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Covers, precovers, and purity. (English) Zbl 1189.16007

A class \(\mathbb{F}\) of modules is (pre)covering if each module \(M\) has an \(\mathbb{F}\)-(pre)cover \(F\to M\) (\(F\in\mathbb{F}\)). If a class \(\mathbb{F}\) is closed under pure quotient modules, then the following conditions are equivalent: (i) \(\mathbb{F}\) is closed under set indexed direct sums; (ii) \(\mathbb{F}\) is precovering; (iii) \(\mathbb{F}\) is covering.
The pair of classes \((\mathbb{F},\mathbb{G})\) is called cotorsion pair if \(\mathbb{F}^\perp=\mathbb{G}\) and \(\mathbb{F}={^\perp\mathbb{G}}\), where \(\mathbb{F}^\perp=\text{Ker\,Ext}^1(\mathbb{F},-)\), \(^\perp\mathbb{G}=\text{Ker\,Ext}^1(-,\mathbb{G})\). If a class \(\mathbb{F}\) contains the ground ring \(R\) and is closed under extensions, set indexed direct sums, pure submodules and pure quotient modules, then \((\mathbb{F},\mathbb{F}^\perp)\) is a perfect cotorsion pair (in particular, \(\mathbb{F}\) is covering and \(\mathbb{F}^\perp\) is enveloping).
As applications some concrete classes of modules are considered: the kernels of homological functors, the torsion free classes for torsion pairs.

MSC:

16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
16D40 Free, projective, and flat modules and ideals in associative algebras
16D90 Module categories in associative algebras
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Full Text: Euclid

References:

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