Formal group: Difference between revisions
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==Definition== | ==Definition== | ||
Let <math>A</math> be a commutative ring. A ''formal group'' in one parameter is a series <math>F\in A[[X,Y]]</math> such that | Let <math>A</math> be a commutative ring. A '''formal group''' in one parameter is a [[formal power series]] <math>F\in A[[X,Y]]</math> such that | ||
#<math>F(X,0)=F(0,X)=X</math> | #<math>F(X,0)=F(0,X)=X</math> | ||
#<math>F(X,Y)=F(Y,X)</math> | #<math>F(X,Y)=F(Y,X)</math> | ||
Latest revision as of 12:24, 9 December 2008
Definition
Let be a commutative ring. A formal group in one parameter is a formal power series such that
![{\displaystyle F\in A[[X,Y]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d560dc668cdffc4fe0237e4361d85f4753ff0319)
- in
- There is a series such that
![{\displaystyle A[[X,Y,Z]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/826428ffe80a3500e6a6878678d46f4be473f6d0)
![{\displaystyle \sigma \in A[[X]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27a91b3e231ba3663d072b852f00b11fc739c90a)
Examples
- The additive formal group:
- The multiplicative formal group: . In this case, .
