Lazard Ring Research Articles

On classifying map of the integral Krichever–Hoehn formal group law

Abstract The classifying map of the integral Krichever–Hoehn formal group law is presented as a quotient map of the Lazard ring by some explicit ideal.

Global group laws and equivariant bordism rings

For every abelian compact Lie group $A$, we prove that the homotopical $A$-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the $A$-equivariant Lazard ring, introduced by Cole--Greenlees--Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian $2$-groups. Our results generalize classical theorems of Quillen (1969) on the connection between non-equivariant bordism rings and formal group laws, and extend the case $A=C_2$ due to Hanke--Wiemeler (2018). We work in the framework of global homotopy theory, which is essential for our proof. In addition to the statements for a fixed group $A$, we also prove a global algebraic universal property that characterizes the collection of all equivariant complex bordism rings simultaneously. We show that they form the universal contravariant functor from abelian compact Lie groups to commutative rings that is equipped with a coordinate; the coordinate is given by the universal Euler class at the circle group. More generally, the ring of $n$-fold cooperations of equivariant complex bordism is shown to be universal among functors equipped with a strict $n$-tuple of coordinates.

An equivariant Quillen theorem

A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which represents the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic Z/2-equivariant unitary bordism ring, introduced by tom Dieck (1970), with the Z/2-equivariant Lazard ring, introduced by Cole–Greenlees–Kriz (2000). Our proof combines a computation of the homotopy theoretic Z/2-equivariant unitary bordism ring due to Strickland (2001) with a detailed investigation of the Z/2-equivariant Lazard ring.

Universal Approach to the Arithmetics of Formal Groups

An approach to the construction of a basis of the Lazard ring by using the coefficients of a universal isogeny is described and examples of the application of this approach are given.

On the structure of algebraic cobordism

In this paper we investigate the structure of algebraic cobordism of Levine–Morel as a module over the Lazard ring with the action of Landweber–Novikov and symmetric operations on it. We show that the associated graded groups of algebraic cobordism with respect to the topological filtration Ω(r)⁎(X) are unions of finitely presented L-modules of very specific structure. Namely, these submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules L/I(p,n)x where deg⁡x≥pn−1p−1. As a corollary we prove the Syzygies Conjecture of Vishik on the existence of certain free L-resolutions of Ω⁎(X), and show that algebraic cobordism of a smooth surface can be described in terms of K0 together with a topological filtration.

Bivariant algebraic cobordism

We associate a bivariant theory to any suitable oriented Borel-Moore homology theory on the category of algebraic schemes or the category of algebraic G-schemes. Applying this to the theory of algebraic cobordism yields operational cobordism rings and operational G-equivariant cobordism rings associated to all schemes in these categories. In the case of toric varieties, the operational T-equivariant cobordism ring may be described as the ring of piecewise graded power series on the fan with coefficients in the Lazard ring.

On numerical equivalence for algebraic cobordism

We define and study the notion of numerical equivalence on algebraic cobordism cycles. We prove that algebraic cobordism modulo numerical equivalence is a finitely generated module over the Lazard ring, and it reproduces the Chow group modulo numerical equivalence. We show this theory defines an oriented Borel–Moore homology theory on schemes and oriented cohomology theory on smooth varieties.We compare it with homological equivalence and smash-equivalence for cobordism cycles. For the former, we show that homological equivalence on algebraic cobordism is strictly finer than numerical equivalence, answering negatively the integral cobordism analogue of the standard conjecture (D). For the latter, using Kimura finiteness on cobordism motives, we partially resolve the cobordism analogue of a conjecture by Voevodsky on rational smash-equivalence and numerical equivalence.

Algebraic Cobordism as a module over the Lazard ring

In this paper we study the structure of the Algebraic Cobordism ring of a variety as a module over the Lazard ring, and show that it has relations in positive codimensions. We actually prove the stronger graded version. This extends the result of Levine and Morel (Algebraic Cobordism. In: Springer Monographs in Mathematics, 2007) claiming that this module has generators in non-negative codimensions. As an application we compute the Algebraic Cobordism ring of a curve. The main tool is Symmetric Operations in Algebraic Cobordism (Vishik, Symmetric operations for all primes and Steenrod operations in Algebraic Cobordism, 2013).

Computing the Krichever genus

Let $\psi$ denote the genus that corresponds to the formal group law having invariant differential $\omega(t)$ equal to $\sqrt{1+p_1t+p_2t^2+p_3t^3+p_4t^4}$ and let $\kappa$ classify the formal group law strictly isomorphic to the universal formal group law under strict isomorphism $x\CP(x)$. We prove that on the rational complex bordism ring the Krichever-H\"ohn genus $\phi_{KH}$ is the composition $\psi\circ \kappa^{-1}$. We construct certain elements $A_{ij}$ in the Lazard ring and give an alternative definition of the universal Krichever formal group law. We conclude that the coefficient ring of the universal Krichever formal group law is the quotient of the Lazard ring by the ideal generated by all $A_{ij}$, $i,j\geq 3$.

From algebraic cobordism to motivic cohomology

Abstract Let S be an essentially smooth scheme over a field of characteristic exponent c. We prove that there is a canonical equivalence of motivic spectra over S MGL/(a 1,a 2,...)[1/c] ≃ Hℤ[1/c] where Hℤ is the motivic cohomology spectrum, MGL is the algebraic cobordism spectrum, and the elements an are generators of the Lazard ring. We discuss several applications including the computation of the slices of ℤ[1/c]-local Landweber exact motivic spectra and the convergence of the associated slice spectral sequences.

