# Formal ternary laws and Buchstaber’s 2-groups

David Coulette  ENS de Lyon, UMPA, UMR 5669, 46 allée d’Italie, 69364 Lyon
Cedex 07, France<EMAIL_ADDRESS>,  Frédéric Déglise  ENS de Lyon,
UMPA, UMR 5669, 46 allée d’Italie, 69364 Lyon Cedex 07, France
<EMAIL_ADDRESS>http://perso.ens-lyon.fr/frederic.deglise/ ,
Jean Fasel  Institut Fourier - UMR 5582, Université Grenoble-Alpes, CS 40700,
38058 Grenoble Cedex 9, France<EMAIL_ADDRESS>https://www-
fourier.univ-grenoble-alpes.fr/ faselj/ and  Jens Hornbostel  Fakultät für
Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr.
20, 42119 Wuppertal, Germany<EMAIL_ADDRESS>http://www2.math.uni-wuppertal.de/ hornbost/

###### Abstract.

We compare formal ternary laws to Buchstaber’s 2-valued formal group laws.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Main results
    3. 1.3 Plan of the paper
  2. 2 On Buchstaber’s multi-valued series and formal group laws
    1. 2.1 Multivalued series
    2. 2.2 Multi-valued formal group laws
    3. 2.3 Buchstaber’s classification of 2-formal groups
  3. 3 Formal ternary laws
    1. 3.1 Algebraic definitions
    2. 3.2 Examples
  4. 4 Comparison
  5. 5 Example: symplectic oriented ring spectra
  6. A Sage implementation of FTL computations
    1. A.1 Main problem characteristics and design principle
    2. A.2 Description of the implemented scheme
    3. A.3 Short description of the code repository
    4. A.4 Sample results 1 : Computation of $\mathcal{W}^{\leq 0}[\frac{1}{2}]$.
    5. A.5 Example 2 : Computation of $\mathcal{W}^{\leq 2}[\frac{1}{2}]/(\epsilon-1)$

##  1\. Introduction

###  1.1. Background

Formal group laws (or FGL for short) are fundamental in many areas of
mathematics. In algebraic topology, since Quillen’s work on complex cobordism,
one dimensional commutative formal group laws have been an important tool when
studying Chern classes for complex oriented cohomology theories $E^{*}$. In
[Qui69], Quillen discovered that the first Chern class of a tensor product of
line bundles is computed by a formal group law canonically associated with the
$E^{*}$. Moreover, he showed that the formal group law associated with complex
cobordism is the universal one, introduced by Lazard in [Laz55]. The work of
Quillen has been hugely influential in algebraic topology.

Among many of his pursuers, Buchstaber has intiatied a program for studying
symplectic cobordism, via the realization of Pontryagin classes of sympectic
bundles in complex cobordism ([BN71, Buh75, Buh76, Buh78], see also [AL75]).
Most notably, he coined down the notion of (commutative one-dimensional)
_2-valued formal group_ over a base ring $R$, formed by certain pairs
$(F_{1}(x,y),F_{2}(x,y))$ of formal power series in two variables with
coefficients in $R$ (see Example 2.2.2 for more details). In fact, he showed
that the Pontryagin polynomial of a tensor product of the two copies of the
universal symplectic bundle gives rise (after taking a “square root”) to a
2-valued formal group. Moreover, the 2-valued group obtained in this way
satisfies a special condition, that Buchstaber calls “type I” (see 2.3.1), and
he finally shows that one obtains the universal 2-valued formal group of type
I, at least after inverting $2$. This is therefore an analogue of Quillen’s
identification of the formal group law of complex cobordism with the universal
one.

More recently, Voevodsky opened a door between algebraic topology and
algebraic geometry, introducing among other things algebraic cobordism,
algebraic Morava K-theory and the motivic Steenrod algebra. After a
formalization of _(algebraic) oriented theories_ in algebraic geometry by
Panin and Smirnov, Levine and Morel [LM07] have extended the work of Quillen
on cobordism and formal group laws, within the setting of oriented theories.
In particular, they show that the algebraic cobordism of any characteristic
$0$ field is isomorphic to the Lazard ring, and the formal group law derived
from Chern classes in algebraic cobordism agrees with the universal formal
group law: _op. cit._ Th. 1.2.6.111This was later generalized by Marc Hoyois
in [Hoy15, Proposition 8.2] for characteristic $p>0$ fields after inverting
$p$, and by Markus Spitzweck in [Spi20, Theorem 6.7] for a local regular ring
after inverting the characteristic exponent of its residue field. The path
opened by these mathematicians has become particularly fecund and inspired
many new results and computations among which: Riemann-Roch formulas ([Pan04],
[Smi06], [Dég18b]), motivic Landweber existence theorem ([NSØ09]), Schubert
calculus ([HK11], [CPZ13], [CZZ15]), operations on oriented theories ([Vis19],
[Mer21]), bivariant formalism ([SY14], [Dég18a]).

However, there are important theories in motivic homotopy, such as Chow-Witt
groups ([BM00], [Fas20]) and higher Grothendieck-Witt groups (encompassing
both orthogonal and hermitian K-theories, see e.g. [Hor05], [ST15] or [Sch17])
which are not oriented, while they possess a well-developed theory of
characteristic classes (Euler and Thom classes). Motivated by these examples,
Panin and Walter introduced a refinement of orientation theory in motivic
homotopy theory in a series of papers [PW10, PW18, PW19b, PW19a]. The idea of
their theory is to restrict the existence of Thom classes to particular kind
of vector bundles, the most important examples being the symplectic bundles
($\mathrm{Sp}$-orientations) and oriented bundles
($\mathrm{SL}$-orientations). The first case seems the most successful
analogue of usual orientation (now $\mathrm{GL}$-orientation), as it allows to
get a theory of _Borel classes_ , completely analogous to that of Chern
classes. Borel classes are the motivic counter-part of Pontryagin classes in
topology.

In the same spirit as Buchstaber’s 2-valued groups, the second and third
authors have developed in [DF21] an initial idea of Walter which proposes to
define an analogue for Borel classes of the formal group laws associated with
Chern classes. This is the theory of _formal ternary laws_. The ternary comes
from the basic fact that a double tensor products of symplectic bundles is not
symplectic, whereas a triple tensor product is. Therefore, formal ternary laws
arise by looking at the Borel classes of triple tensor products of symplectic
bundles of (minimal) ranks $2$. As such a triple tensor product is of rank
$8$, there are $4$ relevant Borel classes which lead in fact to four power
series in three variables (see §3.1).

One interesting features of the theory is that one gets series with
coefficients in the Grothendieck-Witt ring $\operatorname{GW}(\mathbb{Z})$,
rather than just $\mathbb{Z}$ in the classical case. Formal ternary laws are
therefore more complicated than formal group laws, but still seem to play a
classification role for motivic ring spectra analogous to the latter for usual
ring spectra. In fact, the second and third authors successfully in [DF21]
computed the formal ternary law associated with Chow-Witt groups (and its
higher analogue, the MW-motivic ring spectrum, [BCD+20]) while that of
Grothendieck-Witt groups was recently computed in [FH20] – see Example 3.2 for
both cases. According to the nature of these theories, the corresponding
formal ternary laws are the analogue of the additive and multiplicative formal
group laws respectively. As in topology, these results are useful for
computing stable operations. In particular the second and third authors built
the so-called Borel character, analogue of the Chern character, which
corresponds to the exponential isomorphism from the multiplicative to the
additive formal group laws with rational coefficients.

###  1.2. Main results

The aim of this paper is first to develop the algebraic part of the theory of
formal ternary laws with the aim to relate it with Buchstaber’s 2-valued
formal groups.

In order to formulate this comparison, we introduce a generalization of
Buchstaber’s theory, the category $\mathscr{G}_{n,d}(R)$ of _$n$ -valued
$d$-ary formal group laws_ over a ring $R$, or $(n,d)$-groups for short (see
2.2.1). The justification for this extension is that while 2-groups are
$2$-valued secondary formal group laws, formal ternary laws not only are
$4$-valued but also depends on three formal variables. In this general
framework we introduce a natural notion of degree (2.1.5), based on the
analogy with homology. The minimal and maximal degrees of the coefficients of
an $(n,d)$-group roughly measure its complexity. As an example, we could
mention that there is only one formal group law with minimal and maximal
degrees both $0$, the _additive_ one, formal group law of degree $1$ are
exactly the _multuplicative ones_ , and there are no formal group of finite
degree distinct from $0$ and $1$.

We also introduce the analogue of the Lazard ring (2.2.5) for $(n,d)$-groups.
In particular, we let $\mathcal{B}$ be the ring which classifies the 2-groups,
and $\mathcal{B}^{0}$ the quotient ring of $2$-groups whose coefficients have
degree $0$. Then a basic but fundamental observation of Buchstaber can be
restated (2.2.9) by saying that:

$\mathcal{B}^{0}\simeq\mathbb{Z}[\gamma]/\big{(}(\gamma+2)(\gamma-2)^{2}\big{)}.$

This is a fundamental difference with the structure of the Lazard ring, as
this ring is in particular non-reduced.

Next we introduce after Walter the notion of _$4$ -valued formal ternary laws_
(_i.e._ FTL, see 3.1.5), which is an algebraization of the notion appearing in
[DF21, 2.3.2]. Comparing to _loc. cit._ and the original axioms of Walter, the
novelty here is the so-called $\epsilon$-linearity axiom. We refer the reader
to Section 5 for the fact that these FTL are naturally attached to
$\mathrm{Sp}$-oriented cohomology theories.222This improves the results of
_loc. cit._ which treated only the case of $\mathrm{SL}$-oriented theories.

Note that FTL are not exactly $(4,3)$-groups: in fact, they have a _quadratic_
nature which is reflected by the fact that we work over the Grothendieck-Witt
ring of $\mathbb{Z}$,
$\operatorname{GW}(\mathbb{Z})\simeq\mathbb{Z}_{\epsilon}=\mathbb{Z}[\epsilon]/(\epsilon^{2}-1)$
rather than just $\mathbb{Z}$ (see 3.1.6). However we can introduce the
classifying ring $\mathcal{W}$ of FTL, that we call the _Walter ring_. Our
first theorem is a classification result:

######  Theorem 1.

(see 3.1.12) After inverting $2$, there are only two FTLs of degree $\leq 0$.
One of this law is the FTL associated with Chow-Witt groups (see [DF21]).

Therefore, the FTL associated with Chow-Witt groups is the analogue of the
additive FTL. See 3.1.12 for a more precise version in terms of the Walter
ring and 3.1.13 for further explanations.

Formal ternary laws have an extremely high computational complexity, due to
the associativity axiom. More precisely, the Walter ring is naturally defined
as a polynomial algebra with an infinite number of variables modulo an ideal
$\mathcal{R}_{FTL}$, corresponding to the axioms (3.1.9). Nonetheless, this
ideal is very difficult to compute. Therefore the preceding theorem could only
be proved with the help of a computer. More generally, we developed a sage
implementation which computes the Grobner basis of the ideal
$\mathcal{R}_{FTL}$, for a possible range of degrees. See Appendix A for more
details.

The main result of this paper is the following comparison statement below
between the following three categories over an arbitrary commutative ring $R$:

  * •

$\mathcal{FGL}(R)=\mathscr{G}_{1,2}(R)$ the category of formal group laws.

  * •

($2-\mathcal{FGL})_{I}(R)\subset\mathscr{G}_{2,2}(R)$ Buchstaber’s category of
2-groups of the first type.

  * •

$\mathcal{FTL}(R)=$ the category of formal ternary laws.

######  Theorem 2.

There is a functor $C:(2-\mathcal{FGL})_{I}\rightarrow\mathcal{FTL}$ which has
a concrete interpretation in terms of characteristic classes. Composing it
with the functor $N:\mathcal{FGL}\rightarrow(2-\mathcal{FGL})_{I}$ from
Proposition 2.3.12 yields a functor which maps the FGL for Chow groups to the
oriented part (set $\epsilon=-1$) of the FTL for Chow-Witt groups.

As a consequence of this theorem, it is in principle possible to determine all
the FTL associated to oriented theories, including algebraic cobordism. It
also gives an answer to the obvious question to know if FTL can be described
in terms of more basic ingredients such as 2-valued formal groups à la
Buchstaber. The answer is negative in general (e.g. 4.0.3 and the subsequent
discussion). Concretely, this means that in general the Borel classes of a
triple product of symplectic bundles can’t be computed using hypothetic
characteristic classes (satisfying Buchstaber’s axioms) attached to a product
of two symplectic bundles.

To conclude, let us stress some consequences of the universality of the Walter
ring $\mathcal{W}$. This discussion can be taken as a philosophical
explanation of the results of the paper. If $\mathbb{E}$ is an Sp-oriented
motivic ring spectrum over a field $k$, then there is a ring homomorphism

$\mathcal{W}\to\mathbb{E}^{2*,*}(\operatorname{Spec}k)$

defined by mapping the generators of the left-hand side to the corresponding
coefficient in the formal ternary law attached to $\mathbb{E}$. In the
particular case of $\mathbf{MSp}$, recall that there is at present no complete
computation of $\mathbf{MSp}^{2*,*}(\operatorname{Spec}k)$, as well as of the
homotopy groups of its topological counterpart $MSp$. Using complex
realization, we obtain a string of ring homomorphisms

$\mathcal{W}\to\mathbf{MSp}^{2*,*}(\operatorname{Spec}{\mathbb{C}})\to\pi_{-2*}(MSp)$

which factors through a string of ring homomorphisms

(1.2.0.a)
$\mathcal{W}/(\epsilon+1)\to\mathbf{MSp}^{2*,*}(\operatorname{Spec}{\mathbb{C}})\to\pi_{-2*}(MSp)$

(see Remark 5.0.1 for more details). Using real realization instead, we obtain
(using e.g. [BH20, Corollary 4.7]) a sequence

(1.2.0.b)
$\mathcal{W}/(\epsilon-1)\to\mathbf{MSp}^{2*,*}(\operatorname{Spec}{\mathbb{R}})\to\pi_{-*}(MU)$

and the ring $\mathcal{W}$ can then be thought of as providing a (possibly
partial) set of generators and relations for both $\pi_{-2*}(MSp)$ and
$\pi_{-*}(MU)$. At the moment, we are not able to perform integral
computations in the Walter ring, and we leave this question to further work.
After inverting $2$, the computations become much easier. For instance,
Section A.5 shows that

$(\mathcal{W}^{\leq
2}/\langle\epsilon-1\rangle)\otimes\mathbb{Z}[1/2]\simeq\mathbb{Z}[1/2][a^{3}_{111},a^{1}_{111}]/\langle(a^{3}_{111}-8)(a^{3}_{111}+8)\rangle$

which fits well with the computations of $\pi_{-*}(MU)\otimes\mathbb{Z}[1/2]$
(see Remark 3.1.13 for a tentative explanation of the relation
$(a^{3}_{111}-8)(a^{3}_{111}+8)=0$). At present, we do not know if our system
of relations in $\mathcal{W}$ is complete, i.e. that the ring homomorphism
$\mathcal{W}\to\mathbf{MSp}^{2*,*}(\operatorname{Spec}k)$ is an isomorphism
for a reasonable field $k$. We hope to explore this question in future work.

###  1.3. Plan of the paper

In Section 2, we both recall and extend Buchstaber’s theory of multi-valued
series, by introducing possibly more formal variables leading to
$(n,d)$-series and $(n,d)$-groups (resp. Def. 2.1.1 and 2.2.1). The main
technical point is the notion of substitution for these generalized series
(2.1.13). We then recall Buchstaber’s classification of $2$-FGL in our
language.

Section 3 concerns the algebraic notion of formal ternary laws, based on the
language of $(n,d)$-series in Section 1. We state our classification result
(Theorem 1) for low degree FTL, and give examples coming from motivic
homotopy.

In Section 4, we establish in Theorem 2 the comparison between FGL, $2$-FGL
and FTL as stated above.

Finally in Section 5, we give the main steps for constructing the canonical
FTL associated to an $\mathrm{Sp}$-oriented motivic ring spectrum.

