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Actes des rencontres du CIRM

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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
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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.

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