## Actes des rencontres du CIRM

Jesse Elliott
Birings and plethories of integer-valued polynomials
Actes des rencontres du CIRM, 2 no. 2: Third International Meeting on Integer-Valued Polynomials (2010), p. 53-58, doi: 10.5802/acirm.34
Article PDF
Class. Math.: 13G05, 13F20, 13F05, 16W99
Keywords: Biring, plethory, integer-valued polynomial.

Résumé - Abstract

Let $A$ and $B$ be commutative rings with identity. An $A$-$B$-biring is an $A$-algebra $S$ together with a lift of the functor $\operatorname{Hom}_A(S,-)$ from $A$-algebras to sets to a functor from $A$-algebras to $B$-algebras. An $A$-plethory is a monoid object in the monoidal category, equipped with the composition product, of $A$-$A$-birings. The polynomial ring $A[X]$ is an initial object in the category of such structures. The $D$-algebra ${\operatorname{Int}}(D)$ has such a structure if $D = A$ is a domain such that the natural $D$-algebra homomorphism $\theta _n: {\bigotimes _D}_{i = 1}^n {\operatorname{Int}}(D) \rightarrow {\operatorname{Int}}(D^n)$ is an isomorphism for $n = 2$ and injective for $n \le 4$. This holds in particular if $\theta _n$ is an isomorphism for all $n$, which in turn holds, for example, if $D$ is a Krull domain or more generally a TV PVMD. In these cases we also examine properties of the functor $\operatorname{Hom}_D({\operatorname{Int}}(D),-)$ from $D$-algebras to $D$-algebras, which we hope to show is a new object worthy of investigation in the theory of integer-valued polynomials.

Bibliography

[1] G. Bergman and A. Hausknecht, Cogroups and Co-Rings in Categories of Associative Rings, Mathematical Surveys and Monographs, Volume 45, American Mathematical Society, 1996.  MR 1387111 |  Zbl 0857.16001
[2] J. Borger and B. Wieland, Plethystic algebra, Adv. Math. 194 (2005) 246–283.  MR 2139914 |  Zbl 1098.13033
[3] P.-J. Cahen and J.-L. Chabert, Integer-Valued Polynomials, Mathematical Surveys and Monographs, vol. 48, American Mathematical Society, 1997.  MR 1421321 |  Zbl 0884.13010
[4] J. Elliott, Binomial rings, integer-valued polynomials, and $\lambda$-rings, J. Pure Appl. Alg. 207 (2006) 165–185.  MR 2244389 |  Zbl 1100.13026
[5] J. Elliott, Universal properties of integer-valued polynomial rings, J. Algebra 318 (2007) 68–92.  MR 2363125 |  Zbl 1129.13022
[6] J. Elliott, Some new approaches to integer-valued polynomial rings, in Commutative Algebra and its Applications: Proceedings of the Fifth Interational Fez Conference on Commutative Algebra and Applications, Eds. Fontana, Kabbaj, Olberding, and Swanson, de Gruyter, New York, 2009.  MR 2606288 |  Zbl 1177.13053
[7] J. Elliott, Biring and plethory structures on integer-valued polynomial rings, to be submitted for publication.
[8] C. J. Hwang and G. W. Chang, Bull. Korean Math. Soc. 35 (2) (1998) 259–268.  MR 1623691 |  Zbl 0917.13002
[9] D. O. Tall and G. C. Wraith, Representable functors and operations on rings, Proc. London Math Soc. (3) 20 (1970) 619–643.  MR 265348 |  Zbl 0226.13007
Copyright Cellule MathDoc 2019 | Credit | Site Map