1,720,998 research outputs found
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
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
Equivalences between projective and injective modules and Morita duality for Artinian rings
No full textGiven two rings A and R, we study the equivalences between all projective right A-modules and all injective right R-modules. We prove that such equivalences exist if and only if A and R are Artinian with a Morita duality. This naturally generalizes a well-known result on quasi-Frobenius rings
Estimates of global dimension
Full text linksummary: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
Dualities induced by cotilting bimodules
No full textThis paper looks at the dualities induced by cotilting bimodules, which provide a natural generalization of Morita dualities. One of the main goals of the paper is to show that linear compactness also plays a relevant role in this situation, despite the fact that the subcategories involved in cotilting dualities are usually not as wide and do not have closure properties as good as in the Morita case. Ordinary linear compactness does not behave well in this more general context as, for example, the U-reflexive modules with respect to a cotilting bimodule SUR need not be linearly compact. This motivates the study of three generalizations of linear compactness (relative to the bimodule SUR) which are used, among other things, to characterize U-reflexive torsion-free modules. The paper ends with the construction of a cotilting bimodule as a "dual'' of a tilting one and it is shown that this construction is often well behaved from the point of view of linear compactness
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 (∗)
Tilting objects in abelian categories and quasitilted rings
Full text linkIn the paper under review the authors, generalizing classical tilting theory and the theory of quasi-tilted algebras, define and investigate quasi-tilted rings and tilting objects in arbitrary abelian categories.
In more detail, let A be an abelian category and T an object of A such that arbitrary small coproducts of copies of T exist in A. Then T is called a tilting object of A if: (α) T is self-small, that is, the functor HomA(T,−) preserves small coproducts of copies of T; (β) an object A∈A is generated by T if and only if Ext1A(T,A)=0; and (γ) A is the smallest full subcategory of A which contains all objects generated by T and is closed under subobjects. In this setting the authors prove the first main result of the paper, which gives a tilting theorem between A and the module category over the endomorphism ring of T. Then, based on work by D. Happel, I. Reiten and S. O. Smalø [Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88 pp.; MR1327209 (97j:16009)], the authors call a ring R right quasi-tilted provided that there is a split torsion pair in the category Mod-R of right R-modules such that the torsion-free class contains R and the projective dimension of any torsion-free object is at most one. For comparison we recall that an Artin algebra Λ is called quasi-tilted if the category mod-R of finitely generated modules contains a split torsion pair such that the torsion-free class contains Λ and the projective dimension of any torsion-free object is at most one. Then the second main result of the paper characterizes (right) quasi-tilted rings as the endomorphism rings of tilting objects in cocomplete hereditary abelian categories. Further, it is shown that the (right) global dimension of a (right) quasi-tilted ring is at most two and the injective dimension of any torsion module is at most one. Note that all of these results hold true for quasi-tilted Artin algebras. In this connection the authors state and discuss two open problems. The first one asks whether any ring of right global dimension at most two and such that each of its right modules is a direct sum of a module of injective dimension at most one and a module of projective dimension at most one is (right) quasi-tilted (this is true for quasi-tilted Artin algebras by results of Happel, Reiten and Smalø in [op. cit.]). The second problem asks whether a quasi-tilted Artin algebra is quasi-tilted as a ring. For an affirmative answer to the second problem we refer to a recent paper by E. Gregorio ["Every quasitilted algebra is a quasitilted ring'', J. Algebra, to appear].
The paper concludes with an example, further developed and analyzed in the recent paper by the authors and Gregorio [Colloq. Math. 104 (2006), no. 1, 151--156; MR2195804 (2007e:16033)], which shows that the class of right quasi-tilted rings properly extends the class of (right) tilted rings (defined as the endomorphism rings of finitely generated tilting modules over right hereditary rings). Finally, there is an appendix where the authors discuss the behavior of the functor Ext1(−,B) under direct sums in a cocomplete abelian category
Equivalences represented by faithful non-tilting -modules
No full textLet V be an R-module, S=End(RV) and SV∗=HomR(V,Q), where RQ is a fixed cogenerator. Then V is called a ∗-module [C. Menini and A. Orsatti, Rend. Sem. Mat. Univ. Padova 82 (1989), 203--231 (1990); MR1049594 (91h:16026)] if the functors HomR(V,−) and V⊗S− give an equivalence between the category of all R-modules generated by RV and the category of all S-modules cogenerated by SV∗.
It is known that the class of ∗-modules contains faithful quasi-progenerators and tilting modules, and this paper deals with the question whether this containment is proper. The authors prove that if RP is a faithful quasi-progenerator with endomorphism ring A, AT is a tilting module with endomorphism ring S, and both RP and AT are not generators, then RV=RP⊗AT is a faithful ∗-module with endomorphism ring S which is neither a quasi-progenerator nor a tilting module. Moreover, if RP is projective, then RV is a quasi-tilting and a partially tilting module. Note that the above assumptions can be actually satisfied. If A is a ring with a non-projective tilting module AT, PA=A(N)A and R=End(PA), then RV=RP⊗AT is a faithful ∗-module which is neither a quasi-progenerator nor a tilting module, and it is a quasi-tilting and a partially tilting module
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
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