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λ-ring

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. λ k(x+y)= i=0 kλ i(x)λ ki(y)\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]].

Example. For every commutative ring KK, the abelian group W(K)=1+tK[[t]]W(K) = 1+tK[[t]] has the structure of a commutative ring, natural in KK. W(R)W(R) is called the ring of Witt vectors of KK. The multiplicative identity of the ring W(K)W(K) is 1t1-t. The multiplication is completely determined by naturality, formal factorization, and the formula (1rt)*f(t)=f(rt)(1-rt)\ast f(t) = f(rt).

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.