Roos Axiom Holds for Quasi-Coherent Sheaves

Abstract

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.