Example 14.34.7. In Example 14.34.5 we have $X_ n(B) = A[A[\ldots [B]\ldots ]]$ with $n + 1$ brackets. We describe the maps constructed above using a typical element

\[ \xi = \sum \nolimits _ i a_ i [x_{i, 1}] \ldots [x_{i, m_ i}] \in A[A[B]] = X_1(B) \]

where for each $i, j$ we can write

\[ x_{i, j} = \sum a_{i, j, k} [b_{i, j, k, 1}] \ldots [b_{i, j, k, n_{i, j, k}}] \in A[B] \]

Obviously this is horrendous! To ease the notation, to see what the $A$-algebra maps $d_0, d_1 : A[A[B]] \to A[B]$ are doing it suffices to see what happens to the variables $[x]$ where

\[ x = \sum a_ k [b_{k, 1}] \ldots [b_{k, n_ k}] \in A[B] \]

is a general element. For these we get

\[ d_0([x]) = x = \sum a_ k [b_{k, 1}] \ldots [b_{k, n_ k}] \quad \text{and}\quad d_1([x]) = \left[\sum a_ k b_{k, 1} \ldots b_{k, n_ k}\right] \]

The maps $s_0, s_1 : A[A[B]] \to A[A[A[B]]]$ are given by

\[ s_0([x]) = \left[\left[\sum a_ k [b_{k, 1}] \ldots [b_{k, n_ k}]\right]\right] \quad \text{and}\quad s_1([x]) = \left[\sum a_ k [[b_{k, 1}]] \ldots [[b_{k, n_ k}]]\right] \]

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