In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the exi... more In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the existence of a strong relationship between the ΣΣ-pure-injectivity of the cotilting module and the property of the induced cotorsion pair to be of finite type. In particular for cotilting modules of injective dimension at most 1, or for noetherian rings, the two notions are equivalent. On the other hand we prove that a torsion pair is cogenerated by a ΣΣ-pure-injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a ΣΣ-pure-injective cotilting module of injective dimension at most 1 is necessarily coherent. Finally, for noetherian rings, we characterize cotilting torsion pairs induced by ΣΣ-pure-injective cotilting modules.
Http Dx Doi Org 10 1080 00927879208824561, Jul 5, 2007
ABSTRACT We study various features of the lattice LM of linear module topologies on a module M an... more ABSTRACT We study various features of the lattice LM of linear module topologies on a module M and their impact on the structure of the abstract module. A particular emphasis is given to: a) permanence properties of 1- minimal topologies (=atoms in the subset of LM of Hausdorff topologies) in analogy with the theory of minimal topological groups; b) the usual equivalence between linear topologies and the equivalence classes of the atoms; c) characterization of the abstract modules M such that certain classes in LM are singletons (in particular, modules such that each non-discrete linear topology is topologically artinian). Point c) involves the class of modules having all proper quotients artinian.
An abelian category with arbitrary coproducts and a small projective generator is equivalent to a... more An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category \cite{Mit}. A tilting object in a abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts \cite{CGM}. It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By \cite{CGM} the problem simplifies in understanding when, given an associative ring $R$ and a faithful torsion pair $(\X,\Y)$ in the category of right $R$-modules, the \emph{heart of the $t$-structure} $\H(\X,\Y)$ associated to $(\X,\Y)$ is equivalent to a category of modules. In this paper we give a complete answer to this question, proving necessary and sufficient condition on $(\X,\Y)$ for $\H(\X,\Y)$ to be equivalent to a module category. We analyze in detail the case when $R$ is right artinian.
A class $\mathcal F$ of objects of an abelian category $\mathcal A$ is said to define a \emph{hom... more A class $\mathcal F$ of objects of an abelian category $\mathcal A$ is said to define a \emph{homological dimension} if for any object in $\mathcal A$ the length of any $\mathcal F$-resolution is uniquely determined. In the present paper we investigate classes satisfying this property.
Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we ... more Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we prove that there exists a $n$-tilting module $T'_R$ equivalent to $T_R$ which induces a derived equivalence between the unbounded derived category $\D(R)$ and a triangulated subcategory $\mathcal E_{\perp}$ of $\D(\End(T'))$ equivalent to the quotient category of $\D(\End(T'))$ modulo the kernel of the total left derived functor $-\otimes^{\mathbb L}_{S'}T'$. In case $T_R$ is a classical $n$-tilting module, we get again the Cline-Parshall-Scott and Happel's results.
We describe the structure of totally disconnected minimal !- bounded abelian groups by reducing t... more We describe the structure of totally disconnected minimal !- bounded abelian groups by reducing the description to the case of those of them which are subgroups of powers of the p-adic integers Zp .I n this case the description is obtained by means of a functorial correspondence, based on Pontryagin duality, between topological and linearly topologized groups introduced by Tonolo.
Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we ... more Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we prove that there exists a $n$-tilting module $T'_R$ equivalent to $T_R$ which induces a derived equivalence between the unbounded derived category $\D(R)$ and a triangulated subcategory $\mathcal E_{\perp}$ of $\D(\End(T'))$ equivalent to the quotient category of $\D(\End(T'))$ modulo the kernel of the total left derived functor $-\otimes^{\mathbb L}_{S'}T'$. In case $T_R$ is a classical $n$-tilting module, we get again the Cline-Parshall-Scott and Happel's results.
Proceedings of the Royal Society B: Biological Sciences, 1963
The submerged culture production of ferroverdin, an iron-containing green pigment of a novel type... more The submerged culture production of ferroverdin, an iron-containing green pigment of a novel type, by a new species of Streptomyces Wak. is reported. The morphological properties of this micro-organism are described, and the optimum culture conditions for pigmentation in shake flasks and in stirred fermenters are determined. The Preparation and identification of reductive and alkali degradation products of ferroverdin are reported; on the basis of their structures it is suggested that ferroverdin is the ferrous complex of the p -vinylphenylester of 3-nitroso-4-hydroxybenzoic acid.
Proceedings of the American Mathematical Society, 2011
Let T R T_R be a right n n -tilting module over an arbitrary associative ring R R . In this paper... more Let T R T_R be a right n n -tilting module over an arbitrary associative ring R R . In this paper we prove that there exists an n n -tilting module T R ′ T’_R equivalent to T R T_R which induces a derived equivalence between the unbounded derived category D ( R ) \mathcal {D}(R) and a triangulated subcategory E ⊥ \mathcal E_{\perp } of D ( End ( T ′ ) ) \mathcal {D}(\operatorname {End}(T’)) equivalent to the quotient category of D ( End ( T ′ ) ) \mathcal {D}(\operatorname {End}(T’)) modulo the kernel of the total left derived functor − ⊗ S ′ L T ′ -\otimes ^{\mathbb L}_{S’}T’ . If T R T_R is a classical n n -tilting module, we have that T = T ′ T=T’ and the subcategory E ⊥ \mathcal E_{\perp } coincides with D ( End | ( T ) ) \mathcal {D}(\operatorname {End}|(T)) , recovering the classical case.
