Roos axiom holds for quasi-coherent sheaves

Abstract

We show that the Grothendieck abelian category $X\operatorname{\mathsf{--Qcoh}}$ of quasi-coherent sheaves on a quasi-compact semi-separated scheme $X$ satisfies the Roos axiom $\mathrm{AB}4^*$-$n$: the derived functors of infinite product have finite homological dimension in $X\operatorname{\mathsf{--Qcoh}}$, not exceeding the number $n$ of open subschemes in an affine open covering of $X$. The hereditary complete cotorsion pair (very flat quasi-coherent sheaves, contraadjusted quasi-coherent sheaves) in the abelian category $X\operatorname{\mathsf{--Qcoh}}$ plays the key role in our arguments. Simply put, a suitable very flat quasi-coherent sheaf (or alternatively, a suitable direct sum of locally countably presented flat quasi-coherent sheaves) on $X$ is a generator of finite projective dimension for the abelian category $X\operatorname{\mathsf{--Qcoh}}$.