$\lambda$-ring

DEFINITION 8.3   A pre-$\lambda$-ring $A,\lambda_T: A \to \Lambda_{(T)}(A)$ is a $\lambda$-ring if $\lambda_T:A \to \Lambda_{(T)}(A)$ is a $\lambda$-homomorphism.

PROPOSITION 8.4   For any commutative ring $A$, $(\Lambda(A),\lambda_U: \Lambda_{(T)}(A)\to \Lambda_{(U)}\Lambda_{(T)}(A)$ is a $\lambda$-ring.

PROOF.. To avoid some confusion, we use lower case letters for indeterminate variables. Moreover, to distinquish all the lambda's around here, we denote by $\overset{\circ}{\lambda}$ the lambda operation on $\Lambda (A)$:

$$\overset{\circ}{\lambda}_{(t,u)}: \Lambda_{(t)} A \ni [a]_t \mapsto [[a]_t]_u\in \Lambda_{(u)}\Lambda_{(t)} A$$

where $[a]_t$ is the Teichmüller lift of $a\in A$ in $\Lambda_{(t)}A$. We need to verify the commutativity of the following diagram:

$$\xymatrix { \Lambda_{(u)}(A) \ar[r]^{\overset{\circ}{\lambda}_{(t... ...verset{\circ}{\lambda}_{(t,v)}} &\Lambda_{(t)}(\Lambda_{(v)}\Lambda_{(u)}A) }$$

which can be verified by a diagram chasing for generators $[a]_u (a\in A)$:

$$\xymatrix { [a]_{u} \ar[r]^{\overset{\circ}{\lambda}_{(t,u)}} \ar... ...rc}{\lambda}_{(v,u)})}\\ [[a]_u]_v \ar[r]_{\lambda_{(t,v)}} &[[[a]_u]_v]_t }$$

$\qedsymbol$

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