Binomial Ring Research Articles

Binomial rings and homotopy theory

Binomial rings and homotopy theory

Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products

Motivated by the construction of Steenrod cup-i products in the singular cochain algebra of a space and in the algebra of non-commutative differential forms, we define a category of binomial cup-one differential graded algebras over the integers and over prime fields of positive characteristic. The Steenrod and Hirsch identities bind the cup-product, the cup-one product, and the differential in a package that we further enhance with a binomial ring structure arising from a ring of integer-valued rational polynomials. This structure allows us to define the free binomial cup-one differential graded algebra generated by a set and derive its basic properties. It also provides the context for defining restricted triple Massey products, which have a smaller indeterminacy than the classical ones, and hence, give stronger homotopy type invariants.

Idempotent plethories

Let k be a commutative ring with identity. A k-plethory is a commutative k-algebra P together with a comonad structure WP, called the P-Witt ring functor, on the covariant functor that it represents. We say that a k-plethory P is idempotent if the comonad WP is idempotent, or equivalently if the map from the trivial k-plethory k[e] to P is a k-plethory epimorphism. We prove several results on idempotent plethories. We also study the k-plethories contained in K[e], where K is the total quotient ring of k, which are necessarily idempotent and contained in Int(k)={f∈K[e]:f(k)⊆k}. For example, for any ring l between k and K we find necessary and sufficient conditions—all of which hold if k is a integral domain of Krull type—so that the ring Intl(k)=Int(k)∩l[e] has the structure, necessarily unique and idempotent, of a k-plethory with unit given by the inclusion k[e]⟶Intl(k). Our results, when applied to the binomial plethory Int(Z), specialize to known results on binomial rings.

Weil restriction of noncommutative motives

The Weil restriction functor, introduced in the late fifties, was recently extended by Karpenko to the category of Chow motives with integer coefficients. In this article we introduce the noncommutative (=NC) analogue of the Weil restriction functor, where schemes are replaced by dg algebras, and extend it to Kontsevich's categories of NC Chow motives and NC numerical motives. Instead of integer coefficients, we work more generally with coefficients in a binomial ring. Along the way, we extend Karpenko's functor to the classical category of numerical motives, and compare this extension with its NC analogue. As an application, we compute the (NC) Chow motive of the Weil restriction of every smooth projective scheme whose category of perfect complexes admits a full exceptional collection. Finally, in the case of central simple algebras, we describe explicitly the NC analogue of the Weil restriction functor using solely the degree of the field extension. This leads to a “categorification” of the classical corestriction homomorphism between Brauer groups.

Classification and decomposition of the Witt-Burnside ring and Burnside ring of a profinite group

Let G be a profinite group and q be an integer. In this paper, we investigate the structure of the Witt–Burnside ring functor BqG and the generalized Burnside ring functor BGq, which are introduced in Oh [‘q-deformation of Witt–Burnside rings’, Math. Z. 257 (2007) 151–191]. More precisely, we prove a classification theorem as q ranges over the set of integers and a decomposition theorem over the category of binomial rings when G is given by a direct sum of finitely many finite groups whose orders are mutually relatively prime. Also connections between the ring of truncated Witt vectors and Witt–Burnside rings will be dealt with.

Power-Commutative Nilpotent R-Powered Groups

If R is a binomial ring, then a nilpotent R-powered group G is termed power-commutative if for any α ∈ R, [gα, h] = 1 implies [g, h] = 1 whenever gα 6= 1. In this paper, we further contribute to the theory of nilpotent R-powered groups. In particular, we prove that if G is a nilpotent R-powered group of finite type which is not of finite π-type for any prime π ∈ R, then G is PC if and only if it is an abelian R-group.

Ring structures on groups of arithmetic functions

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 .

