Formal Group Law Research Articles

Landweber exact formal group laws and smooth cohomology theories

The main aim of this paper is the construction of a smooth (sometimes called differential) extension \hat{MU} of the cohomology theory complex cobordism MU, using cycles for \hat{MU}(M) which are essentially proper maps W\to M with a fixed U(n)-structure and U(n)-connection on the (stable) normal bundle of W\to M. Crucial is that this model allows the construction of a product structure and of pushdown maps for this smooth extension of MU, which have all the expected properties. Moreover, we show, using the Landweber exact functor principle, that \hat{R}(M):=\hat{MU}(M)\otimes_{MU^*}R defines a multiplicative smooth extension of R(M):=MU(M)\otimes_{MU^*}R whenever R is a Landweber exact MU*-module. An example for this construction is a new way to define a multiplicative smooth K-theory.

Geometric criteria for Landweber exactness

The purpose of this paper is to give a new presentation of some of the main results concerning Landweber exactness in the context of the homotopy theory of stacks. We present two new criteria for Landweber exactness over a flat Hopf algebroid. The first criterion is used to classify stacks arising from Landweber exact maps of rings. Using as extra input only Lazard's theorem and Cartier's classification of p-typical formal group laws, this result is then applied to deduce many of the main results concerning Landweber exactness in stable homotopy theory and to compute the Bousfield classes of certain BP-algebra spectra. The second criterion can be regarded as a generalization of the Landweber exact functor theorem, and we use it to give a proof of the original theorem.

On the classifying ring for Abel formal group laws

The ring over which the universal Abel formal group law is defined is characterized by an integrality condition. Three localizations of this ring considered by Bukhshtaber and Kholodov are given simple descriptions in terms of integer-valued polynomials, and the Abel formal group over one of these rings is shown to be multiplicative.

On the BP-homology of 2e × 2e

AbstractWe study υ0- and υ1-divisibility properties of the [2e]-series associated to the universal 2-typical formal group law. This allows us to identify elements annihilating the toral class τ in BP*(2e × 2e). We conjecture that these elements form a minimal system of generators of the annihilator ideal of τ. This would provide a Landweber-type presentation for the BP-homology of 2e × 2e from which the relation hom:dimBP (Z2e × Z2e) = 2 would be an easy consequence.

Variation of the unit root along the Dwork family of Calabi–Yau varieties

We study the variation of the unit roots of members of the Dwork families of Calabi–Yau varieties over a finite field by the method of Dwork–Katz and also from the point of view of formal group laws. A p-adic analytic formula for the unit roots away from the Hasse locus is obtained.

Extended powers and Steenrod operations in algebraic geometry

Steenrod operations were dened by Voedvodsky in motivic cohomology in order to prove the Milnor and Bloch-Kato conjectures. These operations have also been constructed by Brosnan for Chow rings [5]. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry, for generalized cohomology theories whose formal group law has order two. We adapt the methods used by Bisson-Joyal in studying Steenrod and Dyer-Lashof operations in unoriented cobordism and mod 2 cohomology.

Extrinsic properties of automorphism groups of formal groups

We prove two conjectures on the automorphism group of a one-dimensional formal group law defined over a field of positive characteristic. The first is that if a series commutes with a nontorsion automorphism of the formal group law, then that series is already an automorphism. The second is that the group of automorphisms is its own normalizer in the group of all invertible series over the ground field. A consequence of these results is that a formal group law in positive characteristic is determined by any one of its nontorsion automorphisms.

Classification of the ring of Witt vectors and the necklace ring associated with the formal group law [formula omitted

In this paper, we classify the ring of Witt vectors and the necklace ring associated with the formal group law X + Y − q X Y up to strict-isomorphism as q varies over the set of integers. We also introduce a q-deformation of the Grothendieck ring of formal power series with constant term 1 over rings in which q ⋅ 1 is a unit. Various bijections related to Witt vectors, necklaces, and formal power series are given. As an application, we provide a q-analogue of the celebrated cyclotomic identity and its bijective proof.

Odd primary Steenrod algebra, additive formal group laws, and modular invariants

We give a conceptual clarification of Milnor's theorem, which tells us the Hopf algebra structure of the stable co-operations H * H in the odd primary ordinary cohomology. Directly connecting H * H with the quasi-strict automorphism group of some 1 -dimensional additive formal group law and modular invariants, we give a new proof of this theorem of Milnor.