### Acknowledgments

The first two authors thank David Monniaux and Bruno Salvy for their interest
and help in the computational aspects of this paper. The third author wishes
to thank Michel Brion and Adrien Dubouloz for useful conversations around the
proof of Theorem 5.0.2, and Oliver Röndigs for discussions on faster than
light travel. The authors wish to thank Achim Krause for pointing out he work
of Kochman on the coefficient ring $MSp_{*}$ in topology. The last author’s
research was conducted in the framework of the research training group GRK
2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology.

##  2\. On Buchstaber’s multi-valued series and formal group laws

###  2.1. Multivalued series

The following definition is a variation on Buchstaber’s algebraic theory of
multi-valued power series, compare [Buh75, section 1].

######  Definition 2.1.1.

Let $R$ be a ring, and $n,d>0$ be integers. We let
$\underline{x}=(x_{1},\ldots,x_{d})$ be formal variables. An $n$-valued
$d$-dimensional (power) series with coefficients in $R$, or _$(n,d)$ -series
over $R$_ for short, will be a polynomial $F_{t}(\underline{x})$ in
$R[[\underline{x}]][t]$ of degree $n$ and constant term $1$. We use the
generic notation:

(2.1.1.a)
$F_{t}(\underline{x})=1+F_{1}(\underline{x})t+\ldots+F_{n}(\underline{x})t^{n}.$

We will also consider a different way of writing this type of polynomials,
that we call the _Buchstaber form_ :

$\tilde{F}_{t}(\underline{x}):=t^{n}F_{(-t^{-1})}(\underline{x})=t^{n}-F_{1}(\underline{x})t^{n-1}+\ldots+(-1)^{n}F_{n}(\underline{x}).$

We write $\mathscr{F}_{n,d}(R)$ the corresponding $R$-module.

Let $\varphi:R\rightarrow R^{\prime}$ be a morphism of rings. The obvious
corestriction of scalars functor $\varphi_{*}:R[[\underline{x}]][t]\rightarrow
R^{\prime}[[\underline{x}]][t]$ induces a map:

$\varphi_{*}:\mathscr{F}_{n,d}(R)\rightarrow\mathscr{F}_{n,d}(R^{\prime}),$

which we will still call the _corestriction_ (along $R^{\prime}/R$).

######  Remark 2.1.2.

  1. (1)

Obviously, $\mathscr{F}_{n,d}(R)$ is isomorphic to the $R$-module
$R[[\underline{x}]]^{n}$ and the $(n,d)$-series $F_{t}(\underline{x})$ is
determined by the $n$-tuple of power series in the variables $\underline{x}$:
$F_{1}(\underline{x}),\ldots,F_{n}(\underline{x})$.

In particular, $\mathscr{F}_{1,d}$ is isomorphic to the $R$-module of power
series in $\underline{x}$. Given such a power series $\varphi(\underline{x})$,
we will write $\varphi_{t}(\underline{x})=1+\varphi(\underline{x})t$ the
corresponding object of $\mathscr{F}_{1,n}$.

  2. (2)

The notation (2.1.1.a) is motivated by the theory of characteristic classes,
and especially total classes (Chern, Pontryagin, Borel, etc.). The Buchstaber
form $\tilde{F}_{t}(\underline{x})$, taken from the convention of [Buh75], is
useful when reasoning in terms of roots. We keep the two possible conventions
to help the reader compare our notations with that of Buchstaber, but we
believe that the notation (2.1.1.a) is better suited when dealing with
examples from motivic homotopy theory which are our motivation.

######  Example 2.1.3.

Let $\mathbb{E}$ be an oriented motivic ring spectrum over a scheme $S$ and
let $\mathbb{P}_{S}^{\infty}$ be the infinite projective space (which is the
colimit of $\mathbb{P}_{S}^{n}$ for $n$ increasing). Let
$\mathbb{E}^{**}=\mathbb{E}^{**}(S)$ be the ring of coefficients.

Then it is well-known that
$\mathbb{E}^{**}(\mathbb{P}^{\infty}_{S})=\mathbb{E}^{**}[[x]]$, where $x$ is
the first Chern class of the universal line bundle on
$\mathbb{P}^{\infty}_{S}$. Note further that $x$ is of cohomoloical degree
$(2,1)$. More generally, we have
$\mathbb{E}^{*,*}((\mathbb{P}^{\infty}_{S})^{\times
d})=\mathbb{E}^{**}[[x_{1},\ldots,x_{d}]]$ for any $d\geq 1$, where $x_{i}$
are the first Chern classes of the corresponding line bundles, sitting in
degree $(2,1)$. Let now $G_{d,\infty}$ be the Grassmannian of $d$-dimensional
sub-vector bundles in an infinite dimensional vector bundle over $S$, and let
$V_{d,\infty}$ be the universal $d$-dimensional vector bundle over
$\mathrm{Gr}_{d,\infty}$. The sum of the universal line bundles yields a
morphism

$(\mathbb{P}^{\infty}_{S})^{\times d}\to G_{d,\infty}$

and we may consider the pull-back of the total Chern class (with coefficients
in $\mathbb{E}^{*,*}(G_{d,\infty})$)

$c_{t}(V):=1+\sum_{i=1}^{d}c_{i}(V)t^{i},$

along this morphism. Each $c_{i}(V)$ is a homogeneous power series of degree
$i$ in the variables $x_{1},\ldots,x_{d}$ and we obtain an element of
$\mathscr{F}_{d,d}(\mathbb{E}^{*,*})$. More precisely, we have
$c_{i}(V)=e_{i}(x_{1},\ldots,x_{d})$, the elementary symmetric polynomial of
degree $i$ in the variables $\underline{x}$. Further, consider the map

$(\mathbb{P}^{\infty}_{S})^{\times 2}\to\mathbb{P}_{S}^{\infty}$

classifying the tensor product of the two universal line bundles. Pulling-back
the Chern class $1+xt$ to $(\mathbb{P}^{\infty}_{S})^{\times 2}$, we obtain a
polynomial in $\mathscr{F}_{1,2}(\mathbb{E}^{*,*})$ of the form

$1+F(x_{1},x_{2})t$

where $F(x_{1},x_{2})$ is the first Chern class of the product. This is the
so-called formal group law of $\mathbb{E}^{*,*}$.

######  2.1.4.

We can generically write an $(n,d)$-series $F_{t}(\underline{x})$ with
coefficients in $R$ as

(2.1.4.a)
$F_{t}(\underline{x})=1+\sum_{\underline{\alpha}\in\mathbb{N}^{n},\\\
l=1,\ldots,n}a_{\underline{\alpha}}^{l}\underline{x}^{\underline{\alpha}}t^{l},$

for coefficients $a_{\underline{\alpha}}^{l}\in R$. The preceding example
leads us to consider a grading on $(n,d)$-series, defined by the convention
that the formal variables $x_{1},...,x_{d}$ have degree $+1$ and the
indeterminate $t$ has degree $-1$.

######  Definition 2.1.5.

Consider the above notation. We define the degree (resp. valuation) of the
$(n,d)$-series $F_{t}(\underline{x})$ as the integer $\delta$ (resp. $\nu$)
such that:

$\displaystyle\delta=\sup(A),\text{resp. }\nu=\inf(A),$ $\displaystyle
A=\\{|\underline{\alpha}|-l\mid\underline{\alpha}\in\mathbb{N}^{n},1\leq l\leq
n,a_{\underline{\alpha}}^{l}\neq 0\\}.$

In particular, we view $\mathscr{F}_{n,d}(R)$ as a $\mathbb{Z}$-graded
$R$-module.

By definition, $\delta\geq\nu\geq-n$. The interval $[\nu,\delta]$ measures the
complexity of the $(n,d)$-series $F_{t}(\underline{x})$.

######  Example 2.1.6.

In Example 2.1.3, one notes that the Chern classes $c_{l}(V)$ are of bidegree
$(2l,l)=l(2,1)$. In particular, if one writes:

$c_{l}(V)=\sum_{\underline{\alpha}}a_{\underline{\alpha}}^{l}\underline{c}^{\underline{\alpha}},$

the (cohomological) bidegree of $a_{\underline{\alpha}}^{l}$ is
$(l-|\underline{\alpha}|)(2,1)$, and this legitimates the choice of grading
for $\underline{c}^{\underline{\alpha}}t^{l}$ to be $|\underline{\alpha}|-l$.
In that particular case, we have $\nu=\delta=0$.

######  Remark 2.1.7.

There is an obvious multiplication map:

$\mathscr{F}_{n,d}(R)\times\mathscr{F}_{m,e}(R)\rightarrow\mathscr{F}_{n+m,d+e},F_{t}(\underline{x}),G_{t}(\underline{y})\mapsto
F_{t}(\underline{x}).F_{t}(\underline{y})$

where the product is taken in the ring $R[[\underline{x},\underline{y}]][t]$.
Whereas this product is quite natural, we will rather use the substitution
product of 2.1.13 below.

######  Definition 2.1.8.

Consider an $(n,d)$-series $F_{t}(\underline{x})\in\mathscr{F}_{n,d}(R)$. A
_root_ of $F_{t}(\underline{x})$ will be a root of
$\tilde{F}_{t}(\underline{x})$, as a polynomial in $t$. We say that
$F_{t}(\underline{x})$ is _split_ if $\tilde{F}_{t}(\underline{x})$ is split,
as a polynomial in $t$. In the latter case, we will generically denote by
$F^{[i]}$, $i=1,...,n$ the corresponding roots. It follows that

$F_{t}(\underline{x})=\prod_{i=1}^{n}(1+F^{[i]}t).$

######  Example 2.1.9.

The previous definition is motivated by the splitting principle. As in Example
2.1.3, we have

$c_{t}(V):=1+\sum_{i=1}^{d}c_{i}(V)t^{i}=\prod_{i=1}^{d}(1+x_{i}t)$

In other words, the $n$-dimensional $d$-valued series $c_{t}(V)$ is split in
$\mathbb{E}^{*,*}[[\underline{x}]][t]$ – _i.e._ the polynomial
$\tilde{c}_{t}(V)\in\mathbb{E}^{**}[[\underline{c}]][t]$ splits.

######  Remark 2.1.10.

Beware that if $R$ is not integral, the roots of a split series
$F_{t}(\underline{x})$ are not unique in general.

######  2.1.11.

The algebraic splitting principle. Consider an $(n,d)$-series
$F_{t}(\underline{x})$ with coefficients in $R$. Note that there always exists
a finite ring extension $R[[\underline{x}]]\rightarrow R^{\prime}$ such that
$\tilde{F}_{x}(\underline{x})$ splits in $R^{\prime}$. In fact, there is a
universal one given by the formula

$R^{\prime}=R[[\underline{x}]][\underline{t}]/(e_{i}(\underline{t})-F_{i}(\underline{x}),i=1,\ldots,n)$

where $\underline{t}=(t_{1},...,t_{d})$, $e_{i}(\underline{t})$ is the $i$-th
elementary symmetric polynomial in the variables $\underline{t}$. Given a
finite extension $A/R[[\underline{x}]]$ in which
$\tilde{F}_{x}(\underline{x})$ splits, we denote by $F^{[i]}$ (or sometimes
$F_{A}^{[i]}$ if $A$ is important) its roots so that we have:

$F_{t}(\underline{x})=\prod_{i=1}^{n}(1+F^{[i]}t),$

and call them the _formal roots_ of $F_{t}(\underline{x})$ in $A$. There is a
bijection between the set of pairs $(A,F^{[i]}_{A})$ (with $A$ finite over
$R[[\underline{x}]]$) and the set of finite homomorphisms of
$R[[\underline{x}]]$-algebras $R^{\prime}\to A$.

######  2.1.12.

Substitutions. We will say that a power series $\varphi(x)$ in one variable is
_composable_ if $\varphi(0)=0$. Then for any power series $\psi(x)$, the
composite $\psi(\varphi(x))$ is well-defined. Similar considerations hold when
dealing with more variables and we now extend them in the context of
multivalued series.

Let then $F_{t}(\underline{x})\in\mathscr{F}_{n,d}$ with
$\underline{x}=(x_{1},\ldots,x_{d})$. We fix an integer $1\leq i\leq d$ and
consider the $(d-1)$-tuple of variables
$\underline{x}^{\prime}=(x_{1},...,x_{i-1},x_{i+1},...,x_{d})$. We introduce
another $r$-tuple of variables $\underline{y}=(y_{1},...,y_{r})$ and consider
the expression in $R[[\underline{x}^{\prime},\underline{y}]][t]$:

$\phi_{i}(\underline{x}^{\prime},\underline{y})=\prod_{l=1}^{r}F_{t}(x_{1},...,x_{i-1},y_{l},x_{i+1},...,x_{d}).$

Note that this is a polynomial in $t$ of degree $rn$ and with constant term
$1$. Moreover, $\phi_{i}(\underline{x}^{\prime},\underline{y})$ is invariant
under the action of the group $\mathfrak{S}(\underline{y})$ of permutations of
the variables $\underline{y}$. In particular, there exists a unique element
$\phi_{i}^{\prime}\in R[[\underline{x}^{\prime},\underline{e}]][t]$, where
$\underline{e}=(e_{1},...,e_{r})$, such that

$\phi_{i}(\underline{x},\underline{y})=\phi^{\prime}_{i}(\underline{x},e_{l}=e_{l}(\underline{y}),l=1,\ldots,r),$

where the right-hand side is obtained by the indicated substitution, and
$e_{l}(\underline{y})$ denotes the $l$-th symmetric elementary polynomial in
the variables $\underline{y}$. By construction,
$\phi_{i}^{\prime}(\underline{x}^{\prime},\underline{e})$ is polynomial in the
variable $t$, of degree $rn$, with coefficients formal series in the variables
$\underline{x}^{\prime}$ and $\underline{e}$. In fact, $\phi_{i}^{\prime}$ is
an $(rn,d+r-1)$-series that will serve for the following definition:

######  Definition 2.1.13.

We will say that the multivalued series
$G_{t}(\underline{y})\in\mathscr{F}_{m,r}$ is _composable_ if
$G_{t}(\underline{0})=1$. More precisely, if we write

$G_{t}(\underline{y})=1+\sum_{l=1}^{m}G_{l}(\underline{y})t^{l}.$

then $G_{t}(\underline{y})$ is composable if $G_{l}(\underline{0})=0$ for
$l=1,\ldots,m$. In that case, one defines the _substitution of
$G_{t}(\underline{y})$ in $F_{t}(\underline{x})\in\mathscr{F}_{n,d}$ at the
place $i$_ as the following $(nm,d+r-1)$-series with coefficients in $R$:

$F_{t}\big{(}x_{1},...,x_{i-1},G_{t}(\underline{y}),x_{i+1},...,x_{d}\big{)}:=\phi_{i}^{\prime}\big{(}\underline{x},e_{l}=G_{l}(\underline{y}),l=1,\ldots,r\big{)},$

which is well-defined as by assumption, each formal series
$G_{l}(\underline{y})$ is composable.

Let us write $\mathscr{F}_{n,d}^{\circ}$ the $R$-module of $(n,d)$-series
which are composable. Then we have defined a bilinear pairing of $R$-modules
for any $1\leq i\leq n$:

$\sigma_{i}:\mathscr{F}_{n,d}^{\circ}\times\mathscr{F}_{m,e}^{\circ}\rightarrow\mathscr{F}_{nm,d+e-1}^{\circ}$

Note that with this definition, a power series $\varphi(\underline{x})$ is
composable if an only if the associated $(1,d)$-series
$\varphi_{t}(\underline{x})$ – Remark 2.1.2(1) – is composable.

######  Remark 2.1.14.

Following Buchstaber, one can present the previous definition using the
algebraic splitting principle. Indeed, if one introduces the formal roots
$G^{[l]}$ of $G_{t}(\underline{y})$, we can write suggestively:

$F_{t}\big{(}x_{1},...,x_{i-1},G_{t}(\underline{y}),x_{i+1},...,x_{d}\big{)}=\prod_{l=1}^{m}F_{t}\big{(}x_{1},...,x_{i-1},G^{[l]},x_{i+1},...,x_{d}\big{)}.$

Note that $G^{[l]}$ is in general not a power series in the variables
$\underline{y}$.

As a first application of this general concept, one introduces morphisms of
multivalued series.

