$\Lambda (A)$ as a $\lambda$-ring

The treatment in this subsection essentially follows https://encyclopediaofmath.org/wiki/Lambda-ring. (But a caution is advised: some signatures are different from the article cited above.)

DEFINITION 8.1  

$(A,\lambda_T:A \to \Lambda_T(A))$ is called a pre-$\lambda$-ring if

• $A$ is a commutative ring.
• $\lambda_T$: $A\to \Lambda_{(T)}(A)$ is an additive map.

Let us write $\lambda_T(f)$ for $f\in A$ as $\lambda_T(f)=(\sum_j \lambda^j(f) T^j)_W$. Then the additivity of $\lambda_T$ can be expressed as identities of $\{\lambda^j\}$ of the following form:

• $\lambda^0(f)=1\quad (\forall f \in A)$ .
• $\lambda^1(f)=f \quad (\forall f \in A)$.
• $\lambda^n(f+g)=\sum_{i+j=n} \lambda^(f)\lambda^j(g) \quad\forall f,g \in A$.

(Note that $\lambda^j$ is not a “$j$-th power of $\lambda$” in any sence.)

DEFINITION 8.2   Let $R=(R,\lambda_{(T)}^R:R\to \Lambda_T(R))$, $S=(S,\lambda_{(T)}^S:S\to \Lambda_T(S))$ be pre-lambda rings. Then a $\lambda$-ring homomorphism from $R$ to $S$ is a ring homomorphism $\varphi: R\to$ such that the following diagram commutes.

$$\xymatrix { R \ar[r]^{\lambda_{(T)}^R} \ar[d]_{\varphi}& \Lambda_... ...[d]^{\Lambda_{(T)}(\varphi)}\\ S \ar[r]_{\lambda_{(T)}^S} &\Lambda_{(T)}(S) }$$

The map $\Lambda_{(T)}(\varphi)$ which appears above is defined as follows:

$$\Lambda_{(T)}(\varphi)((\sum a_j T^j)_W)= (\sum \varphi(a_j) T^j)_W \quad (\{a_j\}_{j} \subset A)$$

(Yes, we regard $\Lambda_{(T)}(\bullet)$ as a functor.)

We also note, as a consequence of the definition, that we have the following formula for Teichmüller lifts:

$$\Lambda_{(T)}(\varphi)([a])= [\varphi(a)] \qquad( \forall a \in A)$$