Riccardo Colpi - Academia.edu (original) (raw)
Papers by Riccardo Colpi
Cotilting modules and bimodules over arbitrary associative rings are studied. On the one hand we ... more Cotilting modules and bimodules over arbitrary associative rings are studied. On the one hand we find a connection between reflexive modules with respect to a cotilting (bi)module U and a notion of U-torsionless linear compactness. On the other hand we provide concrete examples of cotilting bimodules over linearly compact noetherian serial rings.
Lecture Notes in Pure and Applied Mathematics, 2004
Transactions of the American Mathematical Society, 2007
D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin K-algebras that... more D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin K-algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring K. Here, employing a notion of * -objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.
J Pure Appl Alg, 2010
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.
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 RRR and a faithful torsion pair (X,Y)(\X,\Y)(X,Y) in the category of right RRR-modules, the \emph{heart of the ttt-structure} H(X,Y)\H(\X,\Y)H(X,Y) associated to (X,Y)(\X,\Y)(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)(\X,\Y)(X,Y) for H(X,Y)\H(\X,\Y)H(X,Y) to be equivalent to a module category. We analyze in detail the case when RRR is right artinian.
Journal of Pure and Applied Algebra, 2015
Firstly we provide a technique to move torsion pairs in abelian categories via adjoint functors a... more Firstly we provide a technique to move torsion pairs in abelian categories via adjoint functors and in particular through Giraud subcategories. We apply this point in order to develop a correspondence between Giraud subcategories of an abelian category C and those of its tilt H(C) i.e., the heart of a t-structure on D b (C) induced by a torsion pair.
Pacific Journal of Mathematics, 2000
Handbook of Tilting Theory, 2007
Journal of Pure and Applied Algebra, 2010
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 Σ-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 Σ-pure injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a Σ-pure injective cotilting module of injective dimension at most 1 is necessarily coherent. Finally, for noetherian rings, we characterize cotilting torsion pairs induced by Σ-pure injective cotilting modules.
Journal of Pure and Applied Algebra, 2007
Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpe... more Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpendicular category of P is equivalent to the full module category Mod-S where S = End( R) and R is the Bongartz complement of P modulo its P-trace. Moreover, there is a ring epimorphism ϕ : R → S. We characterize the case when ϕ is a perfect localization. By [Riccardo Colpi, Alberto Tonolo, Jan Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997) 3225-3237], there exist mutually inverse isomorphisms µ and ν between the interval [GenP, P ⊥ 1 ] in the lattice of torsion classes in Mod-R, and the lattice of all torsion classes in Mod-S. We provide necessary and sufficient conditions for µ and ν to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when R is a Dedekind domain, and if P is finitely presented and R is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of µ and ν contains all injectives.
Journal of Pure and Applied Algebra, 2011
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 [17]). A tilting object in an abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts (Colpi et al. ). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By [8], 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 heart H (X, Y) of the t-structure associated with (X, Y) is equivalent to a category of modules. In this paper, we give a complete answer to this question, proving necessary and sufficient conditions 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.
Journal of Algebra, 1997
Given two rings R and S, we study the category equivalences T T ¡ Y Y, where T T is a torsion cla... more Given two rings R and S, we study the category equivalences T T ¡ Y Y, where T T is a torsion class of R-modules and Y Y is a torsion-free class of S-modules. These Ž . equivalences correspond to quasi-tilting triples R, V, S , where V is a bimodule R S which has, ''locally,'' a tilting behavior. Comparing this setting with tilting bimodules and, more generally, with the torsion theory counter equivalences introduced by Colby and Fuller, we prove a local version of the Tilting Theorem for quasi-tilting triples. A whole section is devoted to examples in case of algebras over a field.