######  Definition 2.1.15.

Let $F_{t}(\underline{x})$ and $G_{t}(\underline{x})$ be composable
$(n,d)$-series with coefficients in $R$. A morphism from
$F_{t}(\underline{x})$ to $G_{t}(\underline{x})$ is a composable power series
$\Theta(x)\in R[[x]]$ such that (in $R[[\underline{x}]][t]$):

$\Theta_{t}(F_{t}(x_{1},\ldots,x_{d}))=G_{t}(\Theta(x_{1}),\ldots,\Theta(x_{d})).$

where $\Theta_{t}(x)$ is the $(1,d)$-series associated with $\Theta(x)$ (cf.
2.1.2(1)).

The composition of such morphisms is defined by the composition of power
series. So in particular, $\mathscr{F}_{n,d}^{\circ}(R)$ becomes a category,
and in fact an $R$-linear category.

It follows from this definition that a morphism $\Theta(x)$ from
$F_{t}(\underline{x})$ to $G_{t}(\underline{x})$ is an isomorphism if and only
if $\Theta(x)=\lambda.x+O(x^{2})$ where $\lambda\in R^{\times}$ (e.g. [Str,
Lemma 2.8]). A usual convention in this setting is to say that $\Theta(x)$ is
a _strict isomorphism_ when $\lambda=1$.

######  Remark 2.1.16.

Explicitly, the equation in the above definition reads:

$\prod_{l=1}^{n}(1+\Theta(F^{[l]})t)=G_{t}(\Theta(x_{1}),\ldots,\Theta(x_{d})).$

If we denote by $G_{\theta}^{[l]}$ the roots of the series
$G_{t}(\Theta(x_{1}),\ldots,\Theta(x_{d}))$, we may also write suggestively
$\Theta(F^{[l]})=G_{\theta}^{[l]}$.

######  2.1.17.

It is clear that given a morphism of rings $\varphi:R\rightarrow R^{\prime}$
the application
$\varphi_{*}:\mathscr{F}_{n,d}^{\circ}(R)\rightarrow\mathscr{F}_{n,d}^{\circ}(R^{\prime})$
is in fact a functor of $\mathbb{Z}$-linear categories, so that we get a
(strict) covariant functor $R\mapsto\mathscr{F}_{n,d}^{\circ}(R)$ from rings
to $\mathbb{Z}$-linear categories. Using the Grothendieck construction, this
can be integrated. Explicitly, we get the following definition.

######  Definition 2.1.18.

We let $\mathscr{F}_{n,d}^{\circ}$ be the category whose objects are pairs
$(R,F_{t}(\underline{x}))$ where $R$ is a ring and $F_{t}(\underline{x})$ is a
composable $(n,d)$-series with coefficients in $R$, and whose morphisms
$(R,F_{t}(\underline{x})))\rightarrow(R^{\prime},G_{t}(\underline{x})))$ are
pairs $(\varphi,\Theta)$ where $\varphi:R\rightarrow R^{\prime}$ is a ring
homomorphism and $\Theta:\varphi_{*}(F_{t}(\underline{x}))\rightarrow
G_{t}(\underline{x})$ a morphism of $(n,d)$-series with coefficients in
$R^{\prime}$. Note that

$(\varphi,\Theta)=(\mathrm{Id},\Theta)\circ(\varphi,\mathrm{Id})$

With this definition, $\mathscr{F}_{n,d}^{\circ}$ is a $\mathbb{Z}$-linear
category, cofibered over the category of rings.

###  2.2. Multi-valued formal group laws

The following definition is a generalization of Buchstaber’s definition which
corresponds to $d=2$, see [Buh75, Definition 1.2].

######  Definition 2.2.1.

Let $R$ be a ring, $n>0$, $d>1$ integers, and
$\underline{x}=(x_{1},...,x_{d})$ formal variables. A $d$-ary $n$-valued
formal group law, $(n,d)$-group for short, is an $(n,d)$-series
$F_{t}(\underline{x})$ satisfying the following properties:

  1. (1)

Neutral element. The $(n,1)$-series $F_{t}(x,0,...,0)$ is split with one root
$x$ of multiplicity $n$, _i.e._ $F_{t}(x,0,...,0)=(1+xt)^{n}$.

  2. (2)

Symmetry. The element $F_{t}(\underline{x})$ of $R[[\underline{x}]][t]$ is a
fixed point under the action of the group $\mathfrak{S}(\underline{x})$
permuting the formal variables.

  3. (3)

Associativity. Given another $(d-1)$-tuple of formal variables
$(x_{d+1},...,x_{2d-1})$, one has the following equality of
$(n^{2},2d-1)$-series:

(2.2.1.a)
$\begin{split}&F_{t}\big{(}F_{t}(x_{1},\ldots,x_{d}),x_{d+1},\ldots,x_{2d-1}\big{)}\\\
&\quad=F_{t}\big{(}x_{1},F_{t}(x_{2},\ldots,x_{d+1}),x_{d+2},\ldots,x_{2d-1}\big{)}\end{split}$

where we have used the substitution operation (2.1.13), valid as by property
(1) $F_{t}(\underline{x})$ is composable.

We will denote by $\mathscr{G}_{n,d}(R)$ the full sub-category of
$\mathscr{F}_{n,d}^{\circ}(R)$ defined by $(n,d)$-groups. The corestriction of
scalars functor obviously respects $(n,d)$-groups so that we can also consider
the sub-category $\mathscr{G}_{n,d}$ of $\mathscr{F}_{n,d}$, cofibered over
the category of rings, formed by pairs $(R,F_{t}(\underline{x}))$ such that
$F_{t}(\underline{x})$ is an $(n,d)$-group.

######  Example 2.2.2.

Our two main examples are the following. The above more general definition
will be useful in the next section (see 3.1.5).

  * •

$\mathcal{FGL}=\mathscr{G}_{1,2}$ is the category of formal group laws.

  * •

$2-\mathcal{FGL}=\mathscr{G}_{2,2}$ is Buchstaber’s category of 2-valued
formal groups (using the terminology of [BN71, Buh75, AL75]).

Indeed, let

$\tilde{F}_{t}(x,y)=t^{2}-F_{1}(x,y)t+F_{2}(x,y)$

and suppose that it satisfies Buchstaber’s axioms. We then have

$F_{t}(x,y)=1+F_{1}(x,y)t+F_{2}(x,y)t^{2}$

and a direct inspection shows that the symmetry and neutral element axioms are
equivalent. Our associativity axiom reads as

$F_{t}(F_{t}(x,y),z)=F_{t}(x,F_{t}(y,z))$

If $F^{[1]}$ and $F^{[2]}$ are the roots of $F$, then this reads as

$F_{t}(F^{[1]},z)F_{t}(F^{[2]},z)=F_{t}(x,F^{[1]})F_{t}(x,F^{[2]}),$

which translates apparently into Buchstaber’s four constraints, see [Buh75,
section 2] or [AL75, p. 299].

######  Remark 2.2.3.

Recall that a formal group law is a commutative group object in one-
dimensional formal schemes together with the choice of a local parameter.
There are several possible definitions of commutative $d$-ary monoids or
groups, and of multivalued variants of these. None of these will lead to the
above definition of $n$-valued $d$-ary laws. First, we did not discuss the
notion of inverses here. (Recall that for formal group laws, their existence
is automatic, see e.g. [Rav86, Proposition A2.1.2]. Second, the associativity
conditions are not separate conditions on the $n$ different values $F_{i}$,
but something more complicated, enforced by the associativity condition for
the “variable” $F(x,y)$.

######  Remark 2.2.4.

  1. (1)

Recall [Haz78, Theorem 6.1] that one-dimensional formal group laws are
commutative over reasonable rings. In general, it is easy to describe non
necessarily commutative multivalued formal group laws.

  2. (2)

Using symplectic bundles and $MU^{*}(\mathrm{H}\mathbb{P}^{\infty})$, one
obtains the motivating example for $2-\mathcal{FGL}=\mathscr{G}_{2,2}$, see
e.g. [AL75, section 2]. A similar construction can be done in the motivic
case.

  3. (3)

It is possible to refine the above notion using the ring
$\mathbb{Z}_{\epsilon}=\mathbb{Z}[\epsilon]/(\epsilon^{2}-1)$ introduced in
Section 3, and obtain in particular the notion of an “unoriented” two-valued
formal group law. Such an object is a $(2,2)$-series $F_{t}(x,y)$ satisfying
the symmetry and associativity axioms, together with the following refined
neutral element axiom: The $(2,1)$-series $F_{t}(x,0)$ is split with roots $x$
and $-\epsilon x$, and the additional condition $F_{t}(-\epsilon
x,y)=F_{-\epsilon t}(x,y)$. Lacking interesting examples of genuine such
objects (i.e. on which $\epsilon$ doesn’t act as $-1$), we refrain from
pursuing this here in detail, but see Proposition 4.0.3 below.

The following result is an obvious generalization of known cases.

######  Proposition 2.2.5.

For any couple of integers $(n,d)$, the category $\mathscr{G}_{n,d}$ admits an
initial object $(\mathcal{A}_{n,d},F_{t}^{\mathrm{univ},n,d})$.

If one writes

$F_{t}^{\mathrm{univ},n,d}(\underline{x})=1+\sum_{\underline{\alpha}\in\mathbb{N}^{d},\\\
l=1,\ldots,n}a_{\underline{\alpha}}^{l}\underline{x}^{\underline{\alpha}}t^{l},$

then the $\mathbb{Z}$-algebra $\mathcal{A}_{n,d}$ is generated by the family
$\\{a_{\underline{\alpha}}^{l},\underline{\alpha}\in\mathbb{N}^{d},0\leq l\leq
n\\}$. It is a finitely presented algebra:

$\mathcal{A}_{n,d}=\mathbb{Z}\big{[}a_{\underline{\alpha}}^{l},(\underline{\alpha},l)\in\mathbb{N}^{d}\times\\{0,\ldots,n\\}\big{]}/\mathcal{R}_{n,d}$

and the ideal of relations $\mathcal{R}_{n,d}$ is generated by homogeneous
elements of the graded polynomial ring
$\mathbb{Z}\big{[}a_{\underline{\alpha}}^{l}\big{]}$ with respect to the
degree defined by

$\deg(a_{\underline{\alpha}}^{l})=|\underline{\alpha}|-l.$

###### Proof.

The proof is well-known: to get the ring $\mathcal{A}_{n,d}$, take the
polynomial over $\mathbb{Z}$ generated by the variables
$a_{\underline{\alpha}}^{l}$ as above, and mod out by the relations coming
from the axioms of $(n,d)$-groups. Then, define
$F_{t}^{\mathrm{univ}}(\underline{x})$ by the above formula. It is an easy
(but lengthy) verification that the relations generating the ideal
$\mathcal{R}_{n,d}$ are homogeneous with respect to this notion of degree. In
particular, the ring $\mathcal{A}_{n,d}$ is naturally $\mathbb{Z}$-graded. ∎

######  Definition 2.2.6.

We call the graded ring $\mathcal{A}_{n,d}$ defined above the universal ring
of $(n,d)$-groups, and $F_{t}^{\mathrm{univ},n,d}(\underline{x})$ — or simply
$F_{t}^{\mathrm{univ}}(\underline{x})$ when $(n,d)$ is clear — the universal
$(n,d)$-group.

Obviously, $\mathcal{L}=\mathcal{A}_{1,2}$ is the Lazard ring. We put
$\mathcal{B}=\mathcal{A}_{2,2}$ and call it the _Buchstaber ring_.

######  Example 2.2.7.

The famous theorem of Lazard ([Laz55, Théorème II]) asserts that $\mathcal{L}$
is isomorphic to a polynomial ring $\mathbb{Z}[u_{1},u_{2},...]$ with an
infinite number of variables $u_{i}$.

######  2.2.8.

From the above definition, the Buchstaber ring $\mathcal{B}$ is generated by
elements of the form $a_{ij}^{l}$, $l=1,2$, $i,j\geq 0$, such that the
universal $2$-FGL is

$F_{t}^{\mathrm{univ}}(x,y)=1+\sum_{i,j}a_{ij}^{1}x^{i}y^{j}t+\sum_{i,j}a_{ij}^{2}x^{i}y^{j}t^{2}.$

Specializing 2.2.1 to $n=d=2$, the symmetry axiom tell us exactly that
$a_{ij}^{l}=a_{ji}^{l}$, and the neutral axiom gives:

$\displaystyle F_{t}^{\mathrm{univ}}(x,0)=(1+tx)^{2}=1+2tx+t^{2}x^{2},$ i.e.
$\displaystyle a^{1}_{i0}=2\delta_{1}^{i},\ a^{2}_{i0}=\delta_{2}^{i}.$

In particular, all elements of $\mathcal{B}$ must have non-negative degree,
and the only element of degree $0$ which is not determined is $a^{2}_{11}$.
Buchstaber denotes this element by $\gamma$ and essentially observed the
following result in [Buh75, Theorem 2.3].333Compared to Buchstaber original
reference, we only add the last observation of the following statement.

######  Proposition 2.2.9.

The Buchstaber ring $\mathcal{B}$ defined above is an $\mathbb{N}$-graded ring
_i.e._ each element has non-negative degree. The sub-ring $\mathcal{B}^{0}$ of
elements of degree $0$ satisfies:

$\mathcal{B}^{0}\simeq\mathbb{Z}[\gamma]/\big{(}(\gamma+2)(\gamma-2)^{2}\big{)}.$

Note in particular that $\mathcal{B}$ is an augmented
$\mathcal{B}^{0}$-algebra.

###### Proof.

The first assertion has already been observed. The second one is now a
consequence of the fact that the associativity relations which are of degree
$0$ reduce to the single relation $(\gamma+2)(\gamma-2)^{2}=0$, and this
concludes. ∎

If we work over $\mathbb{Z}_{\epsilon}$ (compare Remark 2.2.4), the above
computation in degree zero refines to $F^{univ,0}_{\epsilon}$ given by

$(F^{univ,0}_{\epsilon})_{t}(x,y)=1+(1-\epsilon)(x+y)t+\left(-\epsilon(x^{2}+y^{2})+axy\right)t^{2}$

over
$\mathcal{B}^{0}_{\epsilon}\simeq\mathbb{Z}_{\epsilon}[a]/\langle(1+\epsilon)a,(a-(1-\epsilon))^{2}(a+(1-\epsilon))\rangle$.

######  Example 2.2.10.

It follows in particular that all $2$-formal groups $F_{t}(x,y)$ whose degree
(2.1.5) is equal to $0$ are of the form:

$F_{t}^{0}(x,y)=1+2(x+y)t+(x^{2}+\gamma xy+y^{2})t^{2}$

with either $\gamma=-2$ or $\gamma=2$. One can rephrase this by saying that
$(\mathcal{B}^{0},F_{t}^{0})$ is the universal $2$-formal group of degree $0$.
Buchstaber calls the above an _elementary 2-formal group_.

Observe that if $\pi:\mathcal{B}\to\mathcal{B}^{0}$ is the ring homomorphism
obtained by killing the variables $a^{1}_{ij}$ and $a^{2}_{ij}$ of (strictly)
positive degree, then we obtain a morphism

$(\pi,\mathrm{Id}):(\mathcal{B},F^{\mathrm{univ}}_{t})\to(\mathcal{B}^{0},F_{t}^{0})$

in $\mathscr{G}_{2,2}$, which is just the morphism given by the universal
property of $(\mathcal{B},F^{\mathrm{univ}}_{t})$. On the other hand, the ring
homomorphism $\mathcal{B}^{0}\to\mathcal{B}$ cannot be extended to a morphism
in $\mathscr{G}_{2,2}$.

######  Remark 2.2.11.

Note in particular a huge difference between the Lazard ring and the
Buchstaber ring: $\mathcal{B}$ is non reduced. From the above computation
$(\gamma+2)(\gamma-2)$ is a non-zero nilpotent element. In particular,
$\mathcal{B}$ cannot be a polynomial ring.

We close this section with the following elementary lemma (compare [Buh75,
Corollary 2.4]).

######  Lemma 2.2.12.

There is a canonical monomorphism of rings

$\mathcal{B}^{0}\rightarrow\big{(}\mathbb{Z}[\gamma]/(\gamma+2)^{2}\big{)}\times\big{(}\mathbb{Z}[\gamma]/(\gamma-2)\big{)}$

which is an isomorphism after inverting $2$. Moreover, there is canonical
monomorphism of rings

$\mathcal{B}^{0}_{red}\rightarrow\big{(}\mathbb{Z}[\gamma]/(\gamma+2)\big{)}\times\big{(}\mathbb{Z}[\gamma]/(\gamma-2)\big{)}$

which is an isomorphism after inverting $2$.

