thumbnail of Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras

In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast... more In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of the developed theory are given in order to relate the different homological dimensions which have been discussed through the paper.

Research paper thumbnail of The Ø-dimension: A new homological measure

Research paper thumbnail of Corrigendum to: Jacobi–Zariski long nearly exact sequences for associative algebras

Bulletin of the London Mathematical Society

Research paper thumbnail of Split bounded extension algebras and Han’s conjecture

Pacific Journal of Mathematics

Research paper thumbnail of Gerstenhaber structure on Hochschild cohomology of toupie algebras

arXiv (Cornell University), Mar 27, 2018

Research paper thumbnail of Split-by-nilpotent extensions algebras and stratifying systems

arXiv (Cornell University), Apr 18, 2013

Research paper thumbnail of Constructing nearly Frobenius algebras

arXiv (Cornell University), Jun 17, 2013

Research paper thumbnail of Strongly stratifying ideals, Morita contexts and Hochschild homology

arXiv (Cornell University), Mar 30, 2023

We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A... more We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context -or equivalently for a strongly stratifying idealwe show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture. If Han's conjecture is true for local algebras and an algebra Λ admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for Λ.

Research paper thumbnail of Erratum to the article Split bounded extension algebras and Han’s conjecture

Pacific Journal of Mathematics, Aug 1, 2022

Research paper thumbnail of Igusa-Todorov for radical square zero algebras

Research paper thumbnail of Strongly stratifying ideals, Morita contexts and Hochschild homology

Research paper thumbnail of Igusa-Todorov functions for Artin algebras

arXiv (Cornell University), May 31, 2016

In this paper we study the behaviour of the Igusa-Todorov functions for Artin algebras A with fin... more In this paper we study the behaviour of the Igusa-Todorov functions for Artin algebras A with finite injective dimension, and Gorenstein algebras as a particular case. We show that the φ-dimension and ψ-dimension are finite in both cases. Also we prove that monomial, gentle and cluster tilted algebras have finite φ-dimension and finite ψ-dimension.

Research paper thumbnail of The Φ-Dimension: A New Homological Measure

In [15], K. Igusa and G. Todorov introduced two functions φ and ψ, which are natural and importan... more In [15], K. Igusa and G. Todorov introduced two functions φ and ψ, which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become into a powerful tool to understand better the finitistic dimension conjecture. In this paper, for an artin R-algebra A and the Igusa-Todorov function φ, we characterise the φ-dimension of A in terms either of the bi-functors Ext i A (−, −) or Tor's bi-functors Tor A i (−, −). Furthermore, by using the first characterisation of the φ-dimension, we show that the finiteness of the φdimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz's result [3, Corollary 1] as follows: For an artin algebra A, a tilting A-module T and the endomorphism algebra B = End A (T) op , we have that φ dim (A) − pd T ≤ φ dim (B) ≤ φ dim (A) + pd T.

Research paper thumbnail of 1 Layer Lengths, Torsion Theories and the Finitistic Dimension

Let Λ be a left-artinian ring. Generalizing the Loewy length, we propose the layer length associa... more Let Λ be a left-artinian ring. Generalizing the Loewy length, we propose the layer length associated with a torsion theory, which is a new measure for finitely generated Λ-modules. As an application, we obtain a theorem having as corollaries the main results of [3] and [7]. 1. Layer lengths Throughout the paper, we fix the following notation. Λ will be a left-artinian ring and C := mod (Λ) the category of finitely generated left Λ-modules. We also denote by End Z (C) the category of all additive functors from C to C. Furthermore we let rad (resp. soc) denote the Jacobson's radical (resp. socle) lying in End Z (C). Note that the functors rad and soc are both subfunctors of the identity 1 C. Recall that if α and β belong to End Z (C) and α is a subfunctor of β, we have the quotient functor β/α ∈ End Z (C) which is defined as follows: (a) (β/α)(M) := β(M)/α(M) for M ∈ C, and (b) (β/α)(f) (x + α (M)) := β (f) (x) + α (N) for a morphism f : M → N in C. Furthermore, we set top := 1 C /rad ∈ End Z (C). Finally, we also recall that the functors rad and 1 C /soc preserve monomorphisms and epimorphisms in C. Given α ∈ End Z (C), we consider the α-radical functor F α := rad • α and the α-socle quotient functor G α := α/(soc • α) where • is the composition in End Z (C). Furthermore, we consider the classes F α = { M ∈ C : α(M) = 0 } and T α = { M ∈ C : α(M) = M }. Also we set min ∅ := ∞. Definition 1.1. For α and β in End Z (C), we define: (a) the (α, β)-layer length ℓℓ β α : C → N ∪ {∞} ℓℓ β α (M) := min {i ≥ 0 : α • β i (M) = 0 }; (b) the α-radical layer length ℓℓ α := ℓℓ Fα α and the α-socle layer length ℓℓ α := ℓℓ Gα α. Note that ℓℓ α (M) and ℓℓ α (M) are finite for all M in C. Example 1.2. The Loewy length is obtained by taking α = 1 C in 1.1 (b). This yields the usual radical layer length ℓℓ 1C and socle layer length ℓℓ 1C. In this case, it is well known that ℓℓ 1C = ℓℓ 1C .