Motivic Landweber exactness

We prove a motivic Landweber exact functor theorem. The main result shows the assignment given by a Landweber-type formula involving the \mathsf{MGL} -homology of a motivic spectrum defines a homology theory on the motivic stable homotopy category which is representable by a Tate spectrum. Using a universal coefficient spectral sequence we deduce formulas for operations of certain motivic Landweber exact spectra including homotopy algebraic K -theory. Finally we employ a Chern character between motivic spectra in order to compute rational algebraic cobordism groups over fields in terms of rational motivic cohomology groups and the Lazard ring.

Algebraic cobordism revisited

We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative Donaldson–Thomas theory. We use double point cobordism to prove all the degree 0 conjectures in Donaldson–Thomas theory: absolute, relative, and equivariant.

Witt vectors and equivariant ring spectra applied to cobordism

Given a finite group G we show that Dress and Siebeneicher's ring of G-typical Witt vectors on the Lazard ring, that is, on the polynomial ring on countably many indeterminates over the integers, embeds as a subring of the unitary cobordism ring of G-manifolds. We also show that the ring of G-typical Witt vectors on the Lazard ring embeds as a subring of the ring of homotopy groups of the G-fixed point spectrum of the spectrum MU representing cobordism. The above results are derived by exploiting the interaction between restriction, additive transfer and multiplicative transfer. This interaction is described by two Mackey functors satisfying a distributivity relation encoded in a formalism developed by Tambara.

Moduli space of filtered λ‐ringstructures over a filtered ring

Motivated in part by recent works on the genus of classifying spaces of compact Lie groups, here we study the set of filtered λ‐ring structures over a filtered ring from a purely algebraic point of view. From a global perspective, we first show that this set has a canonical topology compatible with the filtration on the given filtered ring. For power series rings R[[x]], where R is between ℤ and ℚ, with the x‐adic filtration, we mimic the construction of the Lazard ring in formal group theory and show that the set of filtered λ‐ring structures over R[[x]] is canonically isomorphic to the set of ring maps from some “universal” ring U to R. From a local perspective, we demonstrate the existence of uncountably many mutually nonisomorphic filtered λ‐ring structures over some filtered rings, including rings of dual numbers over binomial domains, (truncated) polynomial, and power series rings over ℚ‐algebras.

Modified Stirling Numbers and p -Divisibility in the Universal Typical p k -Series

The 1-dimensional universal formal group law is a power series (in two variables and with coefficients in Lazard's ring) carrying a lot of geometrical and algebraic properties. For a prime p, we study the corresponding “p-localized” formal group law through its associated p k-series, [p k](x) = Σ s≥0 a k,s x s(p−1)+1—the p k-fold iterated formal sum of a variable x. The coefficients a k,s lie in the Brown-Peterson ring BP * = ℝ(p)[v 1,v 2,...] and we describe part of their structure as polynomials in the variables v i with p-local coefficients. This is achieved by introducing a family of filtrations {W ϕ}ϕ≥1 in BP * and studying the value of a k,s in each of the associated (bi)graded rings BP */W ϕ. This allows us to identify, among monomials in a k,s of minimal W ϕ-filtration (1 ≤ ϕ ≤ k), an explicit monomial m ϕ,k,s carrying the lowest possible p-divisibility. The p-local coefficient of m ϕ,k,s is described as a Stirling-type number of the second kind and its actual value is computed up to p-local units. It turns out that m k,k,s not only carries the lowest W k -filtration but, more importantly, the lowest p-divisibility among all other monomials in a k,s . In particular, we obtain a complete description of the p-divisibility properties of each a k,s .

Cobordisme algébrique II

For any field k of characteristic 0 we prove for algebraic cobordism the analogue of a theorem of Quillen on complex cobordism: the cobordism ring of the ground field is the Lazard ring L and for any smooth k-variety X, the algebraic cobordism ring Ω ∗(X) is generated, as an L-module, by 1∈Ω 0(X) and the element of positive degrees. This implies Rost's conjectured degree formula. We also give a relation between the Chow ring, the K 0 of a smooth k-variety X and Ω ∗(X) .

Symmetric Functions, Formal Group Laws, and Lazard's Theorem

Lazard's theorem is a central result in formal group theory; it states that the ring over which the universal formal group law is defined (known as the Lazard ring) is a polynomial algebra over the integers with infinitely many generators. This ring also shows up in algebraic topology as the complex cobordism ring. The main aim of this paper is to show that the polynomial structure of the Lazard ring follows from the polynomial structure of a certain subalgebra of symmetric functions with integer coefficients. The connection between symmetric functions and the Lazard ring is provided by a certain Hopf algebra map from symmetric functions to the covariant bialgebra of a formal group law. We study this map by deriving formulas for the images of certain symmetric functions; in passing, we use this map to prove some symmetric function and Catalan number identities. Based on the above results, we prove Lazard's theorem, and present an application to the construction of certainp-typical formal group laws over the integers. Combinatorial methods play a major role throughout this paper.

Algebraic cobordism in mixed characteristic

We compute the geometric part of algebraic cobordism over Dedekind domains of mixed characteristic after inverting the positive residue characteristics and prove cases of a Conjecture of Voevodsky relating this geometric part to the Lazard ring for regular local bases. The method is by analyzing the slice tower of algebraic cobordism, relying on the Hopkins-Morel isomorphism from the quotient of the algebraic cobordism spectrum by the generators of the Lazard ring to the motivic Eilenberg-MacLane spectrum, again after inverting the positive residue characteristics.