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<p>Compactness of the homotopy categories <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{K}(\mathcal {P})\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textrm{K}(\mathcal {I})\)</EquationSource> </InlineEquation> of graded projective and graded injective dg modules over a dg ring are investigated in view of pure acyclic dg modules. For sufficiently nice non-positive dg rings, we show that the two subcategories <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textrm{K}(\mathcal {P})\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\textrm{K}(\mathcal {I})\)</EquationSource> </InlineEquation> are compactly generated.</p>

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Compactness of the Homotopy Categories of Graded Projective and Injective dg Modules

  • 期刊论文

  • 发表日期：

    2025年10月10日

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  • Xiaoyan Yang

摘要

Compactness of the homotopy categories \(\textrm{K}(\mathcal {P})\) and \(\textrm{K}(\mathcal {I})\) of graded projective and graded injective dg modules over a dg ring are investigated in view of pure acyclic dg modules. For sufficiently nice non-positive dg rings, we show that the two subcategories \(\textrm{K}(\mathcal {P})\) and \(\textrm{K}(\mathcal {I})\) are compactly generated.