1,722,657 research outputs found
When the heart of a faithful torsion pair is a module category
No full textAn abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category (Mitchell (1964) [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. (2007) [8]). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By Colpi et al. (2007) [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
When the heart of a fatithful torsion pair is a module category
No full textAn abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category (Mitchell (1964) [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. (2007) [8]). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By Colpi et al. (2007) [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
Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes
No full textLet 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(lR) and lR is the Bongartz complement of P modulo its P-trace. Moreover, there is a ring epimorphism, phi : R -> S. We characterize the case when phi 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 mu' and nu' between the interval [GenP, P-L I I 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 mu' and nu' 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 mu' and nu, contains all injectives
On the Heart of a faithful torsion theory
No full textIt was shown in [R. Colpi and K. R. Fuller, Trans. Amer. Math. Soc. 359 (2007), no. 2, 741--765 (electronic); MR2255195] that for any ring R and faithful torsion theory (X,Y) on Mod-R there is a cocomplete abelian category H(X,Y), the heart associated to (X,Y), and a tilting object V in H(X,Y) with End(V)≃R and which induces a counterequivalence between the torsion theory, (T,F), generated in H(X,Y) by V and (X,Y).
In this paper it is shown first that if A is any abelian category with a tilting object W such that End(W)≃R and which tilts the torsion theory it generates in A to (X,Y), then A must be equivalent to H(X,Y).
Next, it is shown that if H is an abelian category containing a tilting object V then H must have arbitrary coproducts, indeed must be AB4, and both functors HV=HomA(V,−) and HV′=Ext1A(V,−) from H to Mod-R must commute with coproducts. Furthermore, HV commutes with direct limits iff H is Grothendieck. Also, if (X,Y) is a faithful torsion theory in a module category then H(X,Y) has an injective cogenerator iff (X,Y) is cogenerated by a cotilting module.
Further results include that if (X,Y) is a hereditary cotilting torsion theory then (X,Y) is Grothendieck and that, if (X,Y) is faithful, then H(X,Y) is equivalent to a module category iff it is the heart of the t-structure on Db(Mod-R) generated by a tilting complex (in particular this will be so if X is generated by a tilting module)
Cotilting modules and bimodules
No full textCotilting theory (for arbitrary modules over arbitrary unital rings) extends Morita duality in analogy to the way tilting theory extends Morita equivalence. In particular, cotilting modules generalize injective cogenerators similarly as tilting modules generalize progenerators.
Here, right R-module UR is cotilting if UR has injective dimension ≤1, ExtR(Uα,U)=0 for any cardinal α, and KerHomR(−,U)∩KerExtR(−,U)=0. A bimodule SUR is cotilting if SUR is faithfully balanced and both UR and SU are cotilting modules.
Let UR be a cotilting module and CR the class of all modules cogenerated by UR. Then CR is a torsion-free class and every module has a special CR-precover. The key problem of the cotilting theory is to characterize the subclasses of CR and CS formed by all U-reflexive modules (= the modules for which HomR(−,U) and HomS(−,U) induce a duality) in the case when SUR is a cotilting bimodule.
In the classical works of Müller, the problem was solved in the Morita case, that is, in the case when SUR is a Morita bimodule: the U-reflexive modules are exactly the linearly compact ones.
In Section 1, the authors define the notion of a U-torsionless linearly compact (U-tl.l.c.) module. Colpi [in Interactions between ring theory and representations of algebras (Murcia), 81--93, Dekker, New York, 2000; MR1758403 (2001f:16015); see the following review] proved that if SUR is a cotilting bimodule then any U-tl.l.c. bimodule is U-reflexive. The main result of Section 1---Theorem 1.8---then gives a characterization of the U-tl.l.c. modules among the U-reflexive ones. Corollary 1.9 shows that the two classes coincide iff the class of all reflexive S-modules is closed under submodules. Note that applying this corollary, D'Este recently proved that the two classes may be different in general.
Section 2 deals with constructing cotilting bimodules as Morita duals of tilting bimodules. Assume A and R are Morita dual rings via a Morita bimodule AWR, and AV is a tilting module with S=End(AV). Put SXR=HomA(V,W). By Proposition 2.6, SXR is a cotilting bimodule iff (∗) ExtR(Xα,X)=0 for any cardinal α.