In other words, $\mathcal{B}^{0}_{red}$ is isomorphic to
$\mathbb{Z}\times\mathbb{Z}$, such that the projections to the first and
second factors are respectively the evaluations at $\gamma=-2$ and
$\gamma=+2$.

###  2.3. Buchstaber’s classification of 2-formal groups

Given the computation of the degree $0$ part of the Buchstaber ring, and
following [Buh75], one sets the following notation.

######  Definition 2.3.1.

A $2$-formal group $F_{t}(x,y)$ over a ring $R$ will be said to be _of type I_
(resp. _type II_) if the coefficient of $xyt^{2}$, generically denoted by
$\gamma=a_{11}^{2}$, is equal to $-2$ resp. $+2$. Further, we define the two
following rings:

$\mathcal{B}_{I}=\mathcal{B}/(\gamma+2),\quad\mathcal{B}_{II}=\mathcal{B}/(\gamma-2),$

According to what was said before, the ring $\mathcal{B}_{I}$ classifies the
$2$-formal groups of type I. As, according to our conventions,
$\gamma=a_{11}^{2}$ is of pure degree $0$, these two rings are again
$\mathbb{N}$-graded (2.2.9). We write $(2-\mathcal{FGL})_{I}$ for the full
subcategory of $2-\mathcal{FGL}$ whose objects are laws of the first type and
$(2-\mathcal{FGL})_{II}$ for the full subcategory of objects of the second
type.

The following statement is essentially an elaboration of Buchstaber’s original
results.

######  Proposition 2.3.2.

Let $R$ be a ring. The following assignment

$\sigma_{R}:\mathcal{FGL}(R)\rightarrow 2-\mathcal{FGL}(R),F(x,y)\mapsto
F^{2}_{t}(x,y)=(1+F(x,y)t)^{2}$

defines a faithful functor, by taking the identity on morphisms. Its essential
image lies in $2$-formal groups which are split (2.1.8) and of the second
type.

If $2\in R^{\times}$ then $\sigma_{R}$ is full. If in addition $R$ is a
domain, then $\sigma_{R}$ defines an equivalence of categories between formal
group laws over $R$ and the $2$-formal groups over $R$ which are of the second
type.

###### Proof.

The $(2,2)$-series $F^{2}_{t}(x,y)$ obviously satisfies the neutral and
symmetry axioms of 2.2.1, as $F(x,y)$ satisfies the corresponding axioms. As
it is moreover splits, with a double root $F(x,y)$, one easily deduces the
associativity axiom of $2$-formal groups from the associativity axiom of
$F(x,y)$.

Given a morphism $\Theta:F(x,y)\rightarrow G(x,y)$, we get:

(2.3.2.a)
$\begin{split}&\Theta_{t}(F^{2}_{t}(x,y))=(1+\Theta(F(x,y))t)(1+\Theta(F(x,y))t)\\\
&=(1+G(\Theta(x),\Theta(y)))t)(1+G(\Theta(x),\Theta(y)))t)=G_{t}^{2}(\Theta(x),\Theta(y)).\end{split}$

where the first equality holds as $F_{t}^{2}(x,y)$ splits (with double root
$F(x,y)$), and the second one as $\Theta(x)$ is a morphism of FGL. In
particular, $\Theta_{t}$ is indeed a morphism of $2$-FGL.

The assertion on the image of $\sigma_{R}$ is obvious as $F(x,y)\equiv x+y\
(xy)$. Let us assume that $2\in R^{\times}$. We consider a morphism
$\Theta_{t}=1+\Theta(x)t$ from $F^{2}_{t}(x,y)$ to $G_{t}^{2}(x,y)$ and we
check that $\Theta(x)$ is a morphism of FGL from $F$ to $G$. By assumption, we
know that Equation (2.3.2.a) holds. Expanding and taking the coefficient of
$t$, one deduces that

$2\Theta(F(x,y))=2G(\Theta(x),\Theta(y))$

which allows to conclude.

The final assertion was already observed by Buchstaber. If $R$ is integral,
the roots $\theta_{1}(x,y)$ and $\theta_{2}(x,y)$ of a split $2$-formal group
$F_{t}(x,y)$ are unique (as roots of the polynomial $\tilde{F}_{t}$ in $t$,
with coefficients in the integral ring $R[[x,y]]$). One deduces from the
associativity axiom of $2$-formal groups that
$\theta_{1}(x,y)=\theta_{2}(x,y)$ and that is satisfies the axioms of a formal
group law. This implies the essential surjectivity of $\sigma_{R}$. ∎

######  Remark 2.3.3.

The recipe to make the morphism $\mathfrak{S}$ explicit is as follows. We let
$F^{\mathrm{univ}}(x,y)=x+y+\sum_{ij}a_{ij}x^{i}y^{j}$ be the universal FGL.
Then one computes the associated “squared” $2$-formal group law:

$\displaystyle\big{(}1+(x+y+\sum_{ij}a_{ij}x^{i}y^{j})t\big{)}^{2}=1+2(x+y+a_{11}xy+...)t$
$\displaystyle\qquad\qquad+[x^{2}+2xy+y^{2}+2a_{11}(x^{2}y+xy^{2})+(a_{11}^{2}+4a_{12})x^{2}y^{2}+\ldots]t^{2}.$

Then the coefficient of $x^{i}y^{j}t^{l}$ in this expression gives the image
of the element $a_{ij}^{l}\in\mathcal{B}$ under $\mathfrak{S}$. When $l=1$,
this rule is simple: $\mathfrak{S}(a_{ij}^{1})=2a_{ij}$. When $l=2$, one gets
relations like:

$\displaystyle\mathfrak{S}(a_{1i}^{2})=2a_{1,i-1},$
$\displaystyle\mathfrak{S}(a_{22}^{2})=a_{11}^{2}+4a_{21},\
\mathfrak{S}(a_{23}^{2})=2a_{11}a_{12}+2a_{13}+2a_{22},$
$\displaystyle\mathfrak{S}(a_{22}^{2})=2a_{11}a_{22}+2a_{12}^{2}+4a_{23}.$

######  2.3.4.

We now start with preparations which will lead to another more subtle functor
$\mathcal{FGL}\to 2-\mathcal{FGL}$ which produces 2-FGL of the first type and
is more closely related to characteristic classes.

Given a formal group law $F(x,y)$, we will denote by

$\bar{x}=-x+a_{11}x^{2}+(a_{12}-a_{11}-a_{11}^{2})x^{3}+\ldots$

the unique power series in $x$ such that $F(x,\bar{x})=0$ — _i.e._ the formal
inverse, see [Str, Lem. 2.7]. For any composable $f\in R[[x]]$ we denote more
generally by $\overline{f}$ the series

$\bar{f}:=\bar{x}(f)=-f+a_{11}f^{2}+(a_{12}-a_{11}-a_{11}^{2})f^{3}+\ldots$

######  Lemma 2.3.5.

For any composable power series $f\in R[[x]]$, the series $\overline{f}$ is
the unique series such that

$F(f,\overline{f})=0.$

###### Proof.

As $F(x,\bar{x})=0$, it follows immediately, replacing $x$ by $f$ that
$F(f,\overline{f})=0$. Unicity is obvious: If $g\in R[[x]]$ is such that
$F(f,g)=0$, we have

$g=F(g,0)=F(g,F(f,\overline{f}))=F(F(g,f),\overline{f})=\bar{f}.$

∎

For further use, it is worth noting that $\bar{x}$ induces an automorphism of
the formal group law $F$, in the sense that

$\overline{F(x,y)}:=\bar{x}(F(x,y))=F(\bar{x},\bar{y}).$

The following lemma is probably well-known, but we include it for the sake of
completeness.

######  Lemma 2.3.6.

Let $F\in R[[x,y]]$ be a (commutative) formal group law. Then, there exists a
unique power series $G\in R[[x]]$ such that $x+\bar{x}=G(x\bar{x})$.

###### Proof.

We start with a few preliminary observations. We have a ring homomorphism

$R[[x,y]]\to R[[x]]$

such that $x\mapsto x$ and $y\mapsto\bar{x}$. As $\bar{x}=-x+\sum_{i\geq
2}a_{i}x^{i}$, we get $x+\bar{x}=x\bar{x}=0\pmod{x^{2}}$ and then the ideal
$\langle x+y,xy\rangle^{n}$ is mapped under this homomorphism into $\langle
x^{2n}\rangle$. The power series $F$ being symmetric, we may write it as

$F(x,y)=x+y+uxy+\sum_{i+j\geq 2}a_{ij}(x+y)^{i}(xy)^{j}.$

We now construct for any $n\in\mathbb{N}$ polynomials $G_{n}\in R[t]$ of
degree $n$ by induction. We set $G_{0}=0,G_{1}=-ut$ and

$G_{n}=-ut-\sum_{i+j=2}^{n}a_{ij}G_{n-1}(t)^{i}t^{j}\pmod{t^{n+1}}$

for $n\geq 2$. We now check by induction that for any $n\in\mathbb{N}$ the
following conditions are satisfied:

  1. (1)

$x+\bar{x}=G_{n}(x\bar{x})$ modulo $\langle x^{2n+2}\rangle$.

  2. (2)

$G_{n}=G_{n-1}\pmod{t^{n}}$.

For $n=0$, we already know that $x+\bar{x}=0\pmod{x^{2}}$ and the second
relation is empty. For $n=1$, we observe that

$F(x,y)=x+y+uxy\pmod{\langle x+y,xy\rangle^{2}}$

and it follows that

$0=F(x,\bar{x})=x+\bar{x}+ux\bar{x}\pmod{x^{4}}.$

As $G_{1}=-ut$, both properties are satisfied. Let then $n\geq 2$ be such that
properties 1. and 2. are satisfied for $n-1$. Modulo $\langle
x+y,xy\rangle^{n+1}$, we have

$F(x,y)=x+y+uxy+\sum_{i+j=2}^{n}a_{ij}(x+y)^{i}(xy)^{j}$

and consequently

$0=F(x,\bar{x})=x+\bar{x}+ux\bar{x}+\sum_{i+j=2}^{n}a_{ij}(x+\bar{x})^{i}(x\bar{x})^{j}\pmod{x^{2n+2}}.$

By construction $x+\bar{x}=G_{n-1}(x\bar{x})$ modulo $\langle x^{2n}\rangle$,
i.e. $x+\bar{x}=G_{n-1}(x\bar{x})+Hx^{2n}$ for some polynomial $H$. Modulo
$\langle x^{2n+2}\rangle$, we have

$(x+\bar{x})^{i}=G_{n-1}(x\bar{x})^{i}+iHG_{n-1}(x\bar{x})^{i-1}x^{2n}$

and we can use the fact that $G_{n-1}(t)=0\pmod{t}$ and $x\bar{x}=0$ modulo
$\langle x^{2}\rangle$ to get
$(x+\bar{x})^{i}=G_{n-1}(x\bar{x})^{i}\pmod{\langle x^{2n+2}\rangle}$.
Altogether, the above expression reads as

$0=x+\bar{x}+ux\bar{x}+\sum_{i+j=2}^{n}a_{ij}G_{n-1}(x\bar{x})^{i}(x\bar{x})^{j}\pmod{x^{2n+2}}.$

This means that $x+\bar{x}=G_{n}(x\bar{x})$ modulo $x^{2n+2}$, and the first
point is satisfied. We are left with proving that $G_{n}=G_{n-1}$ modulo
$t^{n}$. By definition, we may write

$G_{n}=-ut-\sum_{i+j=2}^{n}a_{ij}G_{n-1}(t)^{i}t^{j}\pmod{t^{n+1}}.$

We know by induction that $G_{n-1}=G_{n-2}\pmod{t^{n-1}}$. It follows that
$G_{n-1}t=G_{n-2}t$ modulo $t^{n}$. On the other hand, $G_{n-2}=0\pmod{t}$ and
we obtain for $i\geq 2$ that

$G_{n-1}^{i}=(G_{n-2}+\alpha t^{n-1})^{i}=G_{n-2}^{i}.$

Modulo $t^{n}$, we then obtain

$G_{n}=-ut-\sum_{i+j=2}^{n}a_{ij}G_{n-1}(t)^{i}t^{j}=-ut-\sum_{i+j=2}^{n}a_{ij}G_{n-2}(t)^{i}t^{j}=G_{n-1}.$

∎

######  Remark 2.3.7.

Obviously, the conclusion of the lemma also holds for $-x\bar{x}$, i.e. there
is a unique power series $G^{\prime}$ such that
$x+\bar{x}=G^{\prime}(-x\bar{x})$. We opted for the previous formulation in
order to avoid sign issues.

######  Remark 2.3.8.

As a consequence of the above lemma, we see that the ring extension
$R[[-x\bar{x}]]\subset R[[x]]$ is finite. Indeed, consider the polynomial

$(T-x)(T-\bar{x})=T^{2}-(x+\bar{x})T+x\bar{x}=T^{2}-G(x\bar{x})T+x\bar{x}$

The extension $R[[-x\bar{x}]]\subset
R[[-x\bar{x}]][T]/(T^{2}-G(x\bar{x})T+x\bar{x})$ is finite and there is an
isomorphism $R[[-x\bar{x}]][T]/(T^{2}-G(x\bar{x})T+x\bar{x})\to R[[x]]$
mapping $T$ to $x$.

######  Lemma 2.3.9.

Let $F$ be a formal group law. Then there exists a unique $(2,2)$-series
$\bar{F}_{t}^{2}(x,y)$ such that the following equality holds in
$R[[x,y]][t]$:

$\bar{F}^{2}_{t}(-x\bar{x},-y\bar{y})=(1-F(x,y)F(\bar{x},\bar{y})t)(1-F(\bar{x},y)F(x,\bar{y})t).$

The series $\bar{F}_{t}^{2}(x,y)$ is of type I.

###### Proof.

As the series $-x\bar{x}$ is of the form $x^{2}\mod(x^{3})$, the endomorphism

$\phi_{F}:R[[x,y]]\rightarrow R[[x,y]],x\mapsto-x\bar{x},y\mapsto-y\bar{y}$

is injective. This proves the uniqueness of $\bar{F}_{t}^{2}(x,y)$. For the
existence, one has to prove that the series:

$F(x,y)F(\bar{x},\bar{y})F(\bar{x},y)F(x,\bar{y})\text{ and
}F(x,y)F(\bar{x},\bar{y})+F(\bar{x},y)F(x,\bar{y})$

belongs to the image of $\phi_{F}$. This is precisely [BN71, Lemma 2.21].

We now prove that $\bar{F}_{t}^{2}(x,y)$ is of type I. For this, it suffices
to prove that $F_{2}(x,x)=0$. Since, $F(\bar{x},x)=0$, we obtain

$\displaystyle\bar{F}^{2}_{t}(-x\bar{x},-x\bar{x})$ $\displaystyle=$
$\displaystyle(1-F(x,x)F(\bar{x},\bar{x}).t)(1-F(\bar{x},x)F(x,\bar{x}).t)$
$\displaystyle=$ $\displaystyle(1-F(x,x)F(\bar{x},\bar{x}).t).$

∎

######  Remark 2.3.10.

In view of Remark 2.3.8, we see that the roots of $\bar{F}^{2}_{t}$ can be
chosen to be $F^{[1]}=-F(x,y)F(\bar{x},\bar{y})$ and
$F^{[2]}=-F(\bar{x},y)F(x,\bar{y})$ in $R[[x,y]]$.

In the following statement involving two formal group laws $F$ and $G$, we
denote by $\overline{x}^{F}$ the formal inverse of $x$ with respect to the law
$F$ and $\overline{x}^{G}$ the one with respect to $G$. Also, given a
composable series $\Theta$ we denote by

$\mathrm{ev}_{\Theta}:R[[x]]\to R[[x]]$

the homomorphism given by $f(x)\mapsto f(\Theta)$. The following results are
mild variations on [Buh78, §3, Lemma 3.1].

