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

TL;DR

This paper develops a topological algebraic framework for understanding additive group actions on affine ind-schemes, extending classical results to infinite-dimensional settings through the concept of topologically integrable derivations.

Contribution

It introduces topologically integrable derivations and establishes a correspondence with additive group actions on affine ind-schemes, extending classical finite-dimensional theory.

Findings

One-to-one correspondence between topologically integrable derivations and additive group actions.

An ind-scheme slice theorem analogous to the classical case.

Examples demonstrating the importance of topological conditions.

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.