Polynomial maps of modules

ABSTRACT

The article focuses on three different notions of polynomiality for maps of modules. In addition to the polynomial maps studied by Eilenberg and Mac Lane and the strict polynomial maps (“lois polynomes”) considered by Roby, we introduce numerical maps and investigate their properties. Even though our notion requires the existence of binomial coefficients in the base ring, we argue that it constitutes the correct way to extend Eilenberg and Mac Lane’s polynomial maps of abelian groups to incorporate modules over more general rings. The main theorem propounds that our maps admit a description corresponding, word by word, to Roby’s definition of strict polynomial maps.

KEYWORDS:

• Binomial ring
• numerical map
• polynomial map

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D10
• Secondary: 13C99

Acknowledgment

This research was carried out at Stockholm University under the supervision of Prof. Torsten Ekedahl. We would also like to thank Dr Christine Vespa for invaluable comments on a draft of this manuscript.