In Kechris's book "Classical Descriputive set theory" Chapter 23 (Exercise 23.4), it claim that there is a $\Sigma^0_\xi$-complete ideal on $\omega$ for each $\xi\geq 3$.

There is a candidate construction as follows: give a $\Sigma^0_\xi$-complete $A$ subset of $2^\omega$, define an ideal $\mathcal I_A$ on ${}^{<\omega}2$ generated by $\{\{x|n:n\in\omega\}:x\in A\}$. I don't underatand why the Borel hierarchy of $\mathcal I_A$ isn't bigger than $\xi$?

Is there a different construction?