Formal Group Law Research Articles

Polynomial generators of $\mathbf{MSU}^\ast [1/2]$ related to classifying maps of certain formal group laws

Polynomial generators of $\mathbf{MSU}^\ast [1/2]$ related to classifying maps of certain formal group laws

Landweber Exactness of the Formal Group Law in $$c_1$$-Spherical Bordism

Landweber Exactness of the Formal Group Law in $$c_1$$-Spherical Bordism

$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory

We study the $SU$-linear operations in complex cobordism and prove that they are generated by the well-known geometric operations $\partial_i$. For the theory $W$ of $c_1$-spherical bordism, we describe all $SU$-linear multiplications on $W$ and projections $MU \to W$. We also analyse complex orientations on $W$ and the corresponding formal group laws $F_W$. The relationship between the formal group laws $F_W$ and the coefficient ring $W_*$ of the $W$-theory was studied by Buchstaber in 1972. We extend his results by showing that for any $SU$-linear multiplication and orientation on $W$, the coefficients of the corresponding formal group law $F_W$ do not generate the ring $W_*$, unlike the situation with complex bordism.

Landweber Exactness of the Formal Group Law in $c_1$-Spherical Bordism

Мы описываем структуру кольца коэффициентов $W^*(pt)=\varOmega_W^*$ теории $c_1$-сферических бордизмов для произвольного $SU$-билинейного умножения. Кроме того, мы доказываем точность по Ландвеберу формальной группы в теории $W^*$ для произвольного $SU$-билинейного умножения. Также мы показываем, что после обращения множества ${\mathcal P}$ простых чисел Ферма существует комплексная ориентация локализованной теории $W^*[{\mathcal P}^{-1}]$, для которой коэффициенты формальной группы порождают все кольцо коэффициентов $\varOmega_W^*[{\mathcal P}^{-1}]$. Библиография: 11 названий.

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.

Vertex F-algebra structures on the complex oriented homology of H-spaces

We give a topological construction of graded vertex F -algebras by generalizing Joyce's vertex algebra construction to complex-oriented homology. Given an H-space X with a B U ( 1 ) -action, a choice of K-theory class, and a complex oriented homology theory E , we build a graded vertex F -algebra structure on E ⁎ ( X ) where F is the formal group law associated with E .

Formal vertex laws associated to Lie conformal algebras

We introduce several definitions within the framework of vertex and conformal algebras that are analogous to some important concepts of the classical Lie theory. Most importantly, we define formal vertex laws, which correspond to the notion of formal group laws. We prove suitable vertex/conformal versions of a number of classical results, such as the Milnor–Moore theorem, Cartier duality, and the equivalence between formal group laws and Lie algebras.

Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum \(E_n\), whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that \(E_n\) is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group \({\mathbb {G}}_n\) of the formal group in question. In this paper we find that the \({\mathbb {G}}_n\)-equivariant dual of \(E_n\) is in fact \(E_n\) twisted by a sphere with a non-trivial (when \(n>1\)) action by \({\mathbb {G}}_n\). This sphere is a dualizing module for the group \({\mathbb {G}}_n\), and we construct and study such an object \(I_{{\mathcal {G}}}\) for any compact p-adic analytic group \({\mathcal {G}}\). If we restrict the action of \({\mathcal {G}}\) on \(I_{{\mathcal {G}}}\) to certain type of small subgroups, we identify \(I_{{\mathcal {G}}}\) with a specific representation sphere coming from the Lie algebra of \({\mathcal {G}}\). This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of \(E_n^{hH}\) for select choices of p and n and finite subgroups H of \({\mathbb {G}}_n\).

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.

On Vlasenko’s formal group laws

Given a Laurent polynomial over a flat \(\mathbf {Z}\)-algebra, Vlasenko defines a formal group law. We identify this formal group law with a coordinate system of a formal group functor. When the “Hasse–Witt matrix” of the Laurent polynomial is invertible, Vlasenko defines a matrix by taking a certain \(p\)-adic limit. We show that this matrix is the Frobenius of the Dieudonné module of this formal group modulo \(p\).

Cohomological realization of the Buchstaber formal group law

Cohomological realization of the Buchstaber formal group law

Picard groups and duality for real Morava E–theories

We show, at the prime 2, that the Picard group of invertible modules over $E_n^{hC_2}$ is cyclic. Here, $E_n$ is the height $n$ Lubin--Tate spectrum and its $C_2$-action is induced from the formal inverse of its associated formal group law. We further show that $E_n^{hC_2}$ is Gross--Hopkins self-dual and determine the exact shift. Our results generalize the well-known results when $n = 1$.

