Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fiber criteria for flatness and homomorphisms of flat affine group schemes

Phùng Hô Hai, Hop D. Nguyen, João Pedro dos Santos

TL;DR

This work advances flatness criteria by removing finiteness assumptions and proving fiberwise flatness implies global flatness under a noetherian base—specifically, for $R$ noetherian, $f:R\to A$ flat, and an $A$-module $M$ that is $R$-flat with all fibers flat, $M$ is flat over $A$. It introduces strengthened fiber criteria using Tor-vanishing and torsion submodules to broaden applicable conditions and establishes purity-based criteria for maps. The authors then apply these results to affine group schemes, obtaining faithful-flatness criteria in terms of purity of Hopf-algebra maps and by analyzing restriction functors on representations, thereby connecting flatness theory with Tannakian duality. Counterexamples illustrate the sharpness of the hypotheses. Overall, the paper bridges fiber criteria, purity, and Tannakian methods to yield practical criteria for group-scheme morphisms and representation-based analyses.

Abstract

A very useful result concerning flatness in Algebraic Geometry is EGA's ``fiber'' criterion. We propose similar fiber criteria to verify flatness of a module while avoiding ``finiteness'' assumptions. Motivated by a Tannakian viewpoint (where the category of representations comes to the front), we derive applications to the theory of affine and flat group schemes.

Fiber criteria for flatness and homomorphisms of flat affine group schemes

TL;DR

This work advances flatness criteria by removing finiteness assumptions and proving fiberwise flatness implies global flatness under a noetherian base—specifically, for noetherian, flat, and an -module that is -flat with all fibers flat, is flat over . It introduces strengthened fiber criteria using Tor-vanishing and torsion submodules to broaden applicable conditions and establishes purity-based criteria for maps. The authors then apply these results to affine group schemes, obtaining faithful-flatness criteria in terms of purity of Hopf-algebra maps and by analyzing restriction functors on representations, thereby connecting flatness theory with Tannakian duality. Counterexamples illustrate the sharpness of the hypotheses. Overall, the paper bridges fiber criteria, purity, and Tannakian methods to yield practical criteria for group-scheme morphisms and representation-based analyses.

Abstract

A very useful result concerning flatness in Algebraic Geometry is EGA's ``fiber'' criterion. We propose similar fiber criteria to verify flatness of a module while avoiding ``finiteness'' assumptions. Motivated by a Tannakian viewpoint (where the category of representations comes to the front), we derive applications to the theory of affine and flat group schemes.
Paper Structure (11 sections, 25 theorems, 48 equations)

This paper contains 11 sections, 25 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Grothendieck's flatness criteria
  4. Affine group schemes and Tannakian duality
  5. Fiber criteria for flatness
  6. Further fiber criteria for flatness
  7. Fiber criteria for purity of maps
  8. Counterexamples
  9. Applications to group schemes
  10. Fiber criteria for group scheme homomorphisms
  11. Flatness criteria involving categories of representations

Key Result

Theorem 1.1

Let $R$ be a noetherian ring and $f:R\longrightarrow A$ be a flat homomorphisms of rings. Let $M$ be an $A$-module. Assume that $M$ is flat over $R$ and Then $M$ is flat over $A$.

Theorems & Definitions (48)

  • Theorem 1.1: = Theorem \ref{['prop_fiber']}
  • Theorem 1.2: = Corollary \ref{['cor_fibercriterion_prime']}
  • Theorem 1.3: = Theorem \ref{['surjective_criterion_group_hom']}
  • Theorem 2.2: Grothendieck's fiber criterion for flatness
  • Lemma 2.3
  • Lemma 2.4: Local criterion for flatness, ega, Mat89
  • Lemma 2.5
  • proof
  • Example 2.6
  • Example 2.7
  • ...and 38 more