Flat morphisms of finite presentation are very flat

Abstract

Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring $R$, $R$-modules built from the rings of functions on principal affine open subschemes in $\mathrm{Spec}R$ using ordinal-indexed filtrations and direct summands are called very flat. The related class of very flat quasi-coherent sheaves over a scheme is intermediate between the classes of locally free and flat sheaves, and has serious technical advantages over both. In this paper we show that very flat modules and sheaves are ubiquitous in algebraic geometry: if $S$ is a finitely presented commutative $R$-algebra which is flat as an $R$-module, then $S$ is a very flat $R$-module. This proves a conjecture formulated in the February 2014 version of the long preprint arXiv:1209.2995. We also show that the (finite) very flatness property of a flat module satisfies descent with respect to commutative ring homomorphisms of finite presentation inducing surjective maps of the spectra.

LaTeX 2e, 56 pages; v.2: small additions in sections 0.1, 0.7, 1.1, 1.7, 5.3, and 5.4, several references added; v.3: small expositional improvements, particularly in section 2; v.4: abstract expanded, section 0.12 inserted; v.5: two references added, several misprints corrected

References in corpus (4)

• On strongly flat and weakly cotorsion modules
• Abelian right perpendicular subcategories in module categories
• Faithfully flat descent for projectivity of modules
• Errata: On the Homology of Completion and Torsion

Cited by in corpus (10)

• $S$-almost perfect commutative rings
• Infinity-tilting theory
• On strongly flat and weakly cotorsion modules
• Abelian right perpendicular subcategories in module categories
• Remarks on derived complete modules and complexes
• Flat commutative ring epimorphisms of almost Krull dimension zero
• Local, colocal, and antilocal properties of modules and complexes over commutative rings
• Countably generated flat modules are quite flat
• Roos axiom holds for quasi-coherent sheaves
• Contraderived categories of CDG-modules