Formal group laws and genera
Taras Panov (Talk | contribs) (→Elements of the theory of formal group laws) |
Taras Panov (Talk | contribs) (→Elements of the theory of formal group laws) |
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Then | Then | ||
# The ring $A$ is multiplicatively generated by the coefficients of the series $F_U$; | # The ring $A$ is multiplicatively generated by the coefficients of the series $F_U$; | ||
− | # The universal formal group law is unique: if | + | # The universal formal group law is unique: if $F'_U$ is another universal formal group law over $A'$, then there |
− | $F'_U$ is another universal formal group law over $A'$, then there | + | |
is an isomorphism $r\colon A\to A'$ such that $F'_U=r(F_U)$. | is an isomorphism $r\colon A\to A'$ such that $F'_U=r(F_U)$. | ||
{{endthm}} | {{endthm}} | ||
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A\to A'$ satisfying $r(F_U)=F_U$. It follows that $ir(F_U)=F_U$. | A\to A'$ satisfying $r(F_U)=F_U$. It follows that $ir(F_U)=F_U$. | ||
This implies that $ir=\mathrm{id}\colon A\to A$ by the uniqueness | This implies that $ir=\mathrm{id}\colon A\to A$ by the uniqueness | ||
− | requirement in the definition of | + | requirement in the definition of $F_U$. Thus $A'=A$. The second |
statement is proved similarly. | statement is proved similarly. | ||
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Contents |
1 Introduction
The theory of formal group laws, originally appeared in algebraic geometry, has been brought into the bordism theory in the pioneering work [Novikov1967]. The applications of formal group laws in algebraic topology are closely connected with the Hirzebruch genera [Hirzebruch1966], one of the most important class of invariants of bordism classes of manifolds.
2 Elements of the theory of formal group laws
Let be a commutative ring with unit.
A formal power series is called a
(commutative one-dimensional) formal group law over
if
it satisfies the following equations:
-
,
;
-
;
-
.
The original example of a formal group law over a field is
provided by the expansion near the unit of the multiplication map
in a one-dimensional algebraic group over
.
This also explains the terminology.
A formal group law over
is called linearisable if
there exists a coordinate change
such that
![\displaystyle g_F(F(u,v))=g_F(u)+g_F(v).](/images/math/7/9/c/79c476308fa074091cdd70a3bbc6a0dc.png)
Note that every formal group law over determines a formal
group law over
.
Theorem 2.1.
Every formal group law is linearisable over
.
Proof. Consider the series . Then
![\displaystyle \omega(F(u,v))=\frac{\partial F(F(u,v),w)}{\partial w}\Bigl|_{w=0}=\frac{\partial F(F(u,w),v)}{\partial F(u,w)}\cdot\frac{\partial F(u,w)}{\partial w}\Bigl|_{w=0}=\frac{\partial F(u,v)}{\partial u}\omega(u).](/images/math/7/5/0/7501bdbad3dcddbac402d33108d8b625.png)
We therefore have
. Set
![\displaystyle g(u)=\int_0^u\frac{dv}{\omega(v)};](/images/math/b/b/d/bbd2161c870c820ada46478e53ec003c.png)
then . This implies that
.
Since
and
, we get
. Thus,
.
A series satisfying the equation
is called
the logarithm of the formal group law
; the above Theorem
shows that a formal group law over
always has a logarithm. Its functional inverse series
is called the exponential of
the formal group law, so that we have
over
. If
does not have torsion (i.e.
is monic), the latter formula shows that a formal group law (as a series with coefficients in
) is fully
determined by its logarithm (which is a series with coefficients in
).
Let be a formal group law over a ring
and
a ring homomorphism. Denote by
the formal series
; then
is a formal group law over
.
A formal group law over a ring
is universal if
for any formal group law
over any ring
there exists a
unique homomorphism
such that
.
Proposition 2.2.
Assume that the universal formal group law over
exists.
Then
- The ring
is multiplicatively generated by the coefficients of the series
;
- The universal formal group law is unique: if
is another universal formal group law over
, then there
is an isomorphism such that
.
Proof. To prove the first statement, denote by the subring in
generated by the coefficients of
. Then there is a
monomorphism
satisfying
. On the
other hand, by universality there exists a homomorphism
satisfying
. It follows that
.
This implies that
by the uniqueness
requirement in the definition of
. Thus
. The second
statement is proved similarly.
Theorem 2.3 ([Lazard1955]).
The universal formal group law exists, and its coefficient
ring
is isomorphic to the polynomial ring
on an infinite number of generators.
3 Formal group law of geometric cobordisms
4 Hirzebruch genera
5 References
- [Hirzebruch1966] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, New York, 1966. MR0202713 (34 #2573) Zbl 0843.14009
- [Lazard1955] M. Lazard, Sur les groupes de Lie formels à un paramètre, Bull. Soc. Math. France 83 (1955), 251–274. MR0073925 (17,508e) Zbl 0068.25703
- [Novikov1967] S. P. Novikov, Methods of algebraic topology from the point of view of cobordism theory, Math. USSR, Izv. 1, (1967) 827–913. MR0221509 (36 #4561) Zbl 0176.52401
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