Divided powers and Kähler differentials

TL;DR

This work extends the theory of divided power ($DP$) algebras to general rings by identifying the universal enveloping algebra $U(A)$ and the Kähler differentials in the $DP$ setting. It proves that the Beck $A$-modules form an abelian category equivalent to left $U(A)$-modules, with $U(A)=A_+ ensor_R U(0)$, and constructs the universal $DP$ derivation $d:A ooldsymbol{ abla}^{DP}_{A/R}$, relating $oldsymbol{ abla}^{DP}_{A/R}$ to the classical $oldsymbol{ abla}^{CA}_{A/R}$ via a quotient of the fold construction. For free DP algebras $A= extGamma_R(V)$, the DP differential module satisfies $oldsymbol{ abla}^{DP}_{A/R}=U(A) ensor V$, and explicit decompositions of $oldsymbol{ abla}_{A/R}$ are obtained, clarifying the role of indecomposables and prime powers in the structure. Overall, the paper generalizes and clarifies previous field-specific results to arbitrary rings and provides concrete, computable descriptions of DP Kähler differentials in key cases.

Abstract

Divided power algebras form an important variety of non-binary universal algebras. We identify the universal enveloping algebra and Kähler differentials associated to a divided power algebra over a general commutative ring, simplifying and generalizing work of Roby and Dokas.

Theorems & Definitions (26)

• Definition 2.1
• Remark 2.2
• Proposition 2.3: roby-pd
• Proposition 2.4: roby-pd
• Proposition 3.1
• proof
• Lemma 3.2
• Definition 4.1
• Proposition 4.2
• proof