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

LaTeX 2e, 27 pages; v.2: the first version v1 corresponds to Section 3 of the updated version v2; new Sections 1, 2, and 4 added; v.3: several misprints corrected, references updated; v.4: several misprints corrected

References in corpus (3)

• Flat comodules and contramodules as directed colimits, and cotorsion periodicity
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• Local, colocal, and antilocal properties of modules and complexes over commutative rings

Cited by in corpus (1)

• The derived $\infty$-category of Frobenius modules