Journal of Algebra, 2007
In [R. Colpi, K.R. Fuller, Tilting objects in abelian categories and quasitilted rings, Trans. Am... more In [R. Colpi, K.R. Fuller, Tilting objects in abelian categories and quasitilted rings, Trans. Amer. Math. Soc., in press] tilting objects in an arbitrary abelian category H are introduced and are shown to yield a version of the classical tilting theorem between H and the category of modules over their endomorphism rings. Moreover, it is shown that given any faithful torsion theory (X , Y) in Mod-R, for a ring R, the corresponding Heart H(X , Y) is an abelian category admitting a tilting object which yields a tilting theorem between the Heart and Mod-R. In this paper we first prove that H(X , Y) is a prototype for any abelian category H admitting a tilting object which tilts to (X , Y) in Mod-R. Then we study AB-type properties of the Heart and commutations with direct limits. This allows us to show, for instance, that any abelian category H with a tilting object is AB4, and to find necessary and sufficient conditions which guarantee that H is a Grothendieck or even a module category. As particular situations, we examine two main cases: when (X , Y) is hereditary cotilting, proving that H(X , Y) is Grothendieck and when (X , Y) is tilting, proving that H(X , Y) is a module category.
Communications in Algebra, 1997
We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary... more We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P1P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rejp-torsion parts, and the elements of L′ by their Trp-torsion-free parts.
Communications in Algebra, 1991
... THE BASIC RING OF A LOCALLY ARTINIAN COMMUTATIVE GROTHENDIECK CATEGORY RICCARDO COLPI, ENRICO... more ... THE BASIC RING OF A LOCALLY ARTINIAN COMMUTATIVE GROTHENDIECK CATEGORY RICCARDO COLPI, ENRICO GREGORIO AND ADALBERTO ORSATTI ... XL, Mem. Mat. (5) 8 (1984), 143-184. [F] KR Fuller, Density and equivalence, J . Algebra 29 (1974), 528-550. ...
Communications in Algebra, 1994
ABSTRACT
Colloquium Mathematicum, 2006
Page 1. COLLOQUIUM MATHEMATICUM VOL. 104 2006 NO. 1 A CLASS OF QUASITILTED RINGS THAT ARE NOT TIL... more Page 1. COLLOQUIUM MATHEMATICUM VOL. 104 2006 NO. 1 A CLASS OF QUASITILTED RINGS THAT ARE NOT TILTED BY RICCARDO COLPI (Padova), KENT R. FULLER (Iowa City, IA) and ENRICO GREGORIO (Verona) Abstract. ...
TRANSACTIONS-AMERICAN MATHEMATICAL …, 2007
D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin K-algebras that... more D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin K-algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring K. Here, employing a notion of * -objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.
Cotilting modules and bimodules over arbitrary associative rings are studied. On the one hand we ... more Cotilting modules and bimodules over arbitrary associative rings are studied. On the one hand we find a connection between reflexive modules with respect to a cotilting (bi)module U and a notion of U-torsionless linear compactness. On the other hand we provide concrete examples of cotilting bimodules over linearly compact noetherian serial rings.
Lecture Notes in Pure and Applied Mathematics, 2004
Transactions of the American Mathematical Society, 2007
D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin K-algebras that... more D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin K-algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring K. Here, employing a notion of * -objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.
J Pure Appl Alg, 2010
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.
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 RRR and a faithful torsion pair (X,Y)(\X,\Y)(X,Y) in the category of right RRR-modules, the \emph{heart of the ttt-structure} H(X,Y)\H(\X,\Y)H(X,Y) associated to (X,Y)(\X,\Y)(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)(\X,\Y)(X,Y) for H(X,Y)\H(\X,\Y)H(X,Y) to be equivalent to a module category. We analyze in detail the case when RRR is right artinian.
Journal of Pure and Applied Algebra, 2015
Firstly we provide a technique to move torsion pairs in abelian categories via adjoint functors a... more Firstly we provide a technique to move torsion pairs in abelian categories via adjoint functors and in particular through Giraud subcategories. We apply this point in order to develop a correspondence between Giraud subcategories of an abelian category C and those of its tilt H(C) i.e., the heart of a t-structure on D b (C) induced by a torsion pair.
Pacific Journal of Mathematics, 2000
Handbook of Tilting Theory, 2007
Journal of Pure and Applied Algebra, 2010
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 Σ-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 Σ-pure injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a Σ-pure injective cotilting module of injective dimension at most 1 is necessarily coherent. Finally, for noetherian rings, we characterize cotilting torsion pairs induced by Σ-pure injective cotilting modules.