Binomial rings, integer-valued polynomials, and [formula omitted]-rings

A commutative ring A is said to be binomial if A is torsion-free (as a Z -module) and the element a ( a − 1 ) ( a − 2 ) ⋯ ( a − n + 1 ) / n ! of A ⊗ Z Q lies in A for every a ∈ A and every positive integer n . Binomial rings were first defined circa 1969 by Philip Hall in connection with his groundbreaking work in the theory of nilpotent groups. They have since had further applications to integer-valued polynomials, Witt vectors, and λ -rings. For any set X ¯ , the ring of integer-valued polynomials in Q [ X ¯ ] is the free binomial ring on the set X ¯ . Thus the binomial property provides a universal property for rings of integer-valued polynomials. We give several characterizations of binomial rings and their homomorphic images. For example, we prove that a binomial ring is equivalently a λ -ring A whose Adams operations are all the identity on A . This allows us to construct a right adjoint Bin U for the inclusion from binomial rings to rings which has several applications in commutative algebra and number theory. For example, there is a natural Bin U ( A ) -algebra structure on the universal λ -ring Λ ( A ) , and likewise on the abelian group of multiplicative A -arithmetic functions. Similarly, there is a natural Bin U ( A ) -module structure on the abelian group 1 + a for any ideal a in A with respect to which A is complete.

Binomial Rings#

Let K be a field. A split basic finite-dimensional K-algebra with left quiver Γ is monomial if it can be represented as the path algebra KΓ modulo an ideal generated by paths in Γ, and binomial if it can be represented as KΓ modulo relations of the form p = λq, where λ ∈ K and p and q are paths in Γ. Burgess et al. [Burgess, W. D., Fuller, K. R., Green, E. L., Zacharia, D. (1993). Left monomial rings–A generalization of monomial algebras. Osaka J. Math. 30(3):543–558] define monomial rings, generalizing monomial algebras; we here take this generalization one step further, introducing and studying the class of binomial rings, which includes the classes of binomial algebras, monomial rings, and square-free rings. We further show that every binomial ring has canonically associated with it a binomial algebra, with which the ring shares many properties.

Tight closure commutes with localization in binomial rings

It is proved that tight closure commutes with localization in any domain which has a module finite extension in which tight closure is known to commute with localization. It follows that tight closure commutes with localization in rings, in particular in semigroup or toric rings. Tight closure, introduced in [HHI], is a closure operation performed on ideals in a commutative, Noetherian ring containing a field. Although it has many important applications to commutative algebra and related areas, some basic questions about tight closure remain open. One of the biggest such open problems is whether or not tight closure commutes with localization. For an exposition of tight closure, including the localization problem, see [Hu]. The purpose of this paper is give a simple proof that tight closure commutes with localization for a class of rings that includes semi-group rings and, slightly more generally, the so-called binomial rings, that is, those that are a quotient of a finitely generated algebra by an ideal generated by binomials. Although this is a modest class of rings, the result here includes coordinate rings of all affine toric varieties, and is general enough to substantially improve previous results of a number a mathematicians, including Moty Katzman, Will Traves, Keith Pardue and Karen Chandler, Irena Swanson and myself. Throughout this note, R denotes a Noetherian commutative ring of prime characteristic. We say that tight closure commutes with localization in R if, for all ideals I in R and all multiplicative systems U in R, I*R[U-1] = (IR[U-1])*. The main result is the following: Theorem. Let R be a ring with the following property: for each minimal prime P of R, the quotient R/P has an integral extension domain in which tight closure commutes with localization. Then tight closure commutes with localization in R. Lemma 1. If tight closure commutes with localization in R/P for each minimal prime P of R, then tight closure commutes with localization in R. Received by the editors January 11, 1999 and, in revised form, May 15, 1999. 1991 Mathematics Subject Classification. Primary 13A35.

The relation between burnside rings and combinatorial species

We describe the close relationship between permutation groups and combinatorial species (introduced by A. Joyal, Adv. in Math. 42, 1981, 1–82). There is a bijection Φ between the set of transitive actions (up to isomorphism) of S n on finite sets and the set of “molecular” species of degree n (up to isomorphism). This bijection extends to a ring isomorphism between B(S n ) (the Burnside ring of the symmetric group) and the ring V S n (of virtual species of degree n).Since permutation groups are well known (and often studied using computers) this helps in finding examples and counterexamples in species. The cycle index series of a molecular species, which is hard to compute directly, is proved to be simply the (Pólya) cycle polynomial of the corresponding permutation group. Conversely, several operations which are hard to define in Π n B(S n ) have a natural description in terms of species. Both situations are extended to coefficients in λ-rings and binomial rings in the last section.