The module theory of divided power algebras

TL;DR

This paper investigates the module theory of divided power algebras, establishing coherence, developing computational tools like Gr"obner bases, and analyzing module structures, with extensions to q-variants and generalized algebras.

Contribution

It introduces a comprehensive module theory for divided power algebras, including coherence, algorithmic methods, and structural insights, extending to broader classes of related algebras.

Findings

D is a coherent ring with a Gr"obner basis theory.

Finitely presented D-modules have well-understood resolutions and invariants.

The divided power algebra in two variables over Z_p is not coherent.

Abstract

We study modules for the divided power algebra $D$ in a single variable over a commutative noetherian ring $k$. Our first result states that $D$ is a coherent ring. In fact, we show that there is a theory of Gr\"obner bases for finitely generated ideals, and so computations with finitely presented $D$-modules are in principle algorithmic. We go on to determine much about the structure of finitely presented $D$-modules, such as: existence of certain nice resolutions, computation of the Grothendieck group, results about injective dimension, and how they interact with torsion modules. Our results apply not just to the classical divided power algebra, but to its $q$-variant as well, and even to a much broader class of algebras we introduce called "generalized divided power algebras." On the other hand, we show that the divided power algebra in two variables over $\mathbf{Z}_p$ is not coherent.