In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the exi... more In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the existence of a strong relationship between the ΣΣ-pure-injectivity of the cotilting module and the property of the induced cotorsion pair to be of finite type. In particular for cotilting modules of injective dimension at most 1, or for noetherian rings, the two notions are equivalent. On the other hand we prove that a torsion pair is cogenerated by a ΣΣ-pure-injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a ΣΣ-pure-injective cotilting module of injective dimension at most 1 is necessarily coherent. Finally, for noetherian rings, we characterize cotilting torsion pairs induced by ΣΣ-pure-injective cotilting modules.
Http Dx Doi Org 10 1080 00927879208824561, Jul 5, 2007
ABSTRACT We study various features of the lattice LM of linear module topologies on a module M an... more ABSTRACT We study various features of the lattice LM of linear module topologies on a module M and their impact on the structure of the abstract module. A particular emphasis is given to: a) permanence properties of 1- minimal topologies (=atoms in the subset of LM of Hausdorff topologies) in analogy with the theory of minimal topological groups; b) the usual equivalence between linear topologies and the equivalence classes of the atoms; c) characterization of the abstract modules M such that certain classes in LM are singletons (in particular, modules such that each non-discrete linear topology is topologically artinian). Point c) involves the class of modules having all proper quotients artinian.
An abelian category with arbitrary coproducts and a small projective generator is equivalent to a... more An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category \cite{Mit}. A tilting object in a abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts \cite{CGM}. It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By \cite{CGM} the problem simplifies in understanding when, given an associative ring $R$ and a faithful torsion pair $(\X,\Y)$ in the category of right $R$-modules, the \emph{heart of the $t$-structure} $\H(\X,\Y)$ associated to $(\X,\Y)$ is equivalent to a category of modules. In this paper we give a complete answer to this question, proving necessary and sufficient condition on $(\X,\Y)$ for $\H(\X,\Y)$ to be equivalent to a module category. We analyze in detail the case when $R$ is right artinian.
A class $\mathcal F$ of objects of an abelian category $\mathcal A$ is said to define a \emph{hom... more A class $\mathcal F$ of objects of an abelian category $\mathcal A$ is said to define a \emph{homological dimension} if for any object in $\mathcal A$ the length of any $\mathcal F$-resolution is uniquely determined. In the present paper we investigate classes satisfying this property.
Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we ... more Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we prove that there exists a $n$-tilting module $T'_R$ equivalent to $T_R$ which induces a derived equivalence between the unbounded derived category $\D(R)$ and a triangulated subcategory $\mathcal E_{\perp}$ of $\D(\End(T'))$ equivalent to the quotient category of $\D(\End(T'))$ modulo the kernel of the total left derived functor $-\otimes^{\mathbb L}_{S'}T'$. In case $T_R$ is a classical $n$-tilting module, we get again the Cline-Parshall-Scott and Happel's results.
We describe the structure of totally disconnected minimal !- bounded abelian groups by reducing t... more We describe the structure of totally disconnected minimal !- bounded abelian groups by reducing the description to the case of those of them which are subgroups of powers of the p-adic integers Zp .I n this case the description is obtained by means of a functorial correspondence, based on Pontryagin duality, between topological and linearly topologized groups introduced by Tonolo.
Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we ... more Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we prove that there exists a $n$-tilting module $T'_R$ equivalent to $T_R$ which induces a derived equivalence between the unbounded derived category $\D(R)$ and a triangulated subcategory $\mathcal E_{\perp}$ of $\D(\End(T'))$ equivalent to the quotient category of $\D(\End(T'))$ modulo the kernel of the total left derived functor $-\otimes^{\mathbb L}_{S'}T'$. In case $T_R$ is a classical $n$-tilting module, we get again the Cline-Parshall-Scott and Happel's results.
Proceedings of the Royal Society B: Biological Sciences, 1963
The submerged culture production of ferroverdin, an iron-containing green pigment of a novel type... more The submerged culture production of ferroverdin, an iron-containing green pigment of a novel type, by a new species of Streptomyces Wak. is reported. The morphological properties of this micro-organism are described, and the optimum culture conditions for pigmentation in shake flasks and in stirred fermenters are determined. The Preparation and identification of reductive and alkali degradation products of ferroverdin are reported; on the basis of their structures it is suggested that ferroverdin is the ferrous complex of the p -vinylphenylester of 3-nitroso-4-hydroxybenzoic acid.
Proceedings of the American Mathematical Society, 2011
Let T R T_R be a right n n -tilting module over an arbitrary associative ring R R . In this paper... more Let T R T_R be a right n n -tilting module over an arbitrary associative ring R R . In this paper we prove that there exists an n n -tilting module T R ′ T’_R equivalent to T R T_R which induces a derived equivalence between the unbounded derived category D ( R ) \mathcal {D}(R) and a triangulated subcategory E ⊥ \mathcal E_{\perp } of D ( End ( T ′ ) ) \mathcal {D}(\operatorname {End}(T’)) equivalent to the quotient category of D ( End ( T ′ ) ) \mathcal {D}(\operatorname {End}(T’)) modulo the kernel of the total left derived functor − ⊗ S ′ L T ′ -\otimes ^{\mathbb L}_{S’}T’ . If T R T_R is a classical n n -tilting module, we have that T = T ′ T=T’ and the subcategory E ⊥ \mathcal E_{\perp } coincides with D ( End | ( T ) ) \mathcal {D}(\operatorname {End}|(T)) , recovering the classical case.
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