Birings and plethories of integer-valued polynomials

Elliott, Jesse

doi:10.5802/acirm.34

Jesse Elliott

Actes des rencontres du CIRM, Volume 2 (2010) no. 2, p. 53-58

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 ${Hom}_{A} (S, -)$ 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 $Int (D)$ has such a structure if $D = A$ is a domain such that the natural $D$ -algebra homomorphism $θ_{n} : {⨂_{D}}_{i = 1}^{n} Int (D) \to Int (D^{n})$ is an isomorphism for $n = 2$ and injective for $n \leq 4$ . This holds in particular if $θ_{n}$ 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 ${Hom}_{D} (Int (D), -)$ 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.

Published online : 2011-10-20
DOI : https://doi.org/10.5802/acirm.34
Classification: 13G05, 13F20, 13F05, 16W99
Keywords: Biring, plethory, integer-valued polynomial.

@article{ACIRM_2010__2_2_53_0,
     author = {Jesse Elliott},
     title = {Birings and plethories of integer-valued polynomials},
     journal = {Actes des rencontres du CIRM},
     publisher = {CIRM},
     volume = {2},
     number = {2},
     year = {2010},
     pages = {53-58},
     doi = {10.5802/acirm.34},
     language = {en},
     url = {https://acirm.centre-mersenne.org/item/ACIRM_2010__2_2_53_0}
}

Elliott, Jesse. Birings and plethories of integer-valued polynomials. Actes des rencontres du CIRM, Volume 2 (2010) no. 2, pp. 53-58. doi : 10.5802/acirm.34. https://acirm.centre-mersenne.org/item/ACIRM_2010__2_2_53_0/

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