Roos axiom holds for quasi-coherent sheaves

TL;DR

This paper proves that the category of quasi-coherent sheaves on certain schemes satisfies the Roos axiom AB4*-n, with two different proofs and implications for cotorsion pairs.

Contribution

It establishes the Roos axiom AB4*-n for quasi-coherent sheaves on specific schemes using elementary and conceptual proofs, and discusses related cotorsion pairs.

Findings

The derived functors of infinite direct product have finite homological dimension.

Two proofs of the main result are provided: one elementary, one conceptual.

Hereditary complete cotorsion pairs are discussed in the context of 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.