######  Lemma 2.3.11.

Let $F(x,y)$ and $G(x,y)$ be formal group laws, and let $\Theta:F\to G$ be a
morphism of formal group laws. Then, there is a unique composable series
$\varphi(\Theta)$ such that the following diagram commutes

$\textstyle{R[[x]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\varphi(\Theta)}}$$\scriptstyle{\mathrm{ev}_{-x\bar{x}^{G}}}$$\textstyle{R[[x]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{-x\bar{x}^{F}}}$$\textstyle{R[[x]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\Theta}}$$\textstyle{R[[x]].}$

Further, given a morphism of formal group laws $\Psi:G\to H$, we have

$\varphi(\Theta\circ\Psi)=\varphi(\Theta)\circ\varphi(\Psi).$

###### Proof.

As $\Theta$ is composable, we have $\Theta(F(x,\overline{x}^{F}))=0$ and then

$G(\Theta,{\Theta}(\overline{x}^{F}))=0$

i.e.
${\Theta}(\overline{x}^{F})=\overline{\Theta}^{G}={\bar{x}}^{G}(\Theta)=\mathrm{ev}_{\Theta}(\bar{x}^{G})$.
It follows that

$-(x\bar{x}^{G})(\Theta)=-\Theta\bar{x}^{G}(\Theta)=-\Theta\Theta(\bar{x}^{F}).$

This power series is obviously symmetric in the variables $x$ and
$\bar{x}^{F}$, and we may write it in terms of the elementary symmetric
polynomials in $x$ and $\bar{x}$. Lemma 2.3.6 then gives a series
$\varphi(\Theta)$ fulfilling the stated property. The unicity statement is
obvious from the fact that $x\mapsto-x\bar{x}^{F}$ is injective. Finally, we
have $\varphi(\Theta)(-x\bar{x}^{F})=-\Theta\Theta(\bar{x}^{F})$, from which
we immediately deduce that $\varphi(\Theta)$ is composable.

The statement about the composite follows from the commutative diagram

$\textstyle{R[[x]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\varphi(\Psi)}}$$\scriptstyle{\mathrm{ev}_{-x\bar{x}^{H}}}$$\textstyle{R[[x]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\varphi(\Theta)}}$$\scriptstyle{\mathrm{ev}_{-x\bar{x}^{G}}}$$\textstyle{R[[x]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{-x\bar{x}^{F}}}$$\textstyle{R[[x]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\Psi}}$$\textstyle{R[[x]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\Theta}}$$\textstyle{R[[x]]}$

and the unicity statement of the previous point. ∎

######  Proposition 2.3.12.

Let $R$ be a ring and consider the notation of the previous lemma. Then the
assignment

$N_{R}:\mathcal{FGL}(R)\rightarrow(2-\mathcal{FGL})_{I}(R),F(x,y)\mapsto\bar{F}^{2}_{t}(x,y)$

can be extended into a functor mapping a morphism $\Theta$ in
$\mathcal{FGL}(R)$ to $\varphi(\Theta)$.

###### Proof.

We already know from Lemma 2.3.9 that the functor is well-defined on objects.
We now consider the commutative diagram

$\textstyle{R[[x,y]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\varphi(\Theta)}}$$\scriptstyle{\phi_{G}}$$\textstyle{R[[x,y]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{F}}$$\textstyle{R[[x,y]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{\Theta}}$$\textstyle{R[[x,y]]}$

where $\phi_{G}(f(x,y))=f(-x\bar{x}^{G},-y\bar{y}^{G})$ (idem for $F$) and

$\mathrm{ev}_{\varphi(\Theta)}(f(x,y))=f(\varphi(\Theta)(x),\varphi(\Theta)(y))$

(idem for $\mathrm{ev}_{\Theta}$). Next, we consider the polynomial

$1+\bar{G}_{1}^{2}(\varphi(\Theta)(x),\varphi(\Theta)(y))t+\bar{G}_{2}^{2}(\varphi(\Theta)(x),\varphi(\Theta)(y))t^{2}.$

Applying to $-x\bar{x}^{F}$, using
$\varphi(\Theta)(-x\bar{x}^{F})=-\Theta\Theta(\bar{x}^{F})$ and
$\Theta(\bar{x}^{F})=\overline{\Theta(x)}^{G}$ we obtain

$(1+G(\Theta(x),\Theta(y))G(\Theta(\bar{x}^{F}),\Theta(\bar{y}^{F}))t)(1+G(\Theta(\bar{x}^{F}),\Theta(y))G(\Theta(x),\Theta(\bar{y}^{F}))t).$

As already observed in the proof of Lemma 2.3.11, we have
$\bar{x}^{G}(\Theta)=\Theta(\bar{x}^{F})$ and we then see that the previous
polynomial is of the form

$(1+\Theta(F(x,y))\Theta(F(\bar{x}^{F},\bar{y}^{F}))t)(1+\Theta(F(\bar{x}^{F},y))\Theta(F(x,\bar{y}^{F}))t).$

As $F(\bar{x}^{F},\bar{y}^{F})=\overline{F(x,y)}^{F}$ and
$F(x,\bar{y}^{F})=\overline{F(\bar{x}^{F},y)}^{F}$, we finally obtain a
polynomial of the form

$(1+\Theta(F(x,y))\Theta(\overline{F(x,y)}^{F})t)(1+\Theta(F(\bar{x}^{F},y))\Theta(\overline{F(\bar{x}^{F},y)}^{F})t).$

As $\Theta\Theta(\bar{x}^{F})=\varphi(\Theta)(x\bar{x}^{F})$, the above
expression becomes

$(1+\varphi(\Theta)(F(x,y)\overline{F(x,y)}^{F})t)(1+\varphi(\Theta)(F(\bar{x}^{F},y)\overline{F(\bar{x}^{F},y)}^{F})t).$

which is $\varphi(\Theta)(\bar{F}^{2}_{t}(-x\bar{x}^{F},-y\bar{y}^{F}))$. We
have then obtained that

$\bar{G}_{t}^{2}(-x\bar{x}^{F},-y\bar{y}^{F})=\varphi(\Theta)(\bar{F}^{2}_{t}(-x\bar{x}^{F},-y\bar{y}^{F}))$

and we deduce from the injectivity of the map $x\mapsto-x\bar{x}^{F}$ that
$\varphi(\Theta)$ is a morphism between $\bar{F}^{2}_{t}$ and
$\bar{G}^{2}_{t}$ (see Definition 2.1.15). The functor respects compositions
by Lemma 2.3.11 and it is straightforward to check that
$\varphi(\mathrm{Id})=\mathrm{Id}$. ∎

######  Remark 2.3.13.

The functor $N_{R}$ is not faithful. Consider the additive formal group law
$F(x,y)=x+y$ and the morphism $\Theta:F\to F$ given by the series
$\Theta(t)=-t$ (i.e. $\Theta=\bar{x}$). Then, we see that the series
$\Theta^{\prime}(t)=t$ and $\Theta$ have the same image under the functor.
Indeed,

$-\Theta(x)\Theta(\bar{x})=--x(-\bar{x})=x^{2},$

showing that $\varphi(\Theta)=t$. The same applies to $\Theta^{\prime}$.

More generally, let $F(x,y)$ be an arbitrary formal group law. Then, we know
that the series $\bar{x}^{F}=\bar{x}$ is an endomorphism of $F$. As
$\bar{x}(\bar{x})=x$, it follows that $\varphi(\bar{t})=t=\varphi(t)$.

######  Remark 2.3.14.

The functor $N$ is not full either. Consider the elementary $2$-formal group
of the first type

$F_{t}(x,y)=1+2(x+y)+(x-y)^{2}t^{2}.$

A straightforward computation shows that it is the image of the additive
formal group law under the functor $N$. Now, consider the series
$\varphi(x)=-x$. It is an endomorphism of $F_{t}$: If $F^{[1]}$ and $F^{[2]}$
are the roots of $F$, we have $F^{[1]}+F^{[2]}=2(x+y)$ and
$F^{[1]}F^{[2]}=(x-y)^{2}$. Now

$(1-F^{[1]}t)(1-F^{[2]}t)=1-2(x+y)t+(x-y)^{2}t^{2}$

which is apparently $F_{t}(-x,-y)$. If $-1$ is not a square in $R$, then
$\varphi$ is not the image of an endomorphism of the additive formal group
law. Indeed, if $\Theta$ is such an endomorphism mapping to $\varphi$, then we
should have

$-x^{2}=\varphi(-x\bar{x})=-\Theta\Theta(\bar{x})$

Writing $\Theta=ax+\sum_{i\geq 2}a_{i}x^{i}$, we see that the above equality
implies $a^{2}=-1$.

These two observations might suggest to restrict the functor to the
subcategories where morphisms are _strict_ isomorphisms. One might ask if the
functor then becomes an equivalence, at least after inverting $2$.

##  3\. Formal ternary laws

###  3.1. Algebraic definitions

######  3.1.1.

Our base ring will be the ring

$\mathbb{Z}_{\epsilon}:=\operatorname{GW}(\mathbb{Z}),$

_i.e._ the Grothendieck-Witt ring associated with the category of finitely
generated projective $\mathbb{Z}$-modules equipped with a non-degenerate
symmetric bilinear form. Let $<-1>$ be the class of the bilinear form

$\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$

given by $(x,y)\mapsto-xy$. Following Morel’s conventions in motivic homotopy
theory, we set $\epsilon:=-<-1>$. Then the canonical map:

$\mathbb{Z}[t]\rightarrow\operatorname{GW}(\mathbb{Z}),t\mapsto\epsilon$

is an epimorphism whose kernel is the ideal $(t^{2}-1)$ (see [MH73, chap. 2]).
One writes suggestively
$\mathbb{Z}_{\epsilon}=\mathbb{Z}[\epsilon]/(\epsilon^{2}-1)$. In particular,
$\mathbb{Z}_{\epsilon}$ is a free $\mathbb{Z}$-module of rank $2$: any element
can uniquely be written as $a+b.\epsilon$ for a couple of integers $(a,b)$.

One defines the plus-part (resp. minus-part) of $\mathbb{Z}_{\epsilon}$ as the
quotient ring:

$\mathbb{Z}_{\epsilon}^{+}=\mathbb{Z}_{\epsilon}/(\epsilon+1)\text{ resp.
}\mathbb{Z}_{\epsilon}^{-}=\mathbb{Z}_{\epsilon}/(\epsilon-1).$

We have an obvious identification $\mathbb{Z}_{\epsilon}^{+}\simeq\mathbb{Z}$
(resp. $\mathbb{Z}_{\epsilon}^{-}\simeq\mathbb{Z}$) obtained by mapping the
class of $a+b\epsilon$ to $a-b$ (resp. $a+b$). The following lemma is now
obvious.

######  Lemma 3.1.2.

The canonical morphism of rings

$\mathbb{Z}_{\epsilon}\rightarrow\mathbb{Z}_{\epsilon}^{+}\times\mathbb{Z}_{\epsilon}^{-}\simeq\mathbb{Z}\times\mathbb{Z}$

is injective and its image is given by the set of pairs $(n,m)$ such that
$n\equiv m\mod 2$. It is an isomorphism after inverting $2$ on both sides.

Geometrically, $\operatorname{Spec}(\mathbb{Z}_{\epsilon})$ is the union of
two closed subschemes whose reductions are $\operatorname{Spec}(\mathbb{Z})$,
and which intersects in a single point,
$\operatorname{Spec}(\mathbb{F}_{2}[t]/(t+1)^{2})$.

######  Corollary 3.1.3.

For any $\mathbb{Z}_{\epsilon}$-algebra $A$, one puts
$A^{\pm}=A\otimes_{\mathbb{Z}_{\epsilon}}\mathbb{Z}_{\epsilon}^{\pm}$.
There is a canonical map: $A\rightarrow A^{+}\times A^{-}$ which is a
monomorphism if $A$ is flat over $\mathbb{Z}_{\epsilon}$, and an isomorphism
if $2$ is invertible in $A$.

######  Example 3.1.4.

The motivation underlying our choice of base ring comes from the following
example. Let $S$ be any scheme, and $\mathbb{E}$ a ring spectrum over $S$.
Then the coefficient ring $\mathbb{E}^{**}(S)$ is canonically a
$\mathbb{Z}_{\epsilon}$-algebra, where the action of $\epsilon$ is given by
the canonical map $\epsilon:\mathbb{G}_{m,S}\rightarrow\mathbb{G}_{m,S}$ seen
in $\operatorname{SH}(S)$ after infinite suspension, induced by the inverse of
the group $S$-scheme $\mathbb{G}_{m,S}$.

The following definition is a variation of the notion of a $(n,d)$-group (Def.
2.2.1) which naturally arise in the theory of characteristic classes related
with $\mathrm{Sp}$-oriented theories (see the next section). It has been
partially introduced in [DF21, section 2.3], with origins in unpublished notes
of Ch. Walter.

######  Definition 3.1.5.

Let $R$ be a $\mathbb{Z}_{\epsilon}$-algebra. A formal ternary law, FTL for
short, with coefficients in $R$ is a $(4,3)$-series with coefficients in $R$

$F_{t}(\underline{x})=1+F_{1}(x,y,z)t+F_{2}(x,y,z)t^{2}+F_{3}(x,y,z)t^{3}+F_{4}(x,y,z)t^{4}$

satisfying the following properties:

  1. (1)

Neutral element. The $(4,1)$-series $F_{t}(x,0,0)$ is split with roots $x$ and
$-\epsilon.x$ each with multiplicity $2$, _i.e._
$F_{t}(x,0,0)=(1+xt)^{2}(1-\epsilon xt)^{2}$.

  2. (2)

Symmetry. The element $F_{t}(x,y,z)$ of $R[[x,y,z]][t]$ is a fixed point under
the action of the group $\mathfrak{S}(x,y,z)$ permuting the formal variables.

  3. (3)

Associativity. Given formal variables $(x,y,z,u,v)$, one has the following
equality of $(16,5)$-series:

(3.1.5.a) $\begin{split}&F_{t}\big{(}F_{t}(x,y,z),u,v\big{)}\\\
&\quad=F_{t}\big{(}x,F_{t}(y,z,u),v\big{)}\end{split}$

where we have used the substitution operation (Definition 2.1.13), valid as
$F_{t}(x,y,z)$ is substituable by property (1).

  4. (4)

$\epsilon$-Linearity. One has the following relation in
$\mathscr{F}_{4,3}(R)$:

$F_{t}(-\epsilon x,y,z)=F_{-\epsilon t}(x,y,z).$

  5. (5)

Weak neutral element. The following relation holds:

$F_{4}(x,x,0)=0.$

We let $\mathcal{FTL}$ be the subcategory of $\mathscr{F}_{4,3}^{\circ}$ whose
objects are as above, and whose morphisms
$(\varphi,\Theta):(R,F_{t})\rightarrow(R^{\prime},F^{\prime}_{t})$ are given
by morphisms of $(4,3)$-series such that $\varphi:R\rightarrow R^{\prime}$ is
$\mathbb{Z}_{\epsilon}$-linear.

######  Remark 3.1.6.

  1. (1)

The symmetry and associativity axioms imply two other associativity axioms:

$\displaystyle\begin{split}F_{t}\big{(}F_{t}(x,y,z),u,v\big{)}=F_{t}\big{(}x,y,F_{t}(z,u,v)\big{)},\\\
F_{t}\big{(}x,F_{t}(y,z,u),v\big{)}=F_{t}\big{(}x,y,F_{t}(z,u,v)\big{)}.\end{split}$

  2. (2)

All FTL arising from motivic symplectic oriented spectra (see section 5 below)
we have studied so far satisfy a stronger version of the weak neutral element
axiom, namely we have $F_{3}(x,x,0)=0$. One may wonder if this stronger
condition holds for all geometric examples and should be added as an axiom.

  3. (3)

Beware that a FTL is not a $(4,3)$-group in general because of the ”Neutral
element” axiom. However, if $\epsilon$ acts as $-1$ on the coefficient ring
$R$, a FTL is precisely a $(4,3)$-group which satisfy the additional “Weak
neutral element” axiom.

######  3.1.7.

Writing

(3.1.7.a) $F_{t}(x,y,z)=1+\sum_{i,j,k\geq 0,1\leq l\leq
4}a_{ijk}^{l}x^{i}y^{j}z^{k}t^{l},$

one can restate the above properties in term of the coefficients $a_{ijk}^{l}$
as follows:

  1. (1)

Symmetry. for all $l$, $a_{ijk}^{l}$ is invariant under permutations of the
triple $(i,j,k)$.

