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Topologically integrable derivations and additive group actions on affine ind-schemes

Roberto Diaz, Adrien Dubouloz, Alvaro Liendo

TL;DR

This paper extends the classical correspondence between additive group actions and locally nilpotent derivations from finite-type varieties to affine ind-schemes by developing a purely algebraic-topological framework. It introduces restricted exponential homomorphisms $ ext{e}: ext{B} o ext{B}igrace T ig rbracket$ and the notion of topologically integrable iterated higher derivations, establishing a one-to-one correspondence between them. A central result is a slice structure theorem: if a restricted exponential admits a slice, then $ ext{Spec}( ext{B})$ is equivariantly isomorphic to $ ext{Spec}( ext{B}^{ ext{e}}) imes ext{A}^1$ with the $ ext{G}_{a}$-action translating along the $ ext{A}^1$-factor. The paper also develops the sliced and localized theory, showing how local slices yield strong structural decompositions, and provides explicit examples on the affine ind-space to demonstrate the necessity of topological hypotheses and to contrast with finite-type behavior.

Abstract

We develop a theory of additive group actions on affine ind-schemes through a purely algebraic and topological framework. Affine ind-schemes are described via complete, second-countable, linearly topologized rings, and actions of the additive group are encoded by restricted exponential homomorphisms. We introduce the notion of a topologically integrable derivation, a continuous derivation whose formal exponential converges in the sense of restricted power series, and show that this notion provides the correct extension of locally nilpotent derivations to the infinite-dimensional setting. Our first main result establishes a one-to-one correspondence between topologically integrable derivations and additive group actions on affine ind-schemes, extending the classical correspondence for affine varieties. We then investigate the structure of such actions admitting a slice. In this context, we prove an ind-scheme analog of the classical slice theorem: if an additive group action admits a slice, then the underlying affine ind-scheme is equivariantly isomorphic to a product with the affine line, and the action is given by translation on the second factor. Several examples illustrate the necessity of the topological hypotheses and highlight phenomena absent in the finite-type case.

Topologically integrable derivations and additive group actions on affine ind-schemes

TL;DR

This paper extends the classical correspondence between additive group actions and locally nilpotent derivations from finite-type varieties to affine ind-schemes by developing a purely algebraic-topological framework. It introduces restricted exponential homomorphisms and the notion of topologically integrable iterated higher derivations, establishing a one-to-one correspondence between them. A central result is a slice structure theorem: if a restricted exponential admits a slice, then is equivariantly isomorphic to with the -action translating along the -factor. The paper also develops the sliced and localized theory, showing how local slices yield strong structural decompositions, and provides explicit examples on the affine ind-space to demonstrate the necessity of topological hypotheses and to contrast with finite-type behavior.

Abstract

We develop a theory of additive group actions on affine ind-schemes through a purely algebraic and topological framework. Affine ind-schemes are described via complete, second-countable, linearly topologized rings, and actions of the additive group are encoded by restricted exponential homomorphisms. We introduce the notion of a topologically integrable derivation, a continuous derivation whose formal exponential converges in the sense of restricted power series, and show that this notion provides the correct extension of locally nilpotent derivations to the infinite-dimensional setting. Our first main result establishes a one-to-one correspondence between topologically integrable derivations and additive group actions on affine ind-schemes, extending the classical correspondence for affine varieties. We then investigate the structure of such actions admitting a slice. In this context, we prove an ind-scheme analog of the classical slice theorem: if an additive group action admits a slice, then the underlying affine ind-scheme is equivariantly isomorphic to a product with the affine line, and the action is given by translation on the second factor. Several examples illustrate the necessity of the topological hypotheses and highlight phenomena absent in the finite-type case.

Paper Structure

This paper contains 23 sections, 33 theorems, 78 equations.

Table of Contents

  1. Preliminaries
  2. Restricted strict affine ind-schemes as complete second-countable linearly topologized rings
  3. Basic definitions, notations and examples
  4. Restricted exponential homomorphisms and topologically integrable derivations
  5. Restricted exponential homomorphisms
  6. Topologically integrable iterated higher derivations
  7. The correspondence
  8. Basic properties of restricted exponential homomorphisms
  9. Rings of invariants and associated restricted exponential homomorphisms
  10. Operations on restricted exponential homomorphism
  11. Restricted exponential homomorphisms and automorphisms
  12. Topologically integrable derivations setting
  13. Sliced restricted exponential homomorphisms
  14. Restricted exponential homomorphisms with a slice
  15. Restricted exponential homomorphism with a local slice
  16. ...and 8 more sections

Key Result

Lemma 2.6

With the notation of Definition Def:IHD, let $c\colon \mathcal{B}\rightarrow\widehat{\mathcal{B}}$ be the separated completion of $\mathcal{B}$. Then for every continuous iterated higher $\mathcal{A}$-derivation $D=\{ D^{(i)}\} _{i\geq0}$ of $\mathcal{B}$, there exists a unique continuous iterated h

Theorems & Definitions (85)

  • Example 1.1: Pro-polynomial rings
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • proof
  • ...and 75 more