Height four formal groups with quadratic complex multiplication

We construct spectral sequences for computing the cohomology of automorphism groups of formal groups with complex multiplication by a $p$-adic number ring. We then compute the cohomology of the group of automorphisms of a height four formal group law which commute with complex multiplication by the ring of integers in the field $\mathbb{Q}_p(\sqrt{p})$, for primes $p>5$. This is a large subgroup of the height four strict Morava stabilizer group. The group cohomology of this group of automorphisms turns out to have cohomological dimension $8$ and total rank $80$. We then run the $K(4)$-local $E_4$-Adams spectral sequence to compute the homotopy groups of the homotopy fixed-point spectrum of this group's action on the Lubin-Tate/Morava spectrum $E_4$.

Equivariant formal group laws and complex-oriented spectra over primary cyclic groups: elliptic curves, Barsotti–Tate groups, and other examples

We explicitly construct and investigate a number of examples of $${\mathbb {Z}}/p^r$$ -equivariant formal group laws and complex-oriented spectra, including those coming from elliptic curves and p-divisible groups, as well as some other related examples.

Models of Lubin–Tate spectra via Real bordism theory

We study certain formal group laws equipped with an action of the cyclic group of order a power of 2. We construct C2n-equivariant Real oriented models of Lubin–Tate spectra Eh at heights h=2n−1m and give explicit formulas of the C2n-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory MUR, and our work examines the height of the formal group laws of the Hill–Hopkins–Ravenel norms of MUR.

THE BOREL CHARACTER

AbstractThe main purpose of this article is to define a quadratic analogue of the Chern character, the so-called Borel character, that identifies rational higher Grothendieck-Witt groups with a sum of rational Milnor-Witt (MW)-motivic cohomologies and rational motivic cohomologies. We also discuss the notion of ternary laws due to Walter, a quadratic analogue of formal group laws, and compute what we call the additive ternary laws, associated with MW-motivic cohomology. Finally, we provide an application of the Borel character by showing that the Milnor-Witt K-theory of a field F embeds into suitable higher Grothendieck-Witt groups of F modulo explicit torsion.

Cokernels in the Category of Formal Group Laws

In a recent article, the authors established an explicit description of kernels in the category of the formal group laws over the ring of Witt vectors over a finite field in terms of Fontaine’s triples. The present research is devoted to an adjacent problem of explicit description of cokernels. The technique developed is applied to a natural monomorphism from $$F_m$$ to the Weil restriction of $$F_m$$ with respect to certain ring extensions. Besides, we investigate some properties of the category of formal group laws over the ring of Witt vectors such as left and right integrability and left and right semi-abelianity.

The Steenrod algebra from the group theoretical viewpoint

In the paper “The Steenrod algebra and its dual” [2], J. Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme Gp represented by the dual Hopf algebra of the mod p Steenrod algebra. Then, Gp assigns a graded commutative algebra A⁎ over a prime field of finite characteristic p to a set of isomorphisms of the additive formal group law over A⁎, whose group structure is given by the composition of formal power series ([4] Appendix). The aim of this paper is to show some group theoretic properties of Gp by making use of this presentation of Gp(A⁎). We give a decreasing filtration of subgroup schemes of Gp which we use for estimating the length of the lower central series of finite subgroup schemes of Gp. We also give a successive quotient maps Gp→ρ0Gp〈1〉→ρ1Gp〈2〉→ρ2⋯→ρk−1Gp〈k〉→ρkGp〈k+1〉→ρk+1⋯ of affine group schemes over a prime field Fp such that the kernel of ρk is a maximal abelian subgroup.

An unstable change of rings for Morava E–theory

The Bousfield-Kan (or unstable Adams) spectral sequence can be constructed for various homology theories such as Brown-Peterson homology theory BP, Johnson-Wilson theory $E(n)$, or Morava $E$-theory $E_n$. For nice spaces the $E_2$-term is given by Ext in a category of unstable comodules. We establish an unstable Morava change of rings isomorphism between $\text{Ext}_{\mathcal{U}_{\Gamma_B}}(B,M)$ and $\text{Ext}_{\mathcal{U}_{E_{n*}E_n}/I_{n}}(E_{n*}/I_{n},E_{n*}\otimes_{BP_*} M)$ where $(B,\Gamma_B)$ denotes the Hopf Algebroid $(v_n^{-1}BP_*/I_{n}, v_n^{-1}BP_*BP/I_{n} )$. We show that the latter groups can be interpreted as $\text{Ext}$ in the category of continuous modules over the profinite monoid of endomorphisms of the Honda formal group law. By comparing this with the cohomology of the Morava stabilizer group we obtain an unstable Morava vanishing theorem when $p-1 \nmid n$.

Formal group laws over

A global analog of Fontaine’s category of triples is introduced. While the latter is antiequivalent to the category of formal group laws over the ring of Witt vectors over a finite field of characteristic p, the introduced category is antiequivalent to the category of formal group laws over Explicit constructions of kernels and cokernels in this category are given, thus proving that it is pre-abelian.