Reflective subcategories
@article{Rada2000ReflectiveS, title={Reflective subcategories}, author={Juan Rada and Manuel Saori{\^a} and Alberto del Valle}, journal={Glasgow Mathematical Journal}, year={2000}, volume={42}, pages={97 - 113} }
Given a full subcategory [Fscr ] of a category [Ascr ], the existence of left [Fscr ]-approximations (or [Fscr ]-preenvelopes) completing diagrams in a unique way is equivalent to the fact that [Fscr ] is reflective in [Ascr ], in the classical terminology of category theory. In the first part of the paper we establish, for a rather general [Ascr ], the relationship between reflectivity and covariant finiteness of [Fscr ] in [Ascr ], and generalize Freyd's adjoint functor theorem (for inclusion…
44 Citations
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References
SHOWING 1-10 OF 31 REFERENCES
Partial cotilting modules and the lattices induced by them
- Mathematics
- 1997
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…
On envelopes with the unique mapping property
- Mathematics
- 1996
We prove that (a) if R is a left coherent ring, then the weak global dimension w D(R) = 2) if and only if every (n – 2)th F–cosyzygy of a finitely presented right R–module has a flat envelope with…
Coherent rings of finite weak global dimension
- Mathematics
- 1998
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,…
Injective and flat covers, envelopes and resolvents
- Mathematics
- 1981
Using the dual of a categorical definition of an injective envelope, injective covers can be defined. For a ringR, every leftR-module is shown to have an injective cover if and only ifR is left…
The Spectrum of a Module Category
- Mathematics
- 2001
Introduction The functor category Definable subcategories Left approximations duality Ideals in the category of finitely presented modules Endofinite modules Krull-Gabriel dimension The infinite…
The Ziegler spectrum of a tame hereditary algebra
- Mathematics
- 1998
Let A be a nite dimensional hereditary algebra of tame representation type. Let Com A be a complete set of indecomposable algebraically compact A-modules (one from each isomorphism class). We are…
Locally Noetherian categories and generalized strictly linearly compact rings. Applications.
- Mathematics
- 1969
Locally Noetherian categories and generalized strictly linearly compact rings. Applications.
Coherence of polynomial rings
- Mathematics
- 1976
The main result is that A (X), the polynomial ring in any number of indeterminates over a coherent ring A of global dimension two, is coherent.
Representation Theory of Artin Algebras II
- Mathematics
- 1974
(1974). Representation Theory of Artin Algebras II. Communications in Algebra: Vol. 1, No. 4, pp. 269-310.