Roos axiom holds for quasi-coherent sheaves
Leonid Positselski
TL;DR
The paper establishes that the abelian category of quasi-coherent sheaves on certain schemes satisfies Roos AB4*-n, a nuanced relaxation of Grothendieck's AB4* condition. It provides two complementary proofs: an elementary Čech coresolution argument for quasi-compact semi-separated schemes, yielding AB4*-n with n equal to the number of affine pieces in a cover, and a co-contra correspondence approach for Noetherian schemes of finite Krull dimension that gives explicit bounds via derived-category techniques. A key technical development is the construction of generators of finite projective dimension using very flat quasi-coherent sheaves, together with a hereditary complete cotorsion pair between very flat and contraadjusted QC sheaves, which underpins the AB4*-n conclusions. The results sharpen our understanding of the derived functors of infinite products in X-Qcoh, provide uniform vanishing bounds for higher direct images, and offer a framework to transfer Roos-type properties through naive co-contra equivalences, with explicit bounds depending on the geometric data (N, D, M). Overall, the work advances both constructive and abstract strategies for controlling infinite products in categories of QC sheaves and informs practical resolution techniques in derived settings.
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.