2024/07/18 by Positselski, Leonid · 1 citation
#Algebraic Geometry (math.AG) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2407.13651
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-Qcoh of quasi-coherent sheaves on X satisfies the Roos axiom AB4^*-n: the derived functors of infinite direct product have finite homological dimension in X-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-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-Qcoh for a quasi-compact semi-separated scheme X is discussed.