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 first author’s long preprint on contraherent cosheaves (Positselski in Contraherent cosheaves, arXiv:1209.2995 [math.CT]). 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.

仿射方案中的主要仿射开放子集是代数几何基础中的重要工具。给定交换环- [R ，- [R -模块从功能的环建立在主仿射开放subschemes在\（{{\，\ mathrm {规格} \，}} r \）使用序数索引的的过滤和直接被加数被称为非常平。方案中非常平坦的准相干滑轮的相关类别介于局部自由滑轮和平坦滑轮的类别之间，并且在两者上均具有明显的技术优势。在本文中，我们证明了非常平坦的模数和滑轮在代数几何中是普遍存在的：如果S是有限表示的可交换R-代数是平坦的R-模块，则S是非常平坦的R-模块。这证明了第一作者在相干的同捆皮草上的长预印本（2014年2月版）中提出的猜想（Positselski in Contraherent cosheaves，arXiv：1209.2995 [math.CT]）。我们还表明，相对于有限表示的可交换环同构，谱的射影图，平坦模块的（有限）非常平坦的特性满足了下降的要求。