# The Azimuth Project λ-ring

## Definition

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

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

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

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

A special $\lambda$-ring is a $\lambda$-ring $K$ such that the group homomorphism $\lambda_t \colon K \to W(K)$ is a $\lambda$-ring homomorphism.

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