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Hereditary cotilting modules
No full textCotilting bimodules over arbitrary rings give rise to a theory which naturally generalizes Morita dualities in the setting of torsion theory. Here we study the case when the torsion theories cogenerated by a cotilting bimodule are hereditary
Generalizing Cotilting Dualities
No full textAbstractA generalization of cotilting bimodules and cotilting dualities is studied. Moreover, we investigate the structure and the properties of the classes involved in this kind of dualities
Complements to projective almost complete tilting modules
No full textAn equivalent version of the Generalized Nakayama Conjecture states that any projective almost complete tilting module admits a finite number of non-isomorphic indecomposable complements. Motivated by this connection, we investigate the number of possible complements of projective almost complete tilting modules for some particular classes of Artin algebras, namely monomial algebras and algebras with exactly two simple modules
Reflexivity in derived categories
No full textAn adjoint pair of contravariant functors between abelian categories can be extended to the adjoint pair of their derived functors in the associated derived categories. We describe the reflexive complexes and interpret the achieved results in terms of objects of the initial abelian categories. In particular we prove that, for functors of any finite cohomological dimension, the objects of the initial abelian categories which are reflexive as stalk complexes form the largest class where a Cotilting Theorem in the sense of Colby and Fuller [CbF1, Ch. 5] works
On classes defining an homological dimension
No full textA class F of objects of an abelian category A is said to define a homological dimension if for any object in A the length of any F-resolution is uniquely determined. In the present paper we investigate classes satisfying this property
Wakamatsu tilting modules
No full textAbstractWe study a generalization of tilting modules to modules of possibly infinite projective dimension, introduced by Wakamatsu in [J. Algebra 114 (1988) 106–114]. In particular, we characterize these modules in terms of suitable subcategories of finitely generated modules and in terms of cotorsion theories
Prüfer modules over Leavitt path algebras
Full text linkLet LK(E) denote the Leavitt path algebra associated to the finite graph E and field K. For any closed path c
in E, we define and investigate the uniserial, artinian, non-Noetherian left LK(E)-module U_{E,c−1}. The unique simple factor of each proper submodule of U_{E,c−1}is isomorphic to the Chen simple module V_[c∞]. In our main result, we classify those closed paths c for which U_{E,c−1}
is injective. In this situation, U_{E,c−1} is the injective hull of V_[c∞]
Cotorsion pairs, torsion pairs, and Sigma-pure-injective cotilting modules.
No full textIn 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
Derived equivalence induced by infinitely generated n-tilting modules
No full textLet TR be a right n-tilting module over an arbitrary associative ring R. In this paper we prove that there exists an n-tilting module TR′ equiva- lent to TR which induces a derived equivalence between the unbounded derived category D(R) and a triangulated subcategory E⊥ of D(End(T′)) equivalent to the quotient category of D(End(T′)) modulo the kernel of the total left derived functor − ⊗LS′ T ′. If TR is a classical n-tilting module, we have that T = T ′ and the subcategory E⊥ coincides with D(End |(T )), recovering the classical case
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
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