  2. (2)

Neutral element. $a_{i00}^{l}=\begin{cases}1&i=l=4,\\\
2(1-\epsilon)&i=l=1,3,\\\ 2(1-2\epsilon)&i=l=2,\\\
0&\text{otherwise.}\end{cases}$

  3. (4)

$\epsilon$-Linearity. The relation $(1+\epsilon)a_{ijk}^{l}$ holds whenever
$l$ and one of the $i,j,k$ don’t have the same parity.

  4. (5)

Weak neutral element. For all $n\geq 0$, $\sum_{i+j=n}a^{4}_{ij0}=0$.

######  Remark 3.1.8.

  1. (1)

The $\epsilon$-linearity axiom exhibits $\mathbb{Z}_{\epsilon}$-torsion
amongst the coefficients on an FTL. When projecting to the plus-part, this
torsion property disappears, and when projecting to the minus-part, it becomes
a $2$-torsion assertion.

  2. (2)

The associativity formula which we omitted in the list is indeed very
difficult to compute, and even to print out! We have worked it out partially
using computers and we refer the interested reader to the appendix. Of course,
this amounts to write explicitly the relations for

$\prod_{i=1}^{4}F_{t}(F^{[i]},u,v)=\prod_{i=1}^{4}F_{t}(x,F^{[i]},v)$

For instance the relations in degree $1$ are given by
$\sum_{i=1}^{4}F_{1}(F^{[i]},u,v)=\sum_{i=1}^{4}F_{1}(x,F^{[i]},v)$.

The following result is an obvious extension of 2.2.5.

######  Proposition 3.1.9.

The category $\mathcal{FTL}$ admits an initial object,
$(\mathcal{W},F_{t}^{univ})$.

If one writes $a_{ijk}^{l}$ the coefficients of the universal FTL
$F_{t}^{univ}(x,y,z)$, the $\mathbb{Z}_{\epsilon}$-algebra $\mathcal{W}$ is
generated by the countable family $\\{a_{ijk}^{l},i\geq j\geq k\geq 0,1\leq
l,\leq 4\\}$. It is a quotient ring of the form

$\mathcal{W}=\mathbb{Z}_{\epsilon}\big{[}a_{ijk}^{l}\big{]}/\mathcal{R}_{FTL}$

and the ideal of relations $\mathcal{R}_{FTL}$ is generated by homogeneous
elements of the polynomial ring
$\mathbb{Z}_{\epsilon}\big{[}a_{ijk}^{l}\big{]}$ with respect to the degree
defined by

$\deg(a_{ijk}^{l})=i+j+k-l,(\text{and $\deg(\lambda)=0$ for
$\lambda\in\mathbb{Z}_{\epsilon}$}).$

The proof works as in _loc. cit._ Note in particular that the supplementary
axioms of $\epsilon$-linearity and weak neutral element readily produce
homogeneous relations as explained in 3.1.7.

######  Definition 3.1.10.

We call the $\mathbb{Z}$-graded ring $\mathcal{W}$ defined above the Walter
ring.

######  Definition 3.1.11.

We say that an object $(R,F)$ in $\mathcal{FTL}$ is _oriented_ provided we
have $\epsilon=-1\in R$.

The term “oriented” is motivated by the fact that oriented spectra lie in the
$+$-part of $\operatorname{SH}(k)$ where $\epsilon=-1$. Our next aim is now to
determine the universal FTL of degree $\leq 0$, i.e. the ring
$\mathcal{W}^{\leq 0}$ with associated law $F^{\leq 0}_{t}(x,y,z)$. The
nonzero coefficients are of the form

$\\{a^{i}_{jkl}|1\leq i\leq 4,0\leq j+k+l\leq i\\}.$

The algorithm described in the appendix allows to find the result with
integral coefficients. There are however 20 generators and 54 relations and
the result would be quite long to state. For this reason, we prefer to state
the result with $\mathbb{Z}[\frac{1}{2}]$-coefficients.

######  Theorem 3.1.12.

Let $\mathcal{W}^{\leq 0}[\frac{1}{2}]$ be the sub-ring of
$\mathcal{W}[\frac{1}{2}]$ generated by variables of non-positive degree.
Then, one has

$\mathcal{W}^{\leq
0}[\frac{1}{2}]\simeq\mathbb{Z}_{\epsilon}[\frac{1}{2}][a]/\langle(a-40)(1-\epsilon),(a-8)(a+8)(1+\epsilon)\rangle$

and the coefficients of the universal FTL are given by

$\begin{array}[]{llll}a^{1}_{100}=2(1-\epsilon)&&&\\\
a^{2}_{200}=2(1-2\epsilon)&a^{2}_{110}=2(1-\epsilon)&&\\\
a^{3}_{300}=2(1-\epsilon)&a^{3}_{210}=-2(1-\epsilon)&a^{3}_{111}=a&\\\
a^{4}_{400}=1&a^{4}_{310}=-2(1-\epsilon)&a^{4}_{220}=2(1-2\epsilon)&a^{4}_{211}=2(1-\epsilon).\end{array}$

###### Proof.

According to Appendix A.4, the parameters have all the stated values, except
for $a:=a^{3}_{111}$ which should satisfy two relations, namely
$a^{2}=832-768\epsilon$ and $(a-40)(\epsilon-1)=0$. In
$\mathbb{Z}_{\epsilon}[\frac{1}{2}]$, we have the idempotents
$e:=\frac{1}{2}(1-\epsilon)$ and $f:=\frac{1}{2}(1+\epsilon)$. One checks
readily that

$a^{2}-832+768\epsilon=((a-40)e+(a-8)f)((a-40)e+(a+8)f)+80(a-40)e$

so that the relevant ideal is $\langle(a-40)e,(a-8)(a+8)f\rangle$. ∎

######  Remark 3.1.13.

The theorem states that there are two laws of degree $\leq 0$ (when $2$ is
inverted). Indeed, one has either $a^{3}_{111}=8(2-3\epsilon)$ or
$a^{3}_{111}=8(3-2\epsilon)$. Setting $F_{t}(x,y,z)$ for the first law (i.e.
with coefficients as above and $a^{3}_{111}=8(2-3\epsilon)$) and
$G_{t}(x,y,z)$ for the second one, we see that

$F_{t}(-\epsilon x,-\epsilon y,-\epsilon z)=G_{t}(x,y,z).$

Comparing with [DF21, Definition 2.1.3, Theorem 3.3.3], see also the next
example, only one of these two laws is known to arise from an Sp-oriented
spectrum and then satisfies the normalization axiom. We may then define a new
Borel class $b_{1}^{\prime}$ as the class $-\epsilon b_{1}$ and keep $t$
fixed. The law $G_{t}(x,y,z)$ is obtained out of $F_{t}(x,y,z)$ in this
fashion, but $F_{t}$ and $G_{t}$ are not isomorphic in the sense of FTL.

###  3.2. Examples

Let $k$ be a field with $6\in k^{\times}$ and consider the ring spectrum
$\mathbf{H}_{\mathrm{MW},k}$ over $k$ representing Milnor-Witt motivic
cohomology, introduced in [BCD+20, Chap. 6]. The FTL $F_{t}(x,y,z)$ associated
to this cohomology theory was computed in [DF21, Theorem 3.3.2]. We have:

$F_{i}=\begin{cases}2(1-\epsilon)\sigma(x),&i=1,\\\
2(1-2\epsilon)\sigma(x^{2})+2(1-\epsilon)\sigma(xy),&i=2,\\\
2(1-\epsilon)\sigma(x^{3})-2(1-\epsilon)\sigma(x^{2}y)+8(2-3\epsilon)xyz,&i=3,\\\
\sigma(x^{4})-2(1-\epsilon)\sigma(x^{3}y)+2(1-2\epsilon)\sigma(x^{2}y^{2})+2(1-\epsilon)\sigma(x^{2}yz),&i=4.\end{cases}$

We note that $F_{t}$ is both of degree and valuation $0$. If we invert $2$, we
observe that this law is the universal one of degree $0$. In a sense, this is
a justification a posteriori of the terminology of [DF21, Definition 3.3.3].

Let now $\mathbf{KO}$ be the absolute ring spectrum representing Hermitian
$K$-theory over $\mathbb{Z}[\frac{1}{2}]$. There exists an element
$\gamma\in\mathbf{KO}^{8,4}(\mathbb{Z}[\frac{1}{2}])$ such that multiplication
by $\gamma$ induces an isomorphism
$\mathbf{KO}^{*,*}\to\mathbf{KO}^{*+8,*+4}$. We denote as usual its inverse by
$\gamma^{-1}\in\mathbf{KO}^{-8,-4}(\mathbb{Z}[\frac{1}{2}])$. On the other
hand, we have
$\mathbf{KO}^{4,2}(\mathbb{Z}[\frac{1}{2}])=\mathrm{GW}^{-}(\mathbb{Z}[\frac{1}{2}])$,
the Grothendieck-Witt group of skew symmetric (non degenerate) bilinear forms.
We have $\mathrm{GW}^{-}(\mathbb{Z}[\frac{1}{2}])=\mathbb{Z}\cdot\tau$, where
$\tau$ is the class of the symplectic form
$\mathbb{Z}[\frac{1}{2}]^{2}\times\mathbb{Z}[\frac{1}{2}]^{2}\to\mathbb{Z}[\frac{1}{2}]$
defined by $((u_{1},v_{1}),(u_{2},v_{2}))\mapsto u_{1}v_{2}-u_{2}v_{1}$. The
following computation can be found in [FH20, Theorem 6.6].

######  Theorem 3.2.1.

The formal ternary laws $F_{i}=F_{i}(x,y,z)$ of Hermitian $K$-theory are

$F_{1}=2(1-\epsilon)\sigma(x)+\tau\gamma^{-1}\sigma(xy)+\gamma^{-1}xyz,\hskip
284.52756pt$
$F_{2}=2(1-2\epsilon)\sigma(x^{2})+2(1-\epsilon)\sigma(xy)+2\tau\gamma^{-1}\sigma(x^{2}y)-3\tau\gamma^{-1}xyz+\gamma^{-1}\sigma(x^{2}y^{2}),\hskip
85.35826pt$
$F_{3}=2(1-\epsilon)\sigma(x^{3})-2(1-\epsilon)\sigma(x^{2}y)+8(2-3\epsilon)xyz+\tau\gamma^{-1}\sigma(x^{3}y)-2\tau\gamma^{-1}\sigma(x^{2}y^{2})+3\tau\gamma^{-1}\sigma(x^{2}yz)+\gamma^{-1}\sigma(x^{3}yz),$
$F_{4}=\sigma(x^{4})-2(1-\epsilon)\sigma(x^{3}y)+2(1-2\epsilon)\sigma(x^{2}y^{2})+2(1-\epsilon)\sigma(x^{2}yz)-\tau\gamma^{-1}\sigma(x^{3}yz)+2\tau\gamma^{-1}\sigma(x^{2}y^{2}z)+\gamma^{-1}\sigma(x^{2}y^{2}z^{2}).$

The computation for $i=1$ is also in unpublished notes of Ch. Walter. As
explained in [FH20, Remark 6.7], setting $\epsilon=1$ and $\tau=0$ one
recovers Ananyevskiy’s computation [Ana17] for Witt cohomology. Setting
$\epsilon=-1$, $\tau=2a^{-2}$ and $\gamma^{-1}=a^{4}$, one recovers the image
of the multiplicative formal group law $x+y-axy$ under the composite of
functors $W=C\circ N$ (see Section 4), which we have calculated independently
by computer (note that the FGL $x+y+axy$ gives the same result).

##  4\. Comparison

We constructed a functor $N:\mathcal{FGL}\rightarrow(2-\mathcal{FGL})_{I}$ in
Proposition 2.3.12. In this section, we will produce a functor
$C:(2-\mathcal{FGL})_{I}\rightarrow\mathcal{FTL}$ and thus obtain a composite
functor

$W:=C\circ N:\mathcal{FGL}\to\mathcal{FTL}.$

The motivation of the construction of $C$ is of geometric nature (compare also
section 5 below). Consider the universal bundles $U_{1},U_{2}$ and $U_{3}$ on
$(\mathrm{H}{\mathbb{P}}^{\infty})^{\times 3}=\mathrm{BSp}_{2}^{\times 3}$. We
write as usual $x$ for the first Borel class of $U_{1}$, $y$ for the first
Borel class of $U_{2}$ and $z$ for the first Borel class of $U_{3}$. Suppose
that we can associate characteristic classes to $U_{1}U_{2}$ under the form of
a reasonable series

$F_{t}(x,y)=1+F_{1}(x,y)t+F_{2}(x,y)t^{2},$

for instance for $F_{t}$ a two-valued formal group law. The roots $F^{[1]}$
and $F^{[2]}$ represent characteristic classes of bundles $V_{1}$ and $V_{2}$
having the property that

$U_{1}U_{2}=V_{1}\oplus V_{2}$

over a suitable extension of $(\mathrm{H}{\mathbb{P}}^{\infty})^{\times 2}$.
We could then write

$U_{1}U_{2}U_{3}=(V_{1}\oplus V_{2})\cdot U_{3}=V_{1}U_{3}\oplus V_{2}U_{3}$

The characteristic classes of the first bundle are given by the polynomial

$1+F_{1}(F^{[1]},z)t+F_{2}(F^{[1]},z)t^{2}$

and the ones of the second bundle by the polynomial

$1+F_{1}(F^{[2]},z)t+F_{2}(F^{[2]},z)t^{2}$

The product of the two polynomials is the well-defined series
$C_{t}(F):=F_{t}(F_{t}(x,y),z)$. Explicitly, we have

$C_{i}(F)(x,y,z)=\begin{cases}F_{1}(F^{[1]},z)+F_{1}(F^{[2]},z)&\text{ if
$i=1$,}\\\
F_{2}(F^{[1]},z)+F_{2}(F^{[2]},z)+F_{1}(F^{[1]},z)F_{1}(F^{[2]},z)&\text{ if
$i=2$,}\\\
F_{1}(F^{[1]},z)F_{2}(F^{[2]},z)+F_{2}(F^{[1]},z)F_{1}(F^{[2]},z)&\text{ if
$i=3$,}\\\ F_{2}(F^{[1]},z)F_{2}(F^{[2]},z)&\text{ if $i=4$.}\end{cases}$

######  Lemma 4.0.1.

Let $F_{t}$ be a 2-valued formal group law of the first type over a ring $R$.
Then, the series $C_{t}(F)(x,y,z):=F_{t}(F_{t}(x,y),z)$ is a FTL over the ring
$R$.

###### Proof.

We start with the symmetry axiom. As $F_{t}(x,y)=F_{t}(y,x)$, we obviously
have $C_{t}(F)(x,y,z)=C_{t}(F)(y,x,z)$. To prove that
$C_{t}(F)(x,y,z)=C_{t}(F)(x,z,y)$, we use the associativity and commutativity
axioms for $F$

$F_{t}(F_{t}(x,y),z)=F_{t}(x,F_{t}(y,z))=F_{t}(x,F_{t}(z,y))=F_{t}(F_{t}(x,z),y).$

For the neutral element, we note that $\epsilon$ acts as $-1$ on $R$, which
implies that we have to prove that

$C_{t}(F)(x,0,0)=1+4xt+6x^{2}t^{2}+4x^{3}t^{3}+x^{4}t^{4}.$

This is a direct computation using $F_{t}(x,0)=1+2xt+x^{2}t^{2}$. For the weak
neutral element axiom, we observe that $F_{2}(x,x)=0$ implies that
$F_{t}(x,x)=1+F_{1}(x,x)t$ so that

$C_{t}(F)(x,x,0)=F_{t}(F_{t}(x,x),0)=F_{t}(F_{1}(x,x),0)=(1+F_{1}(x,x)t)^{2}$

yielding obviously $C_{3}(F)(x,x,0)=C_{4}(F)(x,x,0)=0$. Finally, we have

$\displaystyle C_{t}(F)(C_{t}(F)(x,y,z),v,w)$ $\displaystyle=$ $\displaystyle
C_{t}(F)(F_{t}(F_{t}(x,y),z),v,w)$ $\displaystyle=$ $\displaystyle
F_{t}(F_{t}(F_{t}(F_{t}(x,y),z),v),w)$ $\displaystyle=$ $\displaystyle
F_{t}(F_{t}(F_{t}(x,F_{t}(y,z)),v),w)$ $\displaystyle=$ $\displaystyle
F_{t}(F_{t}(x,F_{t}(F_{t}(y,z),v)),w)$ $\displaystyle=$ $\displaystyle
C_{t}(F)(x,C_{t}(F)(y,z,v),w).$

∎

######  Example 4.0.2.

Consider the two-valued formal group law

$F_{t}(x,y)=1+2(x+y)t+(x-y)^{2}t^{2}.$

which by Remark 2.3.13 is the image of the additive formal group law
(associated to Chow groups) under $N$. Write as usual $F^{+}$ and $F^{-}$ for
the roots of $F_{t}$. We have

$F_{t}(F_{t}(x,y),z)=F_{t}(F^{+},z)F_{t}(F^{-},z)$

and a direct computation yields

$C(F)_{i}=\begin{cases}4\sigma(x)&\text{ if $i=1$.}\\\
6\sigma(x^{2})+4\sigma(xy)&\text{ if $i=2$.}\\\
4\sigma(x^{3})-4\sigma(x^{2}y)+40xyz&\text{ if $i=3$.}\\\
\sigma(x^{4})-4\sigma(x^{3}y)+6\sigma(x^{2}y^{2})+4\sigma(x^{2}yz)&\text{ if
$i=4$.}\end{cases}$

This is the FTL of Chow groups. Note that setting $\epsilon=-1$ in
$\mathbf{H}_{\mathrm{MW}}$ (the first example of subsection 3.2) yields the
same result, as expected.

