All Modules Have Flat Covers

By L. Bican, R. Bashir, Edgar E. Enochs. 2001

416 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d45aec208729f47e73d90e0d82c2b86bb89e9b02

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In Algebra and Number Theory

In Computational Theory and Mathematics

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Citations in the corpus

Cotorsion pairs and model categories (2007, background): “Theauthoralso thanksthe referee forpointing outa subtlety with hereditary cotorsion pairsthatthe authorand G illespie had both m issed.”
On purity and applications to coderived and singularity categories (2014, methodology): “As mentioned, the key ingredient is the solution to the Flat Cover Conjecture from [BEBE01].”
Approximations and adjoints in homotopy categories (2010, background): “It turns out that El Bashir’s proof of the flat cover conjecture [7, 12] leads to a systematic approach yielding such approximations and adjoints.”
Envelopes and Covers by Modules of Finite FP-Injective and Flat Dimensions (2007, neutral): “So the problem has been studied by many authors (see, for example, Aldrich et al., 2001a,b; Angeleri Hügel et al., 2006; Asensio Mayor and Martinez Hernandez, 1988; Bican et al., 2001; Chen and Ding, 1996; Ding, 1996; Eklof and Trlifaj, 2001; Enochs, 1981, 1984; Enochs and Jenda, 2000; Enochs and Oy”
Covers, precovers, and purity (2008, background): “3; the book [10] is a useful reference, but see also [2], [3], [7], [11], and [13].”
Almost free modules and Mittag-Leffler conditions (2009, background): “The class F is κ– deconstructible where κ is the least infinite cardinal ≥ cardR [5], and also F = C where C denotes the class of pure injective modules.”
Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves (2013, methodology): “It came as a byproduct of Enoch’s proof of the Flat Cover Conjecture [8] that F has enough injectives—these are usually called flat cotorsion modules.”
On exact categories and applications to triangulated adjoints and model structures (2010, methodology): “Several versions of El Bashir’s result have appeared in the literature since then, to our best knowledge the most general of them being the one obtained in [40] by Krause.”
Covers and Directed Colimits (2006, background): “For F the class of all flat modules over any ring with unit the existence of F-covers was proved in [2].”
Cotorsion pairs and model structures on Ch(R) (2011, methodology): “Gillespie followed a method analogous to one in [6] to show that the cotorsion pairs in Ch(R) induced by the”
Kaplansky classes and derived categories (2007, background): “By Lemma B.1, |X | 4 ] to find a flat submodule Fx ⊆ F with x ∈ Fx , |Fx | <κ and F/Fx ∈ F. The direct union � x∈X Fx contains X and is flat.”