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Order reduction of $Λ$-marked monomial ideals and weak resolutions

Kai Machida

TL;DR

This work develops a framework for resolving singularities in the category of $\\Lambda$-schemes, marrying $\\mathbb{F}_1$-inspired geometry with classical resolution methods. It constructs compatible $\\Lambda$-structures on blow-ups, both in affine and global settings, by Frobenius lifts and via the universal property of blow-ups (the $W^{\\*}$-monad viewpoint). Central to the approach is the theory of $\\Lambda$-marked monomial ideals and their order reduction, achieved through a two-step induction that reduces from maximal order to monomial data, enabling principalization and a weak resolution for embedded $\\Lambda$-schemes. The methods rely on local blow-up coordinates, SNC divisors, and locally toric structures, giving a robust path to resolutions compatible with $\\Lambda$-structure and their fibers over primes. The results illuminate how $\\Lambda$-equivariant desingularization can be performed in non-standard geometries, with potential extensions to stronger embedded resolutions and broader $\\ ext{F}_1$-geometric frameworks.

Abstract

Borger's theory of $Λ$-spaces imbues algebraic spaces, which include schemes, with an additional structure defined by an extension of the Witt vector functor. Motivated by $\mathbb{F}_1$-geometry, we prove the existence of a weak resolution of singularities in the category of $Λ$-schemes. Our arguments are based on standard arguments in characteristic $0$ using the order reduction of an ideal marked with $Λ$-equivariant data. This paper is based on work from the author's PhD thesis.

Order reduction of $Λ$-marked monomial ideals and weak resolutions

TL;DR

This work develops a framework for resolving singularities in the category of -schemes, marrying -inspired geometry with classical resolution methods. It constructs compatible -structures on blow-ups, both in affine and global settings, by Frobenius lifts and via the universal property of blow-ups (the -monad viewpoint). Central to the approach is the theory of -marked monomial ideals and their order reduction, achieved through a two-step induction that reduces from maximal order to monomial data, enabling principalization and a weak resolution for embedded -schemes. The methods rely on local blow-up coordinates, SNC divisors, and locally toric structures, giving a robust path to resolutions compatible with -structure and their fibers over primes. The results illuminate how -equivariant desingularization can be performed in non-standard geometries, with potential extensions to stronger embedded resolutions and broader -geometric frameworks.

Abstract

Borger's theory of -spaces imbues algebraic spaces, which include schemes, with an additional structure defined by an extension of the Witt vector functor. Motivated by -geometry, we prove the existence of a weak resolution of singularities in the category of -schemes. Our arguments are based on standard arguments in characteristic using the order reduction of an ideal marked with -equivariant data. This paper is based on work from the author's PhD thesis.
Paper Structure (32 sections, 85 theorems, 159 equations)

This paper contains 32 sections, 85 theorems, 159 equations.

Table of Contents

  1. Blow-ups
  2. Affine case
  3. $\Lambda$-structure using Frobenius lifts
  4. $\Lambda$-structure using $W^{\ast}$
  5. General case
  6. Strict transforms
  7. Affine case
  8. General case
  9. Smooth blow-ups and divisors over $\mathbb{Z}$
  10. Local blow-ups
  11. Local blow-up coordinates
  12. Local blow-up coordinates for varieties over perfect fields
  13. Local blow-up coordinates over $\mathbb{Z}$
  14. Simple normal crossings divisors
  15. Locally toric simple normal crossings divisors
  16. ...and 17 more sections

Key Result

Lemma 1.1.1

[lemma]lemma2101 Let $A$ be a ring and $I$ an ideal. $\frac{A}{I^k}$ is $\mathbb{Z}$-flat for all $k$ if and only if the normal cone is $\mathbb{Z}$-flat.

Theorems & Definitions (143)

  • Lemma 1.1.1
  • Lemma 1.1.2
  • Example 1.1.3: Monoid algebras
  • Example 1.1.4: Completions of finitely generated monoid algebras
  • Remark 1.1.5
  • Definition 1.1.6: Graded $\Lambda$-rings
  • Example 1.1.7: Rees algebras of $\Lambda$-rings
  • Lemma 1.1.9
  • Proposition 1.1.10
  • Remark 1.1.11
  • ...and 133 more