Birings and plethories of integer-valued polynomials

Elliott, Jesse

Actes des rencontres du CIRM, Tome 2 (2010) no. 2, pp. 53-58.

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 ${Hom}_A(S,-)$ 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 $A\left[X\right]$ is an initial object in the category of such structures. The $D$-algebra $Int\left(D\right)$ has such a structure if $D=A$ is a domain such that the natural $D$-algebra homomorphism ${\theta }_n:{{⨂}_D}_{i=1}^{n}Int\left(D\right)\to Int\left(D^n\right)$ 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 ${Hom}_D(Int\left(D\right),-)$ 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.

Classification : 13G05, 13F20, 13F05, 16W99
Mots clés : Biring, plethory, integer-valued polynomial.

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