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

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

## Comments (0)

There are also:

• 2 comment(s) on Section 14.34: Standard resolutions

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09CB. Beware of the difference between the letter 'O' and the digit '0'.