Arbeitsgemeinschaft mit aktuellem Thema: Algebraic Cobordism

Over the years, many different types and flavors of cohomology theories for algebraic varieties have been constructed. Theories like étale cohomology or de Rham cohomology provide algebraic versions of the topological theory of singular cohomology. The Chow ring and algebraic K_0 are other (partial) examples, more directly tied to algebraic geometry. The partial theory K_0^{alg} was extended to a full theory with the advent of Quillen's higher algebraic K -theory. It took considerably longer for the Chow ring to be extended to motivic cohomology. In the process of doing so, Voevodsky developed his category of motives, and this construction was put in a more general setting with the development by Morel–Voevodskyof of \mathbb{A}^1 homotopy theory. This enabled a systematic construction of cohomology theories on algebraic varieties, with algebraic K -theory and motivic cohomology being only two fundamental examples. These two cohomology theories have in common the existence of a good theory of push-forward maps for projective morphisms. Not all cohomology theories have this structure, those that do are called oriented . In the Morel–Voevodsky stable homotopy category, the universal oriented theory is represented by the \mathbb{P}^1 -spectrum M\mathbb G\ell , an algebraic version of the classical Thom spectrum MU . The corresponding cohomology theory M\mathbb G\ell^{*,*} is called higher algebraic cobordism . In an attempt to better understand the theory M\mathbb G\ell^{*,*} , Levine and Morel constructed a theory of algebraic cobordism \Omega^* . This is (conjecturally) related to M\mathbb G\ell^{*,*} as the classical Chow ring \mathrm{CH}^* is to motivic cohomology and like \mathrm{CH}^* , \Omega^* has a purely algebro-geometric description. In addition to giving some insight into M\mathbb G\ell^{*,*} , \Omega^* gives a simultaneous presentation of both \mathrm{CH}^* and K_0 , exhibiting K_0 as a deformation of \mathrm{CH}^* . \Omega^* has also been used to give conceptually simple proofs of various “degree formulas” first formulated by Rost. These degree formulas have been used in the study of Pfister quadrics and norm varieties, properties of which are used in the proofs of the Milnor conjecture and the Bloch–Kato conjecture. In this workshop, we describe aspects of the topological theory of complex cobordism which are important for algebraic cobordism (Lectures 1-3) and give the construction of \Omega^* and proofs of its fundamental properties (Lectures 4-7). In lectures 8-11, we show how K_0 and \mathrm{CH}^* are described by \Omega^* , how \Omega^* recovers the universal formal group law, give the proof the generalized degree formula for \Omega^* and use this to proof the degree formula for the Segre class. Additional applications to Steenrod operations, further degree formulas and the use of these in the study of quadrics and other varietes is given in lectures 12 and 13. Lectures 14 and 15 concern the construction of funtorial pull-backs in algebraic cobordism. The two concluding lectures (16 and 17) give a quick sketch of the Morel–Voevodsky \mathbb{A}^1 stable homotopy category and describe what we know about M\mathbb G\ell and its relation to motivic cohomology and \Omega^* . The workshop Algebraic Cobordism ,organised by Marc Levin (Boston) and Fabien Morel (München) was held April 4th–April 8th, 2005. This meeting was well attended with 55 participants.

Morava K-theory ring for a quasi-dihedral group in Chern classes

Morava K-theory ring for a quasi-dihedral group is calculated in terms of Chern characteristic classes and the Honda formal group law.

The Steenrod algebra and the automorphism group of additive formal group law

The Steenrod algebra and the automorphism group of additive formal group law

Formal groups and stable homotopy of commutative rings

We explain a new relationship between formal group laws and ring spectra in stable homotopy theory. We study a ring spectrum denoted DB which depends on a commutative ring B and is closely related to the topological Andre-Quillen homology of B. We present an explicit construction which to every 1-dimensional and commutative formal group law F over B associates a morphism of ring spectra F_*: HZ --> DB from the Eilenberg-MacLane ring spectrum of the integers. We show that formal group laws account for all such ring spectrum maps, and we identify the space of ring spectrum maps between HZ and DB. That description involves formal group law data and the homotopy units of the ring spectrum DB.

Gauss sums and multinomial coefficients

Consider a Gauss sum for a finite field of characteristic p, where p is an odd prime. When such a sum (or a product of such sums) is a p-adic integer we show how it can be realized as a p-adic limit of a sequence of multinomial coefficients. As an application we generalize some congruences of Hahn and Lee to exhibit p-adic limit formulae, in terms of multinomial coefficients, for certain algebraic integers in imaginary quadratic fields related to the splitting of rational primes. We also give an example illustrating how such congruences arise from a p-integral formal group law attached to the p-adic unit part of a product of Gauss sums.

Transferred Chern classes in Morava $K$-theory

Let η be a complex n-plane bundle over the total space of a cyclic covering of prime index p. We show that for k E {1,2,...,np} {p,2p,..., np} the k-th Chern class of the transferred bundle differs from a certain transferred class w k of η by a polynomial in the Chern classes c p ,..., c np of the transferred bundle. The polynomials are defined by the formal group law and certain equalities in K(s)*B(Z/p x U(n)).

Equivariant connective K-theory for compact Lie groups

We construct a complex oriented, multiplicative, Noetherian, G-equivariant analogue of connective K-theory for an arbitrary compact Lie group G. Inverting the Bott element gives Atiyah–Segal equivariant K-theory and completion at the augmentation ideal gives non-equivariant connective K-theory of the Borel construction. We describe its role in the theory of equivariant formal group laws and calculate its coefficient ring for a variety of groups.

On degeneration of one-dimensional formal group laws and applications to stable homotopy theory

In this note we study a certain formal group law over a complete discrete valuation ring F [[ u n -1 ]] of characteristic p > 0 which is of height n over the closed point and of height n -1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law H n -1 of height n - 1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group S n -1 of H n -1 . We show that the automorphism group S n of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of S n -1 to the cohomology of S n with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed.

Computing zeta functions for ordinary formal groups over finite fields

We describe an algorithm to compute the one-dimensional part of the zeta function Z G of an ordinary formal group law G of finite dimension d over a finite field of p N elements and evaluate its time computational complexity. Assume G is given as d formal power series in 2 d variables. Our algorithm computes Z G mod p t with O(d 2p (t−1)(2d+3)N 2( log p) 2) bit operations.

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 .

Morita theory for Hopf algebroids and presheaves of groupoids

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff , the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology T on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasi-coherent sheaves. The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules. We apply this to formal group laws, where we get considerable generalizations of the Miller-Ravenel and Hovey-Sadofsky change of rings theorems in algebraic topology.