Research paper thumbnail of Han's conjecture for bounded extensions

Journal of Algebra, 2022

Let B ⊂ A be a bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long n... more Let B ⊂ 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 bounded extensions are given and we obtain a structure result for the extensions of an algebra given by a quiver and admissible relations.

Research paper thumbnail of Hochschild cohomology of multi-extension zero algebras

The main objective of this paper is to present a theory for computing the Hochschild cohomology o... more 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 long exact sequence is a particular case of the long exact sequence of cohomology that we obtain via the study of trajectories of the quiver. We introduce cohomology along paths, and we compute it under suitable Tor vanishing hypotheses. The cup product on Hochschild cohomology enables us to describe the connecting homomorphism of the long exact sequence. Multi-extension algebras built on the round trip quiver provide square matrix algebras which have two algebras on the diagonal and two bimodules on the corners. If the bimodules are projective, we show that a five-term exact sequences arises. If the bimodules are free of rank one, we...

Research paper thumbnail of Adding or deleting arrows of a bound quiver algebra and Hochschild (co)homology

We describe how the Hochschild (co)homology of a bound quiver algebra changes when adding or dele... more 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.

Research paper thumbnail of Jacobi–Zariski long nearly exact sequences for associative algebras

Bulletin of the London Mathematical Society, 2021

For an extension of associative algebras B ⊂ A over a field and an Abimodule X, we obtain a Jacob... more For an extension of associative algebras B ⊂ A over a field and an Abimodule 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 which converges to the gap of exactness.

Research paper thumbnail of The Phi-dimension: A new homological measure

arXiv: Representation Theory, 2013

K. Igusa and G. Todorov introduced two functions phi\phiphi and psi,\psi,psi, which are natural and importa... more K. Igusa and G. Todorov introduced two functions phi\phiphi and psi,\psi,psi, which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become into a powerful tool to understand better the finitistic dimension conjecture. In this paper, for an artin RRR-algebra AAA and the Igusa-Todorov function phi,\phi,phi, we characterise the phi\phiphi-dimension of AAA in terms either of the bi-functors mathrmExtiA(−,−)\mathrm{Ext}^{i}_{A}(-, -)mathrmExtiA(−,−) or Tor's bi-functors mathrmTorAi(−,−).\mathrm{Tor}^{A}_{i}(-,-).mathrmTorAi(−,−). Furthermore, by using the first characterisation of the phi\phiphi-dimension, we show that the finiteness of the phi\phiphi-dimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz's result as follows: For an artin algebra A,A,A, a tilting AAA-module TTT and the endomorphism algebra B=mathrmEndA(T)op,B=\mathrm{End}_A(T)^{op},B=mathrmEndA(T)op, we have that $\mathrm{Fidim}\,(A)-\mathrm{pd}\,T\leq \mathrm{Fidim...

Research paper thumbnail of 12 Finitistic Dimension Through Infinite Projective Dimension

Abstract. We show that an artin algebra Λ having at most three radical layers of infinite project... more Abstract. We show that an artin algebra Λ having at most three radical layers of infinite projective dimension has finite finitistic dimension, generalizing the known result for algebras with vanishing radical cube. We also give an equivalence between the finiteness of fin.dim.Λ and the finiteness of a given class of Λ-modules of infinite projective dimension. 1. Introduction. Let Λ be an artin algebra, and consider mod Λ the class of finitely generated left Λ-modules. The finitistic dimension of Λ is then defined to be fin.dim. Λ = sup{pdM: M ∈ mod Λ and pdM < ∞}, where pd M denotes the projective dimension of M. It was conjectured by Bass in

Research paper thumbnail of Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras

In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast... more In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of the developed theory are given in order to relate the different homological dimensions which have been discussed through the paper.

Research paper thumbnail of The Ø-dimension: A new homological measure

Research paper thumbnail of Corrigendum to: Jacobi–Zariski long nearly exact sequences for associative algebras

Bulletin of the London Mathematical Society

Research paper thumbnail of Split bounded extension algebras and Han’s conjecture

Pacific Journal of Mathematics