The telescope conjecture for module categories

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Let Λ\Lambda be an artin algebra, and let (A,B)(\mathcal{A},\mathcal{B}) be a hereditary cotorsion pair in the category of all left Λ\Lambda-modules. A cotorsion pair is hereditary when Ext⁡Λi(A,B)=0\operatorname{Ext}^i_\Lambda(\mathcal{A},\mathcal{B})=0 for all i≥2i\geq 2; assume that B\mathcal{B} is closed under filtered colimits. The telescope conjecture for module categories. Every module in A\mathcal{A} is a colimit of a filtered system of finitely generated modules from A\mathcal{A}. This is the paper's module-category formulation of the telescope conjecture; the surrounding discussion relates it to finite-type cotorsion pairs, but the source does not state a resolution here.

References

Primary source

Jan Stovicek, “Telescope conjecture, idempotent ideals, and the transfinite radical”, arXiv:0802.2189 (2008).

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