On flat generators and Matlis duality for quasicoherent sheaves

Abstract

We show that for a quasicompact quasiseparated scheme $X$, the following assertions are equivalent: (1) the category $\mathrm{QCoh}(X)$ of all quasicoherent sheaves on $X$ has a flat generator; (2) for every injective object $\mathcal{E}$ of $\mathrm{QCoh}(X)$, the internal hom functor into $\mathcal{E}$ is exact; (3) the scheme $X$ is semiseparated.

8 pages, added Section 3

References in corpus (2)

• Cotilting Sheaves on Noetherian Schemes
• Non-exactness of direct products of quasi-coherent sheaves

Cited by in corpus (3)

• Cotilting Sheaves on Noetherian Schemes
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• Flat relative Mittag-Leffler modules and Zariski locality