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Giuseppe Metere - One of the best experts on this subject based on the ideXlab platform.
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Bourn-normal Monomorphisms in regular Mal'tsev categories
arXiv: Category Theory, 2016Co-Authors: Giuseppe MetereAbstract:Normal Monomorphisms in the sense of Bourn describe the equivalence classes of an internal equivalence relation. Although the definition is given in the fairly general setting of a category with finite limits, later investigations on this subject often focus on protomodular settings, where normality becomes a property. This paper clarifies the connections between internal equivalence relations and Bourn-normal Monomorphisms in regular Mal'tesv categories with pushouts of split Monomorphisms along arbitrary morphisms, whereas a full description is achieved for quasi-pointed regular Mal'tsev categories with pushouts of split Monomorphisms along arbitrary morphisms.
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Normal Monomorphisms in regular Mal'tsev categories
arXiv: Category Theory, 2016Co-Authors: Giuseppe MetereAbstract:Normal Monomorphisms in the sense of Bourn describe the equivalence classes of an internal equivalence relation. Although the definition is given in the fairly general setting of a category with finite limits, later investigations on this subject often focus on protomodular settings, where normality becomes a property. This paper clarifies the connections between internal equivalence relations and normal Monomorphisms in regular Mal'tesv categories, whereas a full description is achieved for quasi-pointed regular Mal'tsev categories.
Chrysostomos Psaroudakis - One of the best experts on this subject based on the ideXlab platform.
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A functorial approach to Monomorphism categories for species I
arXiv: Representation Theory, 2019Co-Authors: Nan Gao, Julian Külshammer, Sondre Kvamme, Chrysostomos PsaroudakisAbstract:For any generalised species over a locally bounded quiver we investigate abstract versions of the Monomorphism category as studied by Ringel and Schmidmeier. We prove that analogues of the kernel and cokernel functor send almost split sequences over the path algebra and the preprojective algebra to split or almost split sequences in the Monomorphism category.
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Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings
Algebras and Representation Theory, 2016Co-Authors: Nan Gao, Chrysostomos PsaroudakisAbstract:In this paper we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with Monomorphism categories. In the second part, we investigate Monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the Monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.
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Ladders of recollements, categories of Monomorphisms and singularity categories
arXiv: Representation Theory, 2016Co-Authors: Nan Gao, Chrysostomos PsaroudakisAbstract:In this paper we show that the (un)bounded derived categories$\colon$(i) of the Monomorphism category, (ii) of the morphism category and (iii) of the double morphism category, admit a periodic infinite ladder of recollements. These results are based on a characterization that we provide for a recollement of (compactly generated) triangulated categories to admit a ladder of some height going either upwards or downwards. Moreover, we introduce and study the singularity category of the Monomorphism category over an Artin algebra $\Lambda$ and show that there is a periodic infinite ladder that connects this triangulated category with the standard singularity category of $\Lambda$. We also provide sufficient conditions for the Monomorphism categories of two algebras to be singularly equivalent. The last aim of this paper is to study Monomorphisms where the domain has finite projective dimension. We show that the latter category is a Gorenstein subcategory of the Monomorphism category when $\Lambda$ is a Gorenstein Artin algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory, and show that it carries a structure of a Gorenstein abelian category.
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Monomorphism Categories via Morita rings, Gorenstein algebras, AR sequences and derived equivalences
arXiv: Representation Theory, 2015Co-Authors: Nan Gao, Chrysostomos PsaroudakisAbstract:For any ring R the category of Monomorphisms is a full subcategory of the morphsim category over R, where the latter is equivalent to the module category of the triangular matrix ring with entries the ring R. In this work, we consider the Monomorphism category Mono(R) as a full subcategory of the module category over the Morita ring with all entries the ring R and zero bimodule homomorphisms. This approach provides an interesting link between Morita rings and Monomorphism categories. The aim of this paper is two-fold. First, we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work, which is strongly connected with Monomorphism categories that we study in the second part. In particular, following the work of Ringel-Schmidmeier we investigate Auslander-Reiten sequences in the Monomorphism category mono(A) of an Artin algebra A. The last aim of this paper is to show how we can realize the bounded derived category of the Monomorphism category over an Artin algebra A, as the bounded derived category of an abelian category. This is achieved by using the existence of a natural torsion pair in the exact category mono(A), and by constructing tilting objects in the bounded derived category of mono(A) from tilting A-modules.
Ulrich Koschorke - One of the best experts on this subject based on the ideXlab platform.
