Lazard ring

Lazard ring
Name	Lazard ring
Subject	Algebraic topology; Algebraic geometry
Discovered by	Michel Lazard
Year	1955
Related	Formal group law, Complex cobordism, MU (spectrum), Brown–Peterson cohomology

Lazard ring

The Lazard ring is a universal commutative ring corepresenting one-dimensional commutative formal group laws; it is a foundational object in algebraic topology and algebraic geometry. It connects constructions in Michel Lazard's classification of formal groups with the work of Daniel Quillen on complex cobordism and with structures studied by Bourbaki-related schools and researchers at institutions such as École Polytechnique and Institut des Hautes Études Scientifiques. The ring mediates comparisons among cohomology theories developed by groups around J. F. Adams, Douglas Ravenel, and Friedrich Hirzebruch.

Introduction

The Lazard ring arises as the ring of coefficients for the universal one-dimensional commutative formal group law studied by Michel Lazard in the 1950s, influenced by earlier work of Henri Cartan, Jean-Pierre Serre, and Claude Chevalley. Its relevance was amplified by Daniel Quillen's identification with the coefficient ring of the MU (spectrum) complex cobordism theory, linking it to programs by René Thom and collaborations at Institut de Mathématiques de Jussieu. The object mediates between explicit algebraic constructions used by John Milnor, Samuel Eilenberg, and G. W. Whitehead and the abstract frameworks promoted by Alexander Grothendieck and Jean-Louis Koszul.

Construction and definition

Lazard constructed the ring by presenting generators and relations parameterizing coefficients of a formal power series satisfying associativity and commutativity axioms analyzed in the tradition of Emmy Noether and Oscar Zariski. One forms a polynomial ring over Z with countably many indeterminates indexed in the manner of work by Stefan Banach-era analysts and imposes relations coming from equating coefficients of formal identities familiar from treatments by Ivar Fredholm and Émile Picard. Quillen later reinterpreted the construction via homotopy-theoretic methods developed at Princeton University and Harvard University, identifying the resulting graded ring with the coefficient ring of MU (spectrum) as used by Milnor and Adams in stable homotopy theory.

Properties and structure

The Lazard ring is graded, torsion-free, and has a polynomial algebra structure over Z when localized appropriately; these properties echo classification results of Emil Artin and David Mumford about moduli-like objects. Quillen's theorem yields an isomorphism between the graded Lazard ring and the coefficient ring of MU (spectrum), linking generators to complex cobordism classes studied by René Thom and F. Hirzebruch. The ring admits universal maps to coefficient rings of oriented cohomology theories developed by researchers at Massachusetts Institute of Technology and University of Chicago, including theories by Frank Adams, Brown–Peterson cohomology proponents, and later work by Haynes Miller and Mark Hovey.

As a moduli object it represents a functor on CommRing categories studied in contexts influenced by Alexander Grothendieck's moduli theory and by stacks researched by Jean-Michel Bismut and Pierre Deligne. Its structure supports a Hopf algebroid formulation that becomes central in computations carried out by groups at Princeton University and Ohio State University in the stable homotopy community, including uses in the Adams–Novikov spectral sequence developed by S. P. Novikov and J. F. Adams.

Relation to formal group laws

Formal group laws were systematized by Michel Lazard after motivations rooted in work by David Hilbert and classical algebraic geometers such as Oscar Zariski and André Weil. The Lazard ring is the representing object for the functor sending a commutative ring to its set of one-dimensional commutative formal group laws, paralleling representability themes from Grothendieck's approach to moduli of curves pursued by Pierre Deligne and Jean-Pierre Serre. Relations encode the associativity, commutativity, and identity axioms that appear in the algebraic treatments of series by Arthur Cayley and later formalized in modern algebra by Emmy Noether.

Quillen's identification uses the formal group law associated to MU (spectrum) to exhibit the universality: any formal group law over a ring R factors through a unique graded ring map from the Lazard ring to R, an idea resonant with universal properties in category-theoretic work of Saunders Mac Lane and Grothendieck.

Examples and computations

Explicit calculations identify low-degree generators corresponding to cobordism classes such as complex projective spaces studied by René Thom and calculations by Milnor and Stasheff. Over fields like Q, the Lazard ring simplifies, and comparisons with formal group laws such as the additive and multiplicative laws connect to examples examined by Niels Henrik Abel and Évariste Galois in algebraic series contexts. Computational techniques employ spectral sequences introduced by J. F. Adams and modules over Hopf algebroids developed by Ravenel and Greenlees.

Further explicit work by researchers at institutions including University of Cambridge and University of California, Berkeley produced tables of generators and relations in low degrees, drawing on methods pioneered by Frank Adams, Haynes Miller, and Douglas Ravenel in stable homotopy computations.

Applications in topology and algebraic geometry

In algebraic topology, the Lazard ring underpins complex-oriented cohomology theories such as Complex cobordism, Brown–Peterson cohomology, and elliptic cohomology programs advanced by teams at Princeton University and Bonn University. It provides input to the Adams–Novikov spectral sequence and to computations of stable homotopy groups pursued by communities around S. P. Novikov, J. F. Adams, and Mark Mahowald. In algebraic geometry, the ring informs the study of formal groupoids, formal schemes, and the moduli problems investigated by Grothendieck, Deligne, and researchers linked to IHÉS and CNRS.

Connections extend to number-theoretic themes via formal groups in the work of Kurt Hensel and Ernst Witt, and to arithmetic geometry studied by Gerd Faltings and Jean-Pierre Serre. The Lazard ring also appears in modern derived algebraic geometry approaches propagated by Jacob Lurie and Bertrand Toën.

Historical remarks and development

Michel Lazard's original papers in the 1950s established the classification and universality results that named the ring; subsequent recognition came through Quillen's 1960s work linking it to MU (spectrum) and through the flowering of stable homotopy theory in the hands of J. F. Adams, S. P. Novikov, and Douglas Ravenel. Later developments by Ravenel, Mark Hovey, Haynes Miller, and Jack Morava extended applications to chromatic homotopy theory and to formal groups in arithmetic contexts studied by Jean-Pierre Serre and Barry Mazur. Contemporary research engages with derived and motivic generalizations advanced by Jacob Lurie, Vladimir Voevodsky, and Andrei Suslin.

Category:Algebraic topology