Tangent structures for divided power algebras

TL;DR

This paper develops a tangent structure on divided power algebras, introducing a novel approach that connects differential geometry concepts with algebraic structures, including Kähler differentials and vector fields.

Contribution

It constructs a tangent structure on divided power algebras and demonstrates an adjoint structure involving Kähler differentials, linking algebraic and geometric perspectives.

Findings

Established a tangent structure using semidirect products

Connected tangent structures with Kähler differentials and Zariski cotangent space

Analyzed vector fields and differential bundles in this context

Abstract

We build a tangent structure on the category of divided power algebras using a particular notion of semidirect product. We show that this tangent structure admits an adjoint tangent structure, which involves a version of K\"ahler differentials, and which is similar to the Zariski cotangent space for affine schemes. We study vector fields and differential bundles for these two structures, which correspond respectively to a notion of special derivation, and to the category of modules over the underlying commutative algebra of a given divided power algebra.