Constructing Witt–Burnside rings

Elliott, Jesse

doi:10.1016/j.aim.2005.04.014

Cited by 16 publications

(18 citation statements)

References 4 publications

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“…We wish to show that 2 = * . By functoriality and [8,Lemma 4.4], the operation 2 is continuous, where Λ(R) has the Tadic topology. By (2), the subgroup Λ lin (R) of Λ(R) generated by the geometric series forms a subring of Λ(R), ·, 2, and the operations 2 and * agree on this subring.…”

Section: The Ring Of Multiplicative Arithmetic Functionsmentioning

confidence: 99%

Ring structures on groups of arithmetic functions

Elliott

2008

Journal of Number Theory

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Using the theory of Witt vectors, we define ring structures on several well-known groups of arithmetic functions, which in another guise are formal Dirichlet series. The set of multiplicative arithmetic functions over a commutative ring R is shown to have a unique functorial ring structure for which the operation of addition is Dirichlet convolution and the operation of multiplication restricted to the completely multiplicative functions coincides with point-wise multiplication. The group of additive arithmetic functions over R also has a functorial ring structure. In analogy with the ghost homomorphism of Witt vectors, there is a functorial ring homomorphism from the ring of multiplicative functions to the ring of additive functions that is an isomorphism if R is a Q-algebra. The group of rational arithmetic functions, that is, the group generated by the completely multiplicative functions, forms a subring of the ring of multiplicative functions. The latter ring has the structure of a Bin(R)-algebra, where Bin(R) is the universal binomial ring equipped with a ring homomorphism to R. We use this algebra structure to study the order of a rational arithmetic function, as well the powers f α for α ∈ Bin(R) of a multiplicative arithmetic function f . For example, we prove new results about the powers of a given multiplicative arithmetic function that are rational. Finally, we apply our theory to the study of the zeta function of a scheme of finite type over Z.

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Section: The Ring Of Multiplicative Arithmetic Functionsmentioning

confidence: 99%

Ring structures on groups of arithmetic functions

Elliott

2008

Journal of Number Theory

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“…where the right hand side is the Witt-Burnside ring of the polynomial ring Z[X] over G. (Theorem 3.9 in [6], Theorem 1.7 in [4]. See also [3].…”

Section: Polynomial Tambara Functorsmentioning

confidence: 99%

A generalization of the Dress construction for a Tambara functor, and its relation to polynomial Tambara functors

Nakaoka

2013

Advances in Mathematics

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For a finite group G, (semi-)Mackey functors and (semi-)Tambara functors are regarded as G-bivariant analogs of (semi-)groups and (semi-)rings respectively. In fact if G is trivial, they agree with the ordinary (semi-)groups and (semi-)rings, and many naive algebraic properties concerning rings and groups have been extended to these G-bivariant analogous notions.In this article, we investigate a G-bivariant analog of the semi-group rings with coefficients. Just as a coefficient ring R and a monoid Q yield the semigroup ring R[Q], our constrcution enables us to make a Tambara functor T [M ] out of a semi-Mackey functor M , and a coefficient Tambara functor T . This construction is a composant of the Tambarization and the Dress construction.As expected, this construction is the one uniquely determined by the righteous adjoint property. Besides in analogy with the trivial group case, if M is a Mackey functor, then T [M ] is equipped with a natural Hopf structure.Moreover, as an application of the above construction, we also obtain some G-bivariant analogs of the polynomial rings.

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“…Intensive efforts have been made to realize them more naturally. We refer the reader to [3,4,8,10]. The Burnside ring of G has intimate relations with W G .…”

Section: Introductionmentioning

confidence: 97%

Decomposition of the Witt–Burnside ring and Burnside ring of an abelian profinite group

Oh¹

2009

Advances in Mathematics

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Let G, H be abelian profinite groups whose orders are coprime and assume that q ranges over the set of integers. The aim of this paper is to establish an isomorphism of functors W q G • W q H ∼ = W q G×H , where W q G denotes the q-deformed Witt-Burnside ring functor of G introduced in [Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 207 (1) (2007) 151-191]. To do this, we first establish an isomorphism of functors B q G • B q H ∼ = B q G×H , where B q G denotes the q-deformed Burnside ring functor of G which was also introduced in [Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 207 (1) (2007) 151-191]. As an application, we derive a pseudo-multiplicative property of the q-Möbius function associated to the lattice of open subgroups of the direct sum of G and H .

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Constructing Witt–Burnside rings

Cited by 16 publications

References 4 publications

Ring structures on groups of arithmetic functions

Ring structures on groups of arithmetic functions

A generalization of the Dress construction for a Tambara functor, and its relation to polynomial Tambara functors

Decomposition of the Witt–Burnside ring and Burnside ring of an abelian profinite group

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