Alberto del Valle - Academia.edu (original) (raw)
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Papers by Alberto del Valle
Glasgow Mathematical Journal, 2000
Given a full subcategory 3 of a category A, the existence of left 3approximations (or 3-preenvelo... more Given a full subcategory 3 of a category A, the existence of left 3approximations (or 3-preenvelopes) completing diagrams in a unique way is equivalent to the fact that 3 is reflective in A, in the classical terminology of category theory. In the first part of the paper we establish, for a rather general A, the relationship between reflectivity and covariant finiteness of 3 in A, and generalize F'reyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflections with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories. In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcategories of ModR, and has some connections with the "pure-semisimple conjecture".
Communications in Algebra, 1982
The category of left modules over right coherent rings of finite weak global dimension has severa... more The category of left modules over right coherent rings of finite weak global dimension has several nice features. For example, every left module over such a ring has a flat cover (Belshoff, Enochs, Xu) and, if the weak global dimension is at most two, every left module has a flat envelope (Asensio, MartÃnez). We will exploit these features of this category to study its objects. In particular, we will consider orthogonal complements (relative to the extension functor) of several classes of modules in this category. In the case of a commutative ring we describe an idempotent radical on its category of modules which, when the weak global dimension does not exceed 2, can be used to analyze the structure of the flat envelopes and of the ring itself.
Reflective subcategories Given a full subcategory 3 of a category A, the existence of left 3-appr... more Reflective subcategories Given a full subcategory 3 of a category A, the existence of left 3-approximations (or 3-preenvelopes) completing diagrams in a unique way is equivalent to the fact that 3 is reflective in A, in the classical terminology of category theory. In the first part of the paper we establish, for a rather general A, the relationship between reflectivity and covariant finiteness of 3 in A, and generalize F'reyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflec-tions with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories. In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcate...
Communications in Algebra, 1994
Glasgow Mathematical Journal, 2000
Given a full subcategory 3 of a category A, the existence of left 3approximations (or 3-preenvelo... more Given a full subcategory 3 of a category A, the existence of left 3approximations (or 3-preenvelopes) completing diagrams in a unique way is equivalent to the fact that 3 is reflective in A, in the classical terminology of category theory.
Glasgow Mathematical Journal, 2000
Given a full subcategory 3 of a category A, the existence of left 3approximations (or 3-preenvelo... more Given a full subcategory 3 of a category A, the existence of left 3approximations (or 3-preenvelopes) completing diagrams in a unique way is equivalent to the fact that 3 is reflective in A, in the classical terminology of category theory. In the first part of the paper we establish, for a rather general A, the relationship between reflectivity and covariant finiteness of 3 in A, and generalize F'reyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflections with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories. In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcategories of ModR, and has some connections with the "pure-semisimple conjecture".
Communications in Algebra, 1982
The category of left modules over right coherent rings of finite weak global dimension has severa... more The category of left modules over right coherent rings of finite weak global dimension has several nice features. For example, every left module over such a ring has a flat cover (Belshoff, Enochs, Xu) and, if the weak global dimension is at most two, every left module has a flat envelope (Asensio, MartÃnez). We will exploit these features of this category to study its objects. In particular, we will consider orthogonal complements (relative to the extension functor) of several classes of modules in this category. In the case of a commutative ring we describe an idempotent radical on its category of modules which, when the weak global dimension does not exceed 2, can be used to analyze the structure of the flat envelopes and of the ring itself.
Reflective subcategories Given a full subcategory 3 of a category A, the existence of left 3-appr... more Reflective subcategories Given a full subcategory 3 of a category A, the existence of left 3-approximations (or 3-preenvelopes) completing diagrams in a unique way is equivalent to the fact that 3 is reflective in A, in the classical terminology of category theory. In the first part of the paper we establish, for a rather general A, the relationship between reflectivity and covariant finiteness of 3 in A, and generalize F'reyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflec-tions with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories. In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcate...
Communications in Algebra, 1994
Glasgow Mathematical Journal, 2000
Given a full subcategory 3 of a category A, the existence of left 3approximations (or 3-preenvelo... more Given a full subcategory 3 of a category A, the existence of left 3approximations (or 3-preenvelopes) completing diagrams in a unique way is equivalent to the fact that 3 is reflective in A, in the classical terminology of category theory.