If we work more generally with unoriented 2-valued formal group laws over a
$\mathbb{Z}_{\epsilon}$-algebra as in Remark 2.2.4, applying $C$ still is
associative and symmetric, arguing as in the previous lemma. Also,
$\epsilon$-linearity of $F_{t}$ implies $\epsilon$-linearity of $C(F_{t})$,
which in turn implies the $\epsilon$-refinement on the axiom about the neutral
element arguing as in [DF21, Proposition 2.3.4(1)]. However, such laws might
not satisfy the weak neutral element axiom, as the next example shows.

######  Proposition 4.0.3.

Consider the $\mathbb{Z}_{\epsilon}$-variant $F^{univ,0}_{\epsilon}$ over
$\mathcal{B}_{\epsilon}^{0}$ of the degree zero universal two-valued formal
group law as studied after Proposition 2.2.9. Then
$C(\mathcal{B}_{\epsilon}^{0},F^{univ,0}_{\epsilon})$ is given by

$C(\theta_{ue})_{i}=\begin{cases}2(1-\epsilon)\sigma(x),&i=1,\\\
2(1-2\epsilon)\sigma(x^{2})+(a+4)(1-\epsilon)\sigma(xy),&i=2,\\\
2(1-\epsilon)\sigma(x^{3})+2(1-\epsilon)(1+a)\sigma(x^{2}y)+2(1-\epsilon)(a^{2}-a+4)\sigma(xyz),&i=3,\\\
\sigma(x^{4})+2a\sigma(x^{3}y)+(2+a^{2})\sigma(x^{2}y^{2})+2(a^{2}+a)\sigma(x^{2}yz),&i=4.\end{cases}$

###### Proof.

Straightforward computation. Use the relation $(1+\epsilon)a=0$ for $i=2,3,4$
as well as the cubical relation for $a$ in $F_{4}$. ∎

Looking at the oriented part, i.e. setting $\epsilon=-1$, we recover the
previous oriented example. Setting $a=-(1-\epsilon)$, we obtain an unoriented
formal ternary law which is slightly different from the one for MW-motivic
cohomology computed by the first two authors in [DF21] and recalled in Example
3.2 above. Namely, the coefficient of $\sigma(xyz)$ in $F_{3}$ specializes to
$20(1-\epsilon)$ instead of $8(2-3\epsilon)$, and the one of
$\sigma(x^{2}y^{2})$ in $F_{4}$ to $2(2-\epsilon)$ instead of
$2(1-2\epsilon)$. So they differ by multiples of $1+\epsilon$. In particular,
the law of Proposition 4.0.3 does not specialise to the formal ternary law for
$\mathbf{H}_{\mathrm{MW}}$. Moreover, we have
$F_{4}(x,x,0)=2(1+\epsilon)x^{4}$, hence the weak neutral element axiom does
not hold!

######  Proposition 4.0.4.

Let $R$ be a ring or more generally a $\mathbb{Z}_{\epsilon}$-algebra, and
consider the notations of the previous lemma. Then the assignment

$C_{R}:(2-\mathcal{FGL})_{I}(R)\rightarrow\mathcal{FTL},F(x,y)\mapsto
C_{t}(F)(x,y,z)$

extends into a functor which is the identity on morphisms.

###### Proof.

Let $F$ and $G$ be two valued formal groups and let $\Theta$ be a composable
power series such that

$\Theta_{t}(F_{t}(x,y))=G_{t}(\Theta(x),\Theta(y)).$

We then have

$\displaystyle C_{t}(G)(\Theta(x),\Theta(y),\Theta(z))$ $\displaystyle=$
$\displaystyle G_{t}(G_{t}(\Theta(x),\Theta(y)),\Theta(z))$ $\displaystyle=$
$\displaystyle G_{t}(\Theta_{t}(F_{t}(x,y)),\Theta(z))$ $\displaystyle=$
$\displaystyle G_{t}(\Theta(F^{+}),\Theta(z))G_{t}(\Theta(F^{-}),\Theta(z))$
$\displaystyle=$ $\displaystyle\Theta_{t}(F(F^{+},z))\Theta_{t}(F(F^{-},z))$
$\displaystyle=$ $\displaystyle\Theta_{t}(F_{t}(x,y),z).$

∎

To conclude this section, we compute the composite functor

$W:\mathcal{FGL}\xrightarrow{N}(2-\mathcal{FGL})_{I}\xrightarrow{C}\mathcal{FTL}$

by making the roots of $W(F)$ explicit. Let $R$ be a ring and let $F(x,y)$ be
a commutative formal group law. In view of Remark 2.3.10, we may choose the
roots of $N_{t}(F)$ to be
$F^{[1]}=-F(x,y)F(\bar{x},\bar{y})=-F(x,y)\overline{F(x,y)}$ and
$F^{[2]}=-F(\bar{x},y)F(x,\bar{y})=-F(\bar{x},y)\overline{F(\bar{x},y)}$ in
$R[[x,y]]$.

######  Lemma 4.0.5.

Under the finite ring extension $R[[-x\bar{x},-y\bar{y},-z\bar{z}]]\subset
R[[x,y,z]]$, the roots of $W(F)$ are given by

$W(F)^{[i]}=\begin{cases}-F(F(x,y),z)F(F(\bar{x},\bar{y}),\bar{z})&\text{ if
$i=1$.}\\\ -F(F(\bar{x},y),z)F(F(x,\bar{y}),\bar{z})&\text{ if $i=2$.}\\\
-F(F(x,\bar{y}),z)F(F(\bar{x},y),\bar{z})&\text{ if $i=3$.}\\\
-F(F(\bar{x},\bar{y}),z)F(F(x,y),\bar{z})&\text{ if $i=4$.}\end{cases}$

###### Proof.

By construction, we have

$\displaystyle N_{1}(F)(F^{[1]},-z\bar{z})$ $\displaystyle=$ $\displaystyle
N_{1}(F)(-F(x,y)\overline{F(x,y)},-z\bar{z})$ $\displaystyle=$
$\displaystyle-F(F(x,y),z)F(F(\bar{x},\bar{y}),\bar{z})-F(F(\bar{x},\bar{y}),z)F(F(x,y),\bar{z})$

and

$N_{2}(F)(F^{[1]},-z\bar{z})=F(F(x,y),z)F(F(\bar{x},\bar{y}),\bar{z})F(F(\bar{x},\bar{y}),z)F(F(x,y),\bar{z})$

showing that $-F(F(x,y),z)F(F(\bar{x},\bar{y}),\bar{z})$ and
$-F(F(\bar{x},\bar{y}),z)F(F(x,y),\bar{z})$ are roots. We may compute
$N_{1}(F)(F^{[2]},-z\bar{z})$ and $N_{2}(F)(F^{[2]},-z\bar{z})$ in a similar
fashion and then deduce that the roots of

$(1+N_{1}(F)(F^{[1]},-z\bar{z})t+N_{2}(F)(F^{[1]},-z\bar{z})t^{2})(1+N_{1}(F)(F^{[2]},-z\bar{z})t+N_{2}(F)(F^{[2]},-z\bar{z})t^{2})$

are as claimed. ∎

##  5\. Example: symplectic oriented ring spectra

The first two authors have shown that Borel classes for $\mathrm{SL}$-oriented
motivic spectra yield formal ternary laws, see [DF21, section 2]. We now
explain how this result may be extended to $\mathrm{Sp}$-oriented theories.

Let $\mathbb{E}$ be a motivic ring spectrum, that is a (weakly) commutative
monoid in the stable $\mathbb{A}^{1}$-homotopy category over $S$, where is is
a noetherian scheme of finite Krull dimension. The motivic ring spectrum
$\mathbf{MSp}$ has been introduced and studied by Panin and Walter [PW18].
They defined and investigated symplectic oriented cohomology theories and ring
spectra (or $\mathrm{Sp}$-oriented for short). The key result is [PW18,
Theorem 1.1], see also [Ana20, section 3] and the summary provided in [DF21,
section 2.1].

######  Remark 5.0.1.

The definition of the classical topological spectrum $MSp$ is well-known, see
e.g. Switzer’s book [Swi75]. The graded ring
$\pi_{*}(MSp)\otimes\mathbb{Z}[1/2]$ is isomorphic to the (known) polynomial
algebra $MSO^{*}\otimes\mathbb{Z}[1/2]$ by work of Novikov, but integrally
$\pi_{*}(MSp)$ is known only in degrees $\leq 100$ in which the only torsion
is $2$-torsion. The first element of $4$-torsion appears in degree $103$. See
e.g. [Sto67], [Swi75, Remark 20.36], [Ray72, Theorem 5.25] and [Koc93, Section
8]. Over a general base field the bigraded motivic homotopy groups
$\pi_{**}(\mathbf{MSp})$ are not known, either, not even after tensoring with
$\mathbb{Z}[1/2]$, but $\pi_{**}(\mathbf{MSp})[\eta^{-1}]$ is known by the
recent [BH20, Corollary 1.3(2)] over fields, later extended to Dedekind
domains by Bachmann [Bac21]. One may show that $\mathbf{MSp}$ maps to
$MSp^{top}$ under complex realization. We could not find a proof for this in
the literature. One should use that $Sp_{2n}(\mathbb{C})^{an}$ has $Sp(n)$ as
maximal compact subgroup, that Thom spaces realize to Thom spaces and that
$Sp_{2n}$ is “special” in the sense of Serre and others. The following results
also hold for symplectic oriented topological commutative ring spectra, for
which we may set $\epsilon=-1$ everywhere.

The following result of the third author is crucial. Its proof will be
published elsewhere.

######  Theorem 5.0.2 (Fasel).

Let $R$ be a ring, let $U$ be the universal bundle on $\mathrm{BSp}_{2,R}$ and
let $\lambda\in R^{\times}$ be an invertible element. Let further
$\mathrm{th}(U,\varphi):\mathrm{Th}(U)\to\mathbf{MSp}(2)[4]$ be the canonical
Thom class, where $\varphi:U\to U^{\vee}$ is the canonical symplectic form.
Then

$\mathrm{th}(U,\langle\lambda\rangle\varphi)=\langle\lambda\rangle\mathrm{th}(U,\varphi).$

######  Corollary 5.0.3.

For any $\mathrm{Sp}$-oriented motivic ring spectrum $\mathbb{E}$, the Borel
classes yield a formal ternary law as in Definition 3.1.5.

###### Proof.

Commutativity and associativity are obvious. The above theorem for
$\lambda=-1$ implies $\epsilon$-linearity for $\mathbf{MSp}$, hence by [PW18,
Theorem 1.1] for all $\mathrm{Sp}$-oriented ring spectra. Now the remaining
two axioms follow as in [DF21, Proposition 2.3.4]. ∎

##  Appendix A Sage implementation of FTL computations

We describe here the algorithm used to compute a standard (Groebner) basis of
the ideal $\mathcal{R}_{FTL}$ (see Proposition 3.1.9) of and its
implementation using the Sagemath [The21] software together with the
libsingular [DGPS21] back-end for multivariate polynomial computations.

###  A.1. Main problem characteristics and design principle

The inputs of the algorithm are a base numerical ring $R$ (among
$\mathbb{Z},\mathbb{Q},\mathbb{Z}/n\mathbb{Z}$), and a degree range
$(d_{min},d_{max})\in\mathbb{Z}^{2},-4\leq d_{min}\leq d_{max}$. With those
restrictions, the unknown FTL coefficients span a finite set
$a=\\{a^{l}_{ijk},d_{min}\leq deg(a^{l}_{ijk})\leq d_{max},1,\leq l\leq 4\\}$.
The five axioms stated in Definition 3.1.5 induce relations between the FTL
coefficients in the polynomial ring $R[\epsilon,a,\alpha]$, where $\alpha$
stands for one or more optional formal variables. For technical reasons this
is more convenient than to work explicitly in $R_{\epsilon}[a]$. Additional
constraints can be appended to the relation set in order to obtain the desired
quotient ring, (for instance $\epsilon^{2}-1$ for $R_{\epsilon}[a]$ or
$2\alpha-1=0$ for $R[\epsilon,a][\frac{1}{2}]$). The overall purpose of the
algorithm is first to compute the finite set $S_{FTL}=\cup
S_{axiom}=\\{r_{p},1\leq p\leq N,r_{p}\in R[\epsilon,a]\\}$ of relations
stemming from the five FTL axioms and then compute a standard basis $g_{FTL}$
of the ideal generated by $S_{FTL}$. Noting $Gb$ the algorithm used for
Groebner basis computation, a direct resolution scheme would simply read
$g_{FTL}\leftarrow Gb(\mathcal{I}(S_{FTL}))$.

Unlike symmetry, (weak) neutral element, and $\epsilon$-linearity axioms, the
set of relations induced by the associativity axiom tends to be quite large
(hundreds to hundreds of thousands), with total degrees up to a few tens even
for relatively small values ($1$ or $2$) of $d_{max}$. Even using a sparse
storage scheme, the memory footprint of the full relations set can easily grow
to tens of gigabytes. A second issue is the Groebner basis computation, which
can generate numerous intermediate $S$-polynomials with huge coefficients.
Either of memory and processing time costs can easily saturate available
resources, even on ”fat” computing nodes with hundreds of gigabytes of memory.

###  A.2. Description of the implemented scheme

The resolution scheme presented hereafter is designed to mitigate those
technical difficulties and allow to solve the problem for a limited range of
degrees (up to $d_{max}=2$ for some special cases). Two optimization
strategies are used to limit resources consumption. The first is to express
the problem in the most compact manner to avoid representation overhead costs.
The second is divide-and-conquer, i.e. split the whole problem into a sequence
of smaller ones.

A first representation optimization stems from the symmetry axiom, which is
the easiest to deal with. The symmetry relation set is fully generated by
simple relations of the form $a^{l}_{ijk}-a^{l}_{\sigma(ijk)}=0$, with
$\sigma\in S_{3}$. For any fixed $l$, FTL coefficients with the same $l$ value
and whose $(i^{\prime},j^{\prime},k^{\prime})$ indexes share a common orbit
under $S_{3}$ can be replaced by a single coefficient with indices
$(i_{0},j_{0},k_{0})$, which amounts to simply removing them and replacing the
monomial $X^{i}_{0}Y^{j}_{0}Z^{k}_{0}$ by the corresponding monomial symmetric
polynomial in the expression of $F_{t}$. We note $\overline{a}$ this reduced
set of FTL coefficients. Working with relations in $R[\epsilon,\overline{a}]$
with this form for $F$, the symmetry axiom is structurally enforced, with a
significant reduction in the number of variables.
Problem splitting is obtained by partitioning $S_{FTL}$ into $K$ small subsets
$S_{FTL}=\cup_{p\leq K}s_{p}$. In order to avoid storing the full set of
relations, the scheme operates in an iterative manner on an ideal sequence
$\mathcal{I}_{p}$, defined by its generating set $g_{p}$, with initial value
$g_{0}=\emptyset$. A each step $p\leq K$ of the algorithm, the small subset
$s_{p}$ of the relations is generated on the fly. In the current
implementation, $s_{p}$ is the full set of relations for all axioms except
associativity, and a single relation for each of the associativity axiom
relations.
A reduced Groebner basis of $\mathcal{I}(g_{p}\cup s_{p})$ is then computed
and used to replace $g$. A single iteration of the scheme at step $p$ reads
thus $g_{p+1}\leftarrow Gb(\mathcal{I}(g_{p}\cup s_{p}))$. Obviously, this
scheme is more efficient memory-wise than the naive scheme provided that the
intermediate generators sets $g$ size and complexity stay lower than the one
of the full set of relations. While this is true for the final generating set
$g_{K}$ ($S_{FTL}$ has a lot of redundancy), this is not guaranteed for the
intermediate ones, whose complexity depend both on the way $S_{FTL}$ is
partitioned, and the underlying Groebner basis algorithm. From a computation
cost point of view, the scheme is obviously not optimally efficient : indeed
the GB algorithm is only providing partial information about the ideal at each
step, preventing it from operating optimally and leading to quite a lot of
redundant computations. In practice, in many cases the intermediate $g_{p}$
complexity stays rather low, is quite often invariant $g_{p+1}=g_{p}$ due to
the large redundancy of associativity relations.