Journal of Pure and Applied Algebra, 2007
Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpe... more Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpendicular category of P is equivalent to the full module category Mod-S where S = End( R) and R is the Bongartz complement of P modulo its P-trace. Moreover, there is a ring epimorphism ϕ : R → S. We characterize the case when ϕ is a perfect localization. By [Riccardo Colpi, Alberto Tonolo, Jan Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997) 3225-3237], there exist mutually inverse isomorphisms µ and ν between the interval [GenP, P ⊥ 1 ] in the lattice of torsion classes in Mod-R, and the lattice of all torsion classes in Mod-S. We provide necessary and sufficient conditions for µ and ν to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when R is a Dedekind domain, and if P is finitely presented and R is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of µ and ν contains all injectives.
Journal of Pure and Applied Algebra, 2011
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 [17]). A tilting object in an abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts (Colpi et al. ). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By [8], 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 heart H (X, Y) of the t-structure associated with (X, Y) is equivalent to a category of modules. In this paper, we give a complete answer to this question, proving necessary and sufficient conditions 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.
Journal of Algebra, 1997
Given two rings R and S, we study the category equivalences T T ¡ Y Y, where T T is a torsion cla... more Given two rings R and S, we study the category equivalences T T ¡ Y Y, where T T is a torsion class of R-modules and Y Y is a torsion-free class of S-modules. These Ž . equivalences correspond to quasi-tilting triples R, V, S , where V is a bimodule R S which has, ''locally,'' a tilting behavior. Comparing this setting with tilting bimodules and, more generally, with the torsion theory counter equivalences introduced by Colby and Fuller, we prove a local version of the Tilting Theorem for quasi-tilting triples. A whole section is devoted to examples in case of algebras over a field.
Journal of Algebra, 2007
In [R. Colpi, K.R. Fuller, Tilting objects in abelian categories and quasitilted rings, Trans. Am... more In [R. Colpi, K.R. Fuller, Tilting objects in abelian categories and quasitilted rings, Trans. Amer. Math. Soc., in press] tilting objects in an arbitrary abelian category H are introduced and are shown to yield a version of the classical tilting theorem between H and the category of modules over their endomorphism rings. Moreover, it is shown that given any faithful torsion theory (X , Y) in Mod-R, for a ring R, the corresponding Heart H(X , Y) is an abelian category admitting a tilting object which yields a tilting theorem between the Heart and Mod-R. In this paper we first prove that H(X , Y) is a prototype for any abelian category H admitting a tilting object which tilts to (X , Y) in Mod-R. Then we study AB-type properties of the Heart and commutations with direct limits. This allows us to show, for instance, that any abelian category H with a tilting object is AB4, and to find necessary and sufficient conditions which guarantee that H is a Grothendieck or even a module category. As particular situations, we examine two main cases: when (X , Y) is hereditary cotilting, proving that H(X , Y) is Grothendieck and when (X , Y) is tilting, proving that H(X , Y) is a module category.
Communications in Algebra, 1997
We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary... more We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P1P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rejp-torsion parts, and the elements of L′ by their Trp-torsion-free parts.
Communications in Algebra, 1991
... THE BASIC RING OF A LOCALLY ARTINIAN COMMUTATIVE GROTHENDIECK CATEGORY RICCARDO COLPI, ENRICO... more ... THE BASIC RING OF A LOCALLY ARTINIAN COMMUTATIVE GROTHENDIECK CATEGORY RICCARDO COLPI, ENRICO GREGORIO AND ADALBERTO ORSATTI ... XL, Mem. Mat. (5) 8 (1984), 143-184. [F] KR Fuller, Density and equivalence, J . Algebra 29 (1974), 528-550. ...
Communications in Algebra, 1994
ABSTRACT
Colloquium Mathematicum, 2006
Page 1. COLLOQUIUM MATHEMATICUM VOL. 104 2006 NO. 1 A CLASS OF QUASITILTED RINGS THAT ARE NOT TIL... more Page 1. COLLOQUIUM MATHEMATICUM VOL. 104 2006 NO. 1 A CLASS OF QUASITILTED RINGS THAT ARE NOT TILTED BY RICCARDO COLPI (Padova), KENT R. FULLER (Iowa City, IA) and ENRICO GREGORIO (Verona) Abstract. ...
TRANSACTIONS-AMERICAN MATHEMATICAL …, 2007
D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin K-algebras that... more D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin K-algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring K. Here, employing a notion of * -objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.