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antipodal vector bundle Monomorphisms
Israel Journal of Mathematics, 2002Co-Authors: Ulrich KoschorkeAbstract:In this paper we study existence and classification questions concerning antipodal vector bundle Monomorphismsu (i.e.,u is regularly homotopic to its negative −u). In a metastable dimension range the singularity approach yields complete obstructions which, however, have to be weakened usually in order to become computable. In many situations we determine the resulting “weak, stable” invariants completely; a central role here is played by the antipodality obstructionvi(α, β), a curious combination of Stiefel-Whitney classes. Moreover, in some sample cases we describe precisely how much information gets lost by the transition to these weaker invariants. This involves, e.g., identifying some classical second order obstructions. As an application we exhibit a setting where the difference invariantd(u, −u) distinguishes all (and in fact, infinitely many) regular homotopy classes. Also, we give complete existence and enumeration results for nonstable and stable tangent plane fields on complex projective spaces in terms of explicit numerical conditions.
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Complex and real vector bundle Monomorphisms
Topology and its Applications, 1999Co-Authors: Ulrich KoschorkeAbstract:Abstract Given two complex vector bundles over a closed smooth manifold, we compare results concerning the existence and the (homotopy) classification of complex vector bundle Monomorphisms on one hand and of real ones on the other hand. The two theories are related by transition homomorphisms which turn out to fit into an exact Gysin sequence of normal bordism groups. A detailed study reveals astonishing phenomena, e.g., situations where no complex but infinitely many real Monomorphisms exist. Also all possible combinations of finiteness/infiniteness for the following two numbers occur already over products of spheres: 1. (i) the number of complex Monomorphisms which become homotopic as real Monomorphisms, and 2. (ii) the number of real Monomorphisms which are not homotopic to complex ones.
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Nonstable and stable Monomorphisms of vector bundles
Topology and its Applications, 1997Co-Authors: Ulrich KoschorkeAbstract:Abstract In this paper we discuss the existence and classification problem both for nonstable and for stable Monomorphisms between given vector bundles. The singularity method supplies invariants (complete in a metastable dimension range), together with methods how to compute them. Particular attention is also paid to stabilization, e.g., to the question how many different nonstable homotopy classes of Monomorphisms lie in a stable one.
Pu Zhang - One of the best experts on this subject based on the ideXlab platform.
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Bimodule Monomorphism categories and RSS equivalences via cotilting modules
Journal of Algebra, 2018Co-Authors: Bao-lin Xiong, Pu Zhang, Yuehui ZhangAbstract:Abstract The Monomorphism category M ( A , M , B ) induced by a bimodule M B A is the subcategory of Λ-mod consisting of [ X Y ] ϕ such that ϕ : M ⊗ B Y → X is a monic A-map, where Λ = [ A M 0 B ] , and A, B are Artin algebras. In general, M ( A , M , B ) is not the Monomorphism category induced by quivers. It could describe the Gorenstein-projective Λ-modules. This Monomorphism category is a resolving subcategory of Λ-mod if and only if M B is projective. In this case, it has enough injective objects and Auslander–Reiten sequences, and can be also described as the left perpendicular category of a unique basic cotilting Λ-module. If M satisfies the condition (IP) (see Subsection 1.6 ), then the stable category of M ( A , M , B ) admits a recollement of additive categories, which is in fact a recollement of singularity categories if M ( A , M , B ) is a Frobenius category. Ringel–Schmidmeier–Simson equivalence between M ( A , M , B ) and its dual is introduced. If M is an exchangeable bimodule, then an RSS equivalence is given by a Λ–Λ bimodule which is a two-sided cotilting Λ-module with a special property; and the Nakayama functor N Λ gives an RSS equivalence if and only if both A and B are Frobenius algebras.
