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
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
We therefore have . Set
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 ringTex syntax erroris 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 overTex syntax errorexists.
Then
- The ring
Tex syntax error
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 .
Tex syntax error
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
ringTex syntax erroris 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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