In Section 3, the results are applied to the case when R is a Noetherian serial ring with a self-duality (so A=R). By Theorem 3.4, (∗) always holds, so SXR is a cotilting bimodule. By Proposition 3.7, if R is, moreover, hereditary, then any tilting module RX is a finitistic cotilting module in the sense of Colby, and RX satisfies condition (∗)
Estimates of global dimension
No full textsummary:In this note we show that for a -module, in particular, an almost -tilting module, over a ring with such that has finite flat dimension, the upper bound of the global dimension of can be estimated by the global dimension of and hence generalize the corresponding results in tilting theory and the ones in the theory of -modules. As an application, we show that for a finitely generated projective module over a VN regular ring , the global dimension of its endomorphism ring is not more than the global dimension of
Colpi d'arma da fuoco e dintorni : introduzione
No full textIl seminario si è incentrato sulla riflessione e sull'aggiornamento della lesività da arma da fuoco e da esplosioni, illustrando ai partecipanti le principali informazioni teorico-pratiche relative all'attività autoptica e di indagine giudiziaria in tema di lesività da arma da fuoco ed esplosivi. Sono stati altresì illustrati alcuni protocolli di indagine utilizzabili in sede di sopralluogo e di autopsia
Cotilting bimodules and their dualities
No full textA right R-module UR is said to be cotilting if Cog(UR)=⊥UR, where ⊥UR=KerExt1R(,U). So cotilting modules generalize injective cogenerators. If U is cotilting, then T=(KerHom(−,U),Cog(U)) is a torsion theory, the so-called cotilting torsion theory. A bimodule SUR is cotilting if SUR is faithfully balanced and both UR and SU are cotilting modules. So cotilting bimodules generalize Morita bimodules. The main topic of the paper is the study of dualities induced by cotilting bimodules, as a generalization of the classical theory of Morita dualities.
Let SUR be a cotilting bimodule and put Δ=Hom(,U) and Γ=Ext1(,U). Denote by Y the class of all U-reflexive modules, by C the class of all modules of the form K/L where K,L∈Y, and by X the class of all T-torsion modules from C. These classes are studied in the first part of the paper, in order to prove their closure properties, and show that they are sufficiently large. For example, by Proposition 5, C contains all finitely presented modules.
The main result of the paper is the following "cotilting theorem'': If U is a cotilting bimodule, then Δ and Γ realize a duality between the classes Y and X, respectively. Moreover, there is a natural morphism γM:Γ2(M)→M such that the sequence 0→Γ2(M)@>γM>>M@>δM>>Δ2(M)@>>>0 is exact for all M∈C, where δM is the evaluation map.
In the case of Morita dualities, Müller proved that U-reflexive modules coincide with the linearly compact ones. Inspired by this result, the author studies U-torsionless linearly compact (U-tl.l.c.) modules in the case when U is a cotilting module. In Proposition 10, he proves that if SUR is a cotilting bimodule then any U-tl.l.c. bimodule is U-reflexive, and asks whether the converse is true. (By a recent example of D'Este, this is not true in general.)
Finally, the author introduces abstract "duality conditions'' for a pair of torsion theories. These are necessary conditions for the pair to be cogenerated by a cotilting bimodule. The conditions are not sufficient in general. Nevertheless, by Proposition 13, they ensure uniqueness of the representing bimodule
Tilting modules and -modules
No full textC. Menini and A. Orsatti [Rend. Sem. Mat. Univ. Padova 82 (1989), 203--231 (1990); MR1049594 (91h:16026)] introduced ∗-modules in order to characterize equivalences between certain full subcategories of module categories over two rings. If one restricts the study to the case of finite-dimensional algebras over a field k, it was shown by G. D'Este and the reviewer [Rend. Sem. Mat. Univ. Padova 83 (1990), 77--80; MR1066430 (91i:16027)] that faithful ∗-modules are tilting modules in the sense of the reviewer and C. M. Ringel [Trans. Amer. Math. Soc. 274 (1982), no. 2, 399--443; MR0675063 (84d:16027)]. The paper under review now generalizes this characterization to arbitrary rings using the natural generalizations for tilting modules in this case
Tilting in Grothendieck categories
No full textGiven any Grothendieck category G, we study the notion of a tilting object of G, proving some basic facts of tilting theory in this general setting. Our results apply, for instance, to categories of modules over arbitrary rings, as well as to the theory of sheaves in algebraic geometry
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