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On the relation between K-flatness and K-projectivity. (English) Zbl 1419.18021

The author gives an alternative proof of a result of H. Krause [Math. Ann. 353, No. 3, 765–781 (2012; Zbl 1323.18003)], which states that the homotopy category \(\mathbf{K}(R-\mathrm{PureProj})\) of pure projective modules over an associative ring \(R\) is compactly generated and equivalent to the Verdier quotient \(\mathbf{K}(R)/\mathbf{T}(R)\), where \(\mathbf{T}(R)\) is the triangulated sub-category of the homotopy category \(\mathbf{K}(R)\) consisting of the pure acyclic complexes. As an application, it is showed that the K-flat complexes defined in [N. Spaltenstein, Compos. Math. 65, No. 2, 121–154 (1988; Zbl 0636.18006)] are, up to homotopy equivalence, the filtered colimits of K-projective complexes of pure projective modules.

MSC:

18G35 Chain complexes (category-theoretic aspects), dg categories
16D40 Free, projective, and flat modules and ideals in associative algebras
18G05 Projectives and injectives (category-theoretic aspects)
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