The Azimuth Project
λ-ring (Rev #1)

Definition

A commutative ring KK is called a λ\lambda-ring if we are given a family of set operations λ k:KK\lambda^k \colon K \to K for k0k \geq 0 such that for all x,yKx,y \in K

  1. λ 0(x)=1\lambda^0(x) = 1 and λ 1(x)=x\lambda^1(x) = x
  2. $\lambda^k(x+y) = \sum_{i=0}^k \lambda^i(x) \lambda^{k-i}(y).

Note: there is a group homomorphism λ t\lambda_t from the additive group of KK to the multiplicative group W(K)=1+tK[[t]]W(K) = 1+tK[[t]]

A special λ\lambda-ring is a λ\lambda-ring KK such that the group homomorphism λ t:KW(K)\lambda_t \colon K \to W(K) is a λ\lambda-ring homomorphism.

References

  • Charles A. Weibel, The KK-book: An introduction to algebraic KK-theory, American Mathematical Society, Graduate Studies in Mathematics Volume 145, 2013.