Illinois Journal of Mathematics

The module theory of divided power algebras

Rohit Nagpal and Andrew Snowden

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Abstract

We study modules for the divided power algebra $\mathbf{D}$ in a single variable over a commutative Noetherian ring $\mathbf{K}$. Our first result states that $\mathbf{D}$ is a coherent ring. In fact, we show that there is a theory of Gröbner bases for finitely generated ideals, and so computations with finitely presented $\mathbf{D}$-modules are in principle algorithmic. We go on to determine much about the structure of finitely presented $\mathbf{D}$-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.

Article information

Source
Illinois J. Math., Volume 61, Number 3-4 (2017), 287-353.

Dates
Received: 9 January 2017
Revised: 9 March 2018
First available in Project Euclid: 22 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1534924829

Digital Object Identifier
doi:10.1215/ijm/1534924829

Mathematical Reviews number (MathSciNet)
MR3845723

Zentralblatt MATH identifier
06932506

Subjects
Primary: 13C 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 16Z05: Computational aspects of associative rings [See also 68W30] 14F30: $p$-adic cohomology, crystalline cohomology

Citation

Nagpal, Rohit; Snowden, Andrew. The module theory of divided power algebras. Illinois J. Math. 61 (2017), no. 3-4, 287--353. doi:10.1215/ijm/1534924829. https://projecteuclid.org/euclid.ijm/1534924829


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