Based on this observation, a slight variation of the iteration scheme has also
been implemented : after generation of $s_{p}$, $s_{p}$ is reduced by
$\mathcal{I}$ before attempting computation of the Groebner basis. The scheme
thus reads $g_{p+1}\leftarrow Gb(\mathcal{I}(g_{p}\cup
s_{p}/\mathcal{I}(g_{p})))$. In cases when $g_{p+1}=g_{p}$, $s_{p}$ actually
reduces to $0$ so that there is no need for an additional Groebner basis
computation. Though this pre-reduction step is formally redundant, it appears
empirically to ease the work of the Groebner algorithm when working over
$\mathbb{Z}$ and limit coefficient swelling.
A the end of each step, a second use of the representation sparsity
optimization is then performed. Once the new set of generators $g_{p+1}$ is
computed, trivial relations of the form $r_{u}=a^{l}_{ijk}-c=0,c\in
R[\epsilon]$ are searched for. Whenever such a relation is found, the
corresponding FTL coefficient is replaced by the unique solution $c$, and
$a^{l}_{ijk}$ removed from the pool of active variables of the working ring.
While this is formally completely equivalent to keeping $r_{u}$ in the
generating set $g_{p}$, it has a non-negligible impact on memory consumption
and computation complexity. A first guaranteed reduction is due to the fact
that the number of active variables drives the size of all multiindexes
representing monomial exponents in the polynomial representation of elements
of $R[\epsilon,\overline{a}]$). This implies memory savings proportional to
the number of variables that can be removed that way, at the risk of a slight
increase in the complexity of the coefficients. Second, in practice, the
solutions $c$ are usually fairly simple and often zero, leading in a drastic
reduction of the complexity of generating relations in the restricted
polynomial ring. Most such substitutions stem from the neutral, weak neutral
element and linearity axioms, which are treated first in the resolution
scheme, and occur early in the computation.

###  A.3. Short description of the code repository

The the core algorithm and data structures are regrouped in a single Python
module (ftl_comp.py) meant to be imported and used from Sage. Basic usage
examples are provided in the form of Jupyter notebooks for light computations
and post-computation work using saved results, and batch python scripts for
non-interactive heavier computations. A selection of computations results
saved in json format is also provided.

###  A.4. Sample results 1 : Computation of $\mathcal{W}^{\leq
0}[\frac{1}{2}]$.

The following results were obtained in the polynomial ring
$\mathbb{Z}[\epsilon,a]$, with the additional relations $2\alpha-1=0$ and
$\epsilon^{2}-1=0$, which is equivalent to working in
$\mathbb{Z}_{\epsilon}[\frac{1}{2}][a]$. For this range of degrees ($-4$ to
$0$) there are $24$ FTL coefficients. For $23$ of those, fixed values
solutions were found,

$\left\\{\begin{array}[]{l}a^{4}_{000}=a^{3}_{000}=a^{4}_{100}=a^{2}_{000}=a^{3}_{100}=a^{4}_{200}=a^{1}_{000}=0\\\
a^{2}_{100}=a^{3}_{200}=a^{4}_{300}=a^{4}_{110}=a^{4}_{210}=a^{3}_{110}=a^{4}_{111}=0\\\
a^{4}_{400}=1\\\ a^{2}_{200}=a^{4}_{220}=-4\epsilon+2\\\
a^{3}_{210}=a^{4}_{310}=2\epsilon-2\\\
a^{1}_{100}=a^{3}_{300}=a^{2}_{110}=a^{4}_{211}=-2\epsilon+2\end{array}\right.,$

and $a^{3}_{111}$ is the ony non-fixed FTL coefficient, with the remaining
relations

$\left\\{\begin{array}[]{l@{\ = \ }c}(a^{3}_{111})^{2}+768\epsilon-832&0\\\
(a^{3}_{111}-40)(\epsilon-1)&0\\\ \end{array}\right..$

With the same parameters, removing the $2$-invertibility assumption yields the
solution for $\mathcal{W}^{\leq 0}$. In that case only $15$ FTL coefficients
are fixed, $9$ remain free, with $134$ remaining relations. They are too
numerous to list here, but can be recomputed or reloaded from the saved runs
provided in the code repository:

https://plmlab.math.cnrs.fr/coulette/ftlcomp.

###  A.5. Example 2 : Computation of $\mathcal{W}^{\leq
2}[\frac{1}{2}]/(\epsilon-1)$

While the computation of $\mathcal{W}^{\leq 2}$ and $\mathcal{W}^{\leq
2}[\frac{1}{2}]$ have proven untractable so far, a fast and quite simple
solution can be obtained when $\epsilon=1$. In that case $52$ of the $57$
coefficients have fixed values

$\left\\{\begin{array}[]{l}a^{4}_{000}=a^{3}_{000}=a^{4}_{100}=a^{2}_{000}=a^{3}_{100}=a^{4}_{200}=a^{1}_{000}=0\\\
a^{2}_{100}=a^{3}_{200}=a^{4}_{300}=a^{1}_{100}=a^{3}_{300}=a^{1}_{200}=a^{2}_{300}=0\\\
a^{3}_{400}=a^{4}_{500}=a^{1}_{300}=a^{2}_{400}=a^{3}_{500}=a^{4}_{600}=a^{4}_{110}=0\\\
a^{4}_{210}=a^{3}_{110}=a^{4}_{111}=a^{2}_{110}=a^{3}_{210}=a^{4}_{211}=a^{4}_{310}=0\\\
a^{1}_{110}=a^{2}_{111}=a^{2}_{210}=a^{3}_{211}=a^{3}_{220}=a^{3}_{310}=a^{4}_{221}=0\\\
a^{4}_{311}=a^{4}_{320}=a^{4}_{410}=a^{1}_{210}=a^{2}_{211}=a^{2}_{310}=a^{3}_{221}=0\\\
a^{3}_{320}=a^{3}_{410}=a^{4}_{321}=a^{4}_{330}=a^{4}_{411}=a^{4}_{420}=a^{4}_{510}=0\\\
a^{4}_{400}=1\\\ a^{2}_{200}=a^{4}_{220}=-2\end{array}\right.,$

with the five coefficients
$a^{3}_{111},a^{1}_{111},a^{2}_{220},a^{3}_{311},a^{4}_{222}$ and the
remaining constraints

$\left\\{\begin{array}[]{l@{ = \quad} l}(a^{3}_{111})^{2}-64&0\\\
a^{3}_{111}a^{3}_{311}+8a^{4}_{222}&0\\\
(a^{3}_{311})^{2}-(a^{4}_{222})^{2}&0\\\
a^{3}_{111}a^{4}_{222}+8a^{3}_{311}&0\\\ a^{1}_{111}-a^{3}_{311}&0\\\
a^{2}_{220}-a^{4}_{222}&0\\\ \end{array}\right.$

## References

  * [AL75] J. F. Adams and A. Liulevicius, _Buhstaber’s work on two-valued formal groups_ , Topology 14 (1975), no. 4, 291–296. MR 391134
  * [Ana17] A. Ananyevskiy, _Stable operations and cooperations in derived Witt theory with rational coefficients_ , Ann. K-Theory 2 (2017), no. 4, 517–560. MR 3681106
  * [Ana20] by same author, _SL-oriented cohomology theories_ , Motivic homotopy theory and refined enumerative geometry, Contemp. Math., vol. 745, Amer. Math. Soc., Providence, RI, [2020] ©2020, pp. 1–19. MR 4071210
  * [Bac21] T. Bachmann, _$\eta$ -periodic motivic stable homotopy theory over Dedkind domains_, arXiv:2006.02086, 2021.
  * [BCD+20] T. Bachmann, B. Calmès, F. Déglise, J. Fasel, and P.A. Østvæ r, _Milnor-Witt Motives_ , arXiv: 2004.06634, 2020.
  * [BH20] T. Bachmann and M. J. Hopkins, _$\eta$ -periodic motivic stable homotopy theory over fields_, arXiv: 2005.06778, 2020.
  * [BM00] Jean Barge and Fabien Morel, _Groupe de Chow des cycles orientés et classe d’Euler des fibrés vectoriels_ , C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 4, 287–290. MR MR1753295 (2000m:14004)
  * [BN71] V. M. Buhštaber and S. P. Novikov, _Formal groups, power systems and Adams operators_ , Mat. Sb. (N.S.) 84(126) (1971), 81–118, English translation: Math. USSR-Sb. 13 (1971), 80–116. MR 0291159
  * [Buh75] V. M. Buhštaber, _Two-valued formal groups. Algebraic theory and applications to cobordism. I_ , Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 5, 1044–1064, 1219. MR 0469927
  * [Buh76] by same author, _Two-valued formal groups. Algebraic theory and applications to cobordism. II_ , Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 2, 289–325, 469. MR 0480545
  * [Buh78] by same author, _Topological applications of the theory of two-valued formal groups_ , Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 1, 130–184, 215\. MR 0494160
  * [CPZ13] B Calmès, V. Petrov, and K. Zainoulline, _Invariants, torsion indices and oriented cohomology of complete flags_ , Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 3, 405–448 (2013). MR 3099981
  * [CZZ15] B. Calmès, K. Zainoulline, and C. Zhong, _Equivariant oriented cohomology of flag varieties_ , Doc. Math. (2015), no. Extra vol.: Alexander S. Merkurjev’s sixtieth birthday, 113–144. MR 3404378
  * [Dég18a] F. Déglise, _Bivariant theories in motivic stable homotopy_ , Doc. Math. 23 (2018), 997–1076.
  * [Dég18b] by same author, _Orientation theory in arithmetic geometry_ , $K$-Theory—Proceedings of the International Colloquium, Mumbai, 2016, Hindustan Book Agency, New Delhi, 2018, pp. 239–347.
  * [DF21] F. Déglise and J. Fasel, _The Borel character_ , Journal of the IMJ (2021), Firstview.
  * [DGPS21] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, _Singular 4-2-0.p3 — A computer algebra system for polynomial computations_, http://www.singular.uni-kl.de, 2021.
  * [Fas20] J. Fasel, _Lectures on Chow-Witt groups_ , Motivic homotopy theory and refined enumerative geometry, Contemp. Math., vol. 745, Amer. Math. Soc., [Providence], RI, [2020] ©2020, pp. 83–121.
  * [FH20] J. Fasel and O. Haution, _The stable Adams operations on Hermitian K-theory_ , 2020.
  * [Haz78] M. Hazewinkel, _Formal groups and applications_ , Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506881
  * [HK11] J. Hornbostel and V. Kiritchenko, _Schubert calculus for algebraic cobordism_ , J. Reine Angew. Math. 656 (2011), 59–85. MR 2818856
  * [Hor05] J. Hornbostel, _$A^{1}$ -representability of Hermitian $K$-theory and Witt groups_, Topology 44 (2005), no. 3, 661–687.
  * [Hoy15] M. Hoyois, _From algebraic cobordism to motivic cohomology_ , J. Reine Angew. Math. 702 (2015), 173–226.
  * [Koc93] S. Kochman, _Symplectic cobordism and the computation of stable stems_ , Mem. Amer. Math. Soc. 104 (1993), no. 496, x+88.
  * [Las63] R. Lashof, _Poincaré duality and cobordism_ , Trans. Amer. Math. Soc. 109 (1963), 257–277.
  * [Laz55] M. Lazard, _Sur les groupes de Lie formels à un paramètre_ , Bull. Soc. Math. France 83 (1955), 251–274.
  * [LM07] M. Levine and F. Morel, _Algebraic cobordism_ , Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2286826
  * [Mer21] A. S. Merkurjev, _Additive operations between connective $K$-theory and Chow theory_, Adv. Math. 388 (2021), Paper No. 107911, 25.
  * [MH73] J. W. Milnor and D. H. Husemoller, _Symmetric bilinear forms_ , vol. 73, Springer-Verlag, Berlin, 1973.
  * [NSØ09] N. Naumann, M. Spitzweck, and P. A. Østvær, _Motivic Landweber exactness_ , Doc. Math. 14 (2009), 551–593.
  * [Pan04] I. Panin, _Riemann-Roch theorems for oriented cohomology_ , Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., Dordrecht, 2004, pp. 261–333.
  * [PW10] I. Panin and C. Walter, _Quaternionic Grassmannians and Pontryagin classes in algebraic geometry_ , arXiv:1011.0649, 11 2010.
  * [PW18] by same author, _On the algebraic cobordism spectra $\mathbf{MSL}$ and $\mathbf{MSp}$_, arXiv: 1011.0651, 2018.
  * [PW19a] I. Panin and C. Walter, _On the motivic commutative ring spectrum $\mathbf{BO}$_, St. Petersbg. Math. J. 30 (2019), no. 6, 933–972, arXiv: 1011.0650.
  * [PW19b] I. Panin and C. Walter, _On the relation of symplectic algebraic cobordism to Hermitian $K$-theory_, Proc. Steklov Inst. Math. 307 (2019), 162–173.
  * [Qui69] D. Quillen, _On the formal group laws of unoriented and complex cobordism theory_ , Bull. Amer. Math. Soc. 75 (1969), 1293–1298.
  * [Rav86] D. C. Ravenel, _Complex cobordism and stable homotopy groups of spheres_ , Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR 860042
  * [Ray72] N. Ray, _The symplectic bordism ring_ , Proc. Cambridge Philos. Soc. 71 (1972), 271–282. MR 290384
  * [Sch17] Marco Schlichting, _Hermitian $K$-theory, derived equivalences and Karoubi’s fundamental theorem_, J. Pure Appl. Algebra 221 (2017), no. 7, 1729–1844, arXiv:1209.0848.
  * [Smi06] A. L. Smirnov, _The Riemann-Roch theorem for operations in the cohomology of algebraic varieties_ , Algebra i Analiz 18 (2006), no. 5, 210–236.
  * [Spi20] M. Spitzweck, _Algebraic cobordism in mixed characteristic_ , Homology Homotopy Appl. 22 (2020), no. 2, 91–103.
  * [ST15] M. Schlichting and G. S. Tripathi, _Geometric models for higher Grothendieck-Witt groups in $\mathbb{A}^{1}$-homotopy theory_, Math. Ann. 362 (2015), no. 3-4, 1143–1167.
  * [Sto67] R. E. Stong, _Some remarks on symplectic cobordism_ , Ann. of Math. (2) 86 (1967), 425–433. MR 219079
  * [Str] N. Strickland, _Formal groups_ , http://neil-strickland.staff.shef.ac.uk/courses/formalgroups/fg.pdf.
  * [Swi75] R. M. Switzer, _Algebraic topology—homotopy and homology_ , Springer-Verlag, New York-Heidelberg, 1975, Die Grundlehren der mathematischen Wissenschaften, Band 212. MR 0385836
  * [SY14] J. Schürmann and S. Yokura, _Motivic bivariant characteristic classes_ , Adv. Math. 250 (2014), 611–649.
  * [The21] The Sage Developers, _Sagemath, the Sage Mathematics Software System (Version 9.4)_ , 2021, https://www.sagemath.org.
  * [Vis19] Alexander Vishik, _Stable and unstable operations in algebraic cobordism_ , Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), no. 3, 561–630.