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<p>Let <i>X</i> be either a quasi-compact semi-separated scheme, or a Noetherian scheme of finite Krull dimension. We show that the Grothendieck abelian category <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(X{{\mathsf {-Qcoh}}}\)</EquationSource> </InlineEquation> of quasi-coherent sheaves on <i>X</i> satisfies the Roos axiom <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textrm{AB}4^*\)</EquationSource> </InlineEquation>-<i>n</i>: the derived functors of infinite direct product have finite homological dimension in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(X{{\mathsf {-Qcoh}}}\)</EquationSource> </InlineEquation>. In each of the two settings, two proofs of the main result are given: a more elementary one, based on the Čech coresolution, and a more conceptual one, demonstrating existence of a generator of finite projective dimension in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(X{{\mathsf {-Qcoh}}}\)</EquationSource> </InlineEquation> in the semi-separated case and using the co-contra correspondence (with contraherent cosheaves) in the Noetherian case. The hereditary complete cotorsion pair (very flat quasi-coherent sheaves, contraadjusted quasi-coherent sheaves) in the abelian category <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(X{{\mathsf {-Qcoh}}}\)</EquationSource> </InlineEquation> for a quasi-compact semi-separated scheme <i>X</i> is discussed.</p>

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Roos Axiom Holds for Quasi-Coherent Sheaves

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  • 发表日期：

    2026年2月18日

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  • Leonid Positselski

摘要

Let X be either a quasi-compact semi-separated scheme, or a Noetherian scheme of finite Krull dimension. We show that the Grothendieck abelian category \(X{{\mathsf {-Qcoh}}}\) of quasi-coherent sheaves on X satisfies the Roos axiom \(\textrm{AB}4^*\) -n: the derived functors of infinite direct product have finite homological dimension in \(X{{\mathsf {-Qcoh}}}\) . In each of the two settings, two proofs of the main result are given: a more elementary one, based on the Čech coresolution, and a more conceptual one, demonstrating existence of a generator of finite projective dimension in \(X{{\mathsf {-Qcoh}}}\) in the semi-separated case and using the co-contra correspondence (with contraherent cosheaves) in the Noetherian case. The hereditary complete cotorsion pair (very flat quasi-coherent sheaves, contraadjusted quasi-coherent sheaves) in the abelian category \(X{{\mathsf {-Qcoh}}}\) for a quasi-compact semi-separated scheme X is discussed.