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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# $F(u,v)=F(v,u)$. | # $F(u,v)=F(v,u)$. | ||
− | The original example of a formal group law over a field $\ | + | The original example of a formal group law over a field $\mathbf k$ is |
provided by the expansion near the unit of the multiplication map | provided by the expansion near the unit of the multiplication map | ||
− | $G\times G\to G$ in a one-dimensional algebraic group over $\ | + | $G\times G\to G$ in a one-dimensional algebraic group over $\mathbf k$. |
This also explains the terminology. | This also explains the terminology. | ||
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{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
− | Every formal group law $F$ is linearisable over $R\otimes\Q$. | + | Every formal group law $F$ is linearisable over $R\otimes\mathbb Q$. |
{{endthm}} | {{endthm}} | ||
''Proof.'' Consider the series $\omega(u)=\frac{\partial F(u,w)}{\partial | ''Proof.'' Consider the series $\omega(u)=\frac{\partial F(u,w)}{\partial | ||
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$f(t)\in R\otimes\mathbb Q[[t]]$ is called the <i>exponential</i> of | $f(t)\in R\otimes\mathbb Q[[t]]$ is called the <i>exponential</i> of | ||
the formal group law, so that we have $F(u,v)=f(g(u)+g(v))$ | the formal group law, so that we have $F(u,v)=f(g(u)+g(v))$ | ||
− | over $R\otimes\Q$. If $R$ does not have torsion ( | + | over $R\otimes\mathbb Q$. If $R$ does not have torsion (i.e. $R\to |
− | R\otimes\mathbb Q$ is monic), the latter formula shows that a | + | R\otimes\mathbb Q$ is monic), the latter formula shows that a formal group law (as a series with coefficients in $R$) is fully |
− | formal group law (as a series with coefficients in $R$) is fully | + | determined by its logarithm (which is a series with coefficients in $R\otimes\mathbb Q$). |
− | determined by its logarithm (which is a series with coefficients | + | |
− | in $R\otimes\mathbb Q$). | + | |
Let $F=\sum_{k,l}a_{kl}u^kv^l$ be a formal group law over a ring | Let $F=\sum_{k,l}a_{kl}u^kv^l$ be a formal group law over a ring | ||
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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}} | ||
− | <i>Proof</i> | + | <i>Proof.</i> To prove the first statement, denote by $A'$ the subring in $A$ |
− | To prove the first statement, denote by $A'$ the subring in $A$ | + | |
generated by the coefficients of $F_U$. Then there is a | generated by the coefficients of $F_U$. Then there is a | ||
monomorphism $i\colon A'\to A$ satisfying $i(F_U)=F_U$. On the | monomorphism $i\colon A'\to A$ satisfying $i(F_U)=F_U$. On the |
Revision as of 20:06, 1 April 2010
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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 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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