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Bimodule Monomorphism categories and RSS equivalences via cotilting modules
2017Co-Authors: Xiong Bao-lin, Pu Zhang, Zhang Yue-huiAbstract:The Monomorphism category $\mathscr{S}(A, M, B)$ induced by a bimodule $_AM_B$ is the subcategory of $\Lambda$-mod consisting of $\left[\begin{smallmatrix} X\\ Y\end{smallmatrix}\right]_{\phi}$ such that $\phi: M\otimes_B Y\rightarrow X$ is a monic $A$-map, where $\Lambda=\left[\begin{smallmatrix} A&M\\0&B \end{smallmatrix}\right]$. In general, it is not the Monomorphism categories induced by quivers. It could describe the Gorenstein-projective $\m$-modules. This Monomorphism category is a resolving subcategory of $\modcat{\Lambda}$ if and only if $M_B$ is projective. In this case, it has enough injective objects and Auslander-Reiten sequences, and can be also described as the left perpendicular category of a unique basic cotilting $\Lambda$-module. If $M$ satisfies the condition ${\rm (IP)}$, then the stable category of $\mathscr{S}(A, M, B)$ admits a recollement of additive categories, which is in fact a recollement of singularity categories if $\mathscr{S}(A, M, B)$ is a {\rm Frobenius} category. Ringel-Schmidmeier-Simson equivalence between $\mathscr{S}(A, M, B)$ and its dual is introduced. If $M$ is an exchangeable bimodule, then an {\rm RSS} equivalence is given by a $\Lambda$-$\Lambda$ bimodule which is a two-sided cotilting $\Lambda$-module with a special property; and the Nakayama functor $\mathcal N_\m$ gives an {\rm RSS} equivalence if and only if both $A$ and $B$ are Frobenius algebras
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Monomorphism Operator and Perpendicular Operator
Communications in Algebra, 2014Co-Authors: Keyan Song, Fan Kong, Pu ZhangAbstract:For a quiver Q, a k-algebra A, and an additive full subcategory 𝒳 of A-mod, the Monomorphism category Mon(Q, 𝒳) is introduced. The main result says that if T is an A-module such that there is an exact sequence 0 → T m → … → T 0 → D(A A ) → 0 with each T i ∈ add(T), then Mon(Q, ⊥ T) =⊥(kQ ⊗ k T); and if T is cotilting, then kQ ⊗ k T is a unique cotilting Λ-module, up to multiplicities of indecomposable direct summands, such that Mon(Q, ⊥ T) =⊥(kQ ⊗ k T). As applications, the category of the Gorenstein-projective (kQ ⊗ k A)-modules is characterized as Mon(Q, 𝒢𝒫(A)) if A is Gorenstein; the contravariantly finiteness of Mon(Q, 𝒳) can be described; and a sufficient and necessary condition for Mon(Q, A) being of finite type is given.
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Monomorphism operator and perpendicular operator
arXiv: Representation Theory, 2013Co-Authors: Keyan Song, Pu ZhangAbstract:For a quiver $Q$, a $k$-algebra $A$, and a full subcategory $\mathcal X$ of $A$-mod, the Monomorphism category ${\rm Mon}(Q, \mathcal X)$ is introduced. The main result says that if $T$ is an $A$-module such that there is an exact sequence $0\rightarrow T_m\rightarrow...\rightarrow T_0\rightarrow D(A_A)\rightarrow 0$ with each $T_i\in {\rm add} (T)$, then ${\rm Mon}(Q, \ ^\perp T) = \ ^\perp (kQ\otimes_k T)$; and if $T$ is cotilting, then $kQ\otimes_k T$ is a unique cotilting $\m$-module, up to multiplicities of indecomposable direct summands, such that ${\rm Mon}(Q, \ ^\perp T)= \ ^\perp (kQ \otimes_k T)$. As applications, the category of the Gorenstein-projective $(kQ\otimes_kA)$-modules is characterized as ${\rm Mon}(Q, \mathcal{GP}(A))$ if $A$ is Gorenstein; the contravariantly finiteness of ${\rm Mon}(Q, \mathcal X)$ can be described; and a sufficient and necessary condition for ${\rm Mon}(Q, A)$ being of finite type is given.
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Auslander-Reiten translations in Monomorphism categories
arXiv: Representation Theory, 2011Co-Authors: Bao-lin Xiong, Pu Zhang, Yuehui ZhangAbstract:We generalize Ringel and Schmidmeier's theory on the Auslander-Reiten translation of the submodule category $\mathcal S_2(A)$ to the Monomorphism category $\mathcal S_n(A)$. As in the case of $n=2$, $\mathcal S_n(A)$ has Auslander-Reiten sequences, and the Auslander-Reiten translation $\tau_{\mathcal{S}}$ of $\mathcal S_n(A)$ can be explicitly formulated via $\tau$ of $A$-mod. Furthermore, if $A$ is a selfinjective algebra, we study the periodicity of $\tau_{\mathcal{S}}$ on the objects of $\mathcal S_n(A)$, and of the Serre functor $F_{\mathcal S}$ on the objects of the stable Monomorphism category $\underline{\mathcal{S}_n(A)}$. In particular, $\tau_{\mathcal S}^{2m(n+1)}X\cong X$ for $X\in\mathcal{S}_n(\A(m, t))$; and $F_{\mathcal S}^{m(n+1)}X\cong X$ for $X\in\underline{\mathcal{S}_n(\A(m, t))}$, where $\A(m, t), \ m\ge1, \ t\ge2,$ are the selfinjective Nakayama algebras.
