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 $\mathrm{AB}4^*$-$n$: 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 Cech 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.