A Formalization of Divided Powers in Lean

Abstract

Given an ideal $I$ in a commutative ring $A$, a divided power structure on $I$ is a collection of maps $\{γ_n \colon I \to A\}_{n \in \mathbb{N}}$, subject to axioms that imply that it behaves like the family $\{x \mapsto \frac{x^n}{n!}\}_{n \in \mathbb{N}}$, but which can be defined even when division by factorials is not possible in $A$. Divided power structures have important applications in diverse areas of mathematics, including algebraic topology, number theory and algebraic geometry. In this article we describe a formalization in Lean 4 of the basic theory of divided power structures, including divided power morphisms and sub-divided power ideals, and we provide several fundamental constructions, in particular quotients and sums. This constitutes the first formalization of this theory in any theorem prover. As a prerequisite of general interest, we expand the formalized theory of multivariate power series rings, endowing them with a topology and defining evaluation and substitution of power series.