Yuehui Zhang - One of the best experts on this subject based on the ideXlab platform.
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Bimodule Monomorphism categories and RSS equivalences via cotilting modules
Journal of Algebra, 2018Co-Authors: Bao-lin Xiong, Pu Zhang, Yuehui ZhangAbstract:Abstract The Monomorphism category M ( A , M , B ) induced by a bimodule M B A is the subcategory of Λ-mod consisting of [ X Y ] ϕ such that ϕ : M ⊗ B Y → X is a monic A-map, where Λ = [ A M 0 B ] , and A, B are Artin algebras. In general, M ( A , M , B ) is not the Monomorphism category induced by quivers. It could describe the Gorenstein-projective Λ-modules. This Monomorphism category is a resolving subcategory of Λ-mod if and only if M B is projective. In this case, it has enough injective objects and Auslander–Reiten sequences, and can be also described as the left perpendicular category of a unique basic cotilting Λ-module. If M satisfies the condition (IP) (see Subsection 1.6 ), then the stable category of M ( A , M , B ) admits a recollement of additive categories, which is in fact a recollement of singularity categories if M ( A , M , B ) is a Frobenius category. Ringel–Schmidmeier–Simson equivalence between M ( A , M , B ) and its dual is introduced. If M is an exchangeable bimodule, then an RSS equivalence is given by a Λ–Λ bimodule which is a two-sided cotilting Λ-module with a special property; and the Nakayama functor N Λ gives an RSS equivalence if and only if both A and B are Frobenius algebras.
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Injective objects of Monomorphism categories
Frontiers of Mathematics in China, 2016Co-Authors: Keyan Song, Yuehui ZhangAbstract:For an acyclic quiver Q and a finite-dimensional algebra A, we give a unified form of the indecomposable injective objects in the Monomorphism category Mon(Q,A) and prove that Mon(Q,A) has enough injective objects.
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Injective Objects of Monomorphism Categories
arXiv: Representation Theory, 2013Co-Authors: Keyan Song, Zhanping Wang, Yuehui ZhangAbstract:For an acyclic quiver $Q$ and a finite-dimensional algebra $A$, we give a unified form of the indecomposable injective objects in the Monomorphism category ${\rm Mon}(Q,A)$ and prove that ${\rm Mon}(Q, A)$ has enough injective objects. As applications, we show that for a given self-injective algebra $A$, a tilting object in the stable category $\underline{A}$-mod induces a natural tilting object in the stable Monomorphism category $\underline{\rm Mon}(Q,A)$. We also realize the singularity category of the algebra $kQ\otimes_k A$ as the stable Monomorphism category of the module category of $A$.
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Auslander-Reiten translations in Monomorphism categories
arXiv: Representation Theory, 2011Co-Authors: Bao-lin Xiong, Pu Zhang, Yuehui ZhangAbstract:We generalize Ringel and Schmidmeier's theory on the Auslander-Reiten translation of the submodule category $\mathcal S_2(A)$ to the Monomorphism category $\mathcal S_n(A)$. As in the case of $n=2$, $\mathcal S_n(A)$ has Auslander-Reiten sequences, and the Auslander-Reiten translation $\tau_{\mathcal{S}}$ of $\mathcal S_n(A)$ can be explicitly formulated via $\tau$ of $A$-mod. Furthermore, if $A$ is a selfinjective algebra, we study the periodicity of $\tau_{\mathcal{S}}$ on the objects of $\mathcal S_n(A)$, and of the Serre functor $F_{\mathcal S}$ on the objects of the stable Monomorphism category $\underline{\mathcal{S}_n(A)}$. In particular, $\tau_{\mathcal S}^{2m(n+1)}X\cong X$ for $X\in\mathcal{S}_n(\A(m, t))$; and $F_{\mathcal S}^{m(n+1)}X\cong X$ for $X\in\underline{\mathcal{S}_n(\A(m, t))}$, where $\A(m, t), \ m\ge1, \ t\ge2,$ are the selfinjective Nakayama algebras.