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Power Towers: Purely Inseparable Galois Theory and Foliations in Positive Characteristic

Przemysław Grabowski

TL;DR

This work delivers a unified, Galois-type framework for purely inseparable phenomena in positive characteristic by tying purely inseparable morphisms to finite subalgebras of differential operators. It introduces power towers as foliation-like objects that organize subfields and operator data, and proves Galois-type correspondences for both fields and varieties, with explicit constructions such as Jacobson sequences and saturated subalgebras. A canonical divisor pullback formula for arbitrary purely inseparable morphisms is established, enabling new non-Lüroth-type conclusions in high characteristic. By connecting Jacobson–Bourbaki, Ekedahl, and higher-derivation approaches within a single theory, the paper opens new directions for positive-characteristic geometry and the study of foliations via differential-operator algebras.

Abstract

We build a purely inseparable Galois theory using non-derived commutative algebra. Our theory works on fields and on normal varieties. It says that a purely inseparable morphism corresponds to a finite (saturated) subalgebra of differential operators. Our approach unifies most of the literature about purely inseparable morphisms and shows new research directions. Indeed, our theory extends to a theory that covers all (saturated) subalgebras of differential operators. The extended theory gives a correspondence between the subalgebras and a new notion of \textbf{power towers}. A power tower is an object analogous to a foliation from differential geometry. In particular, it admits its own versions of many key results about foliations, such as a ``fibrations inject into foliations'' and a ``Frobenius theorem''. The latter has a nontrivial twist: the local structure is trivial at every point, but it may differ between points! In general, our analogy between power towers and foliations explains why purely inseparable morphisms are foliation-like, because these morphisms are power towers in an explicit way. Finally, we use our theory to produce a formula for pullbacks of canonical divisors for arbitrary purely inseparable morphisms. We use it to conclude some ``not-L{ü}roth theorems'': if the characteristic is high enough, then any variety purely inseparably covered by an $n$-dimensional projective space has Iitaka dimension equal to $-\infty$ or $n$, i.e., it is not zero.

Power Towers: Purely Inseparable Galois Theory and Foliations in Positive Characteristic

TL;DR

This work delivers a unified, Galois-type framework for purely inseparable phenomena in positive characteristic by tying purely inseparable morphisms to finite subalgebras of differential operators. It introduces power towers as foliation-like objects that organize subfields and operator data, and proves Galois-type correspondences for both fields and varieties, with explicit constructions such as Jacobson sequences and saturated subalgebras. A canonical divisor pullback formula for arbitrary purely inseparable morphisms is established, enabling new non-Lüroth-type conclusions in high characteristic. By connecting Jacobson–Bourbaki, Ekedahl, and higher-derivation approaches within a single theory, the paper opens new directions for positive-characteristic geometry and the study of foliations via differential-operator algebras.

Abstract

We build a purely inseparable Galois theory using non-derived commutative algebra. Our theory works on fields and on normal varieties. It says that a purely inseparable morphism corresponds to a finite (saturated) subalgebra of differential operators. Our approach unifies most of the literature about purely inseparable morphisms and shows new research directions. Indeed, our theory extends to a theory that covers all (saturated) subalgebras of differential operators. The extended theory gives a correspondence between the subalgebras and a new notion of \textbf{power towers}. A power tower is an object analogous to a foliation from differential geometry. In particular, it admits its own versions of many key results about foliations, such as a ``fibrations inject into foliations'' and a ``Frobenius theorem''. The latter has a nontrivial twist: the local structure is trivial at every point, but it may differ between points! In general, our analogy between power towers and foliations explains why purely inseparable morphisms are foliation-like, because these morphisms are power towers in an explicit way. Finally, we use our theory to produce a formula for pullbacks of canonical divisors for arbitrary purely inseparable morphisms. We use it to conclude some ``not-L{ü}roth theorems'': if the characteristic is high enough, then any variety purely inseparably covered by an -dimensional projective space has Iitaka dimension equal to or , i.e., it is not zero.

Paper Structure

This paper contains 69 sections, 85 theorems, 280 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A Brief Literature Overview
  3. What Is a Power Tower? Why Is It Like a Foliation?
  4. The Basic Example
  5. Galois Theory for Power Towers
  6. Foliations, $1$-Foliations, and $\infty$-Foliations
  7. Canonical Divisor Formula
  8. Preliminaries
  9. Definitions and Basic Facts about Fields
  10. Purely Inseparable
  11. $p$-basis and separable transcendental basis
  12. Purely Inseparable Morphisms and Frobenius Morphisms
  13. Saturation
  14. Differential Operators
  15. Inductive Definition
  16. ...and 54 more sections

Key Result

Theorem 1.10

Each of the following data from Example intro - example can be computed from any other: Explicitly, these computations are done by the following operations: Above, we used the notation: $X=\operatorname{Spec}(k[x,y]), Y_n=\operatorname{Spec}(k[x^{p^n},y])$.

Figures (4)

  • Figure 1: A visualization of a subalgebra $\mathcal{D}\subset \operatorname{Diff}_k(K)$ for orders satisfying $n>r_1>r_2=r_3>r_4=r_5=\ldots>0$, where $p^{r_i}=[W_{i}:W_{i-1}]$ for $i>0$.
  • Figure 2: A visualisation of a subalgebra corresponding to a $\infty$-foliation.
  • Figure 3: The Jacobson sequence for $k[x^{p^2}, y^p, y^{p+1}-x^p]\to k[x,y]$.
  • Figure 4: A visualization of being non-Ekedahl for $k[x^{p^2},y^p, y^{p+1}-x^p]\to k[x,y]$.

Theorems & Definitions (292)

  • Definition 1.1: Purely Inseparable Ring Map
  • Definition 1.2: Purely Inseparable Morphism
  • Definition 1.5: Up to $p^n$-powers.
  • Definition 1.6: Power Tower
  • Example 1.9
  • Theorem 1.10: Galois-Type Correspondence for a Power Tower
  • Theorem 1.11: Slogan version of Theorem \ref{['lifting => extension - thm']}
  • Theorem 1.12: Canonical Divisor Formula: Theorems \ref{['Canonical Divisor Formula - theorem']} and \ref{['canonical divisor for n-foli and ekedahl - prop']}
  • Theorem 1.14: Theorem \ref{['big not-luroth theorem']}
  • Definition 2.1
  • ...and 282 more