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In Homology operations for $H_\infty$ and $H_n$ spectra (pdf), Steinberger makes the computation of the Dyer-Lashof operations in $H\mathbb F_p$, and at some point uses the following "basic fact"

Lemma 6.1. The following equalities hold in $\mathcal A_*$. For $p\geq2$ and $i\geq0$, $$P^r_*\chi\xi_i=\left\{\begin{array}{ll}-\chi\xi_{i-k}^{p^k}&\text{if }r=\frac{p^k-1}{p-1}\\0&\text{otherwise}\end{array}\right.$$ [...]

I have tried my best to show it by hands but I couldn't manage to make the computation myself. Also I've been looking for a reference without success...

Would someone know a reference where this is made explicit ?

Pierre Elis
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    A quick comment: $\xi_1$ must be a typo — it should be $\xi_i$. (This typo is present in the original paper.) – John Palmieri Feb 05 '21 at 19:41
  • Thank you, I copy pasted the original article without paying attention, but fortunately I had noticed the error when I tried the computation myself – Pierre Elis Feb 05 '21 at 20:21

1 Answers1

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Let me give this a shot. I'm bound to mess up signs, so assume $p = 2$. Write $\langle \cdot, \cdot \rangle$ for the pairing between $\mathcal{A}_*$ and $\mathcal{A}^*$.

Lemma. $\newcommand{\Sq}{\mathrm{Sq}}\langle \chi \xi_k, \Sq^r \rangle = \delta_{r = 2^k-1}$.

Proof. We have \begin{align} \langle \chi \xi_k, \Sq^r \rangle &= \langle \xi_1^{2^k-1} \bmod{(\xi_2, \xi_3, \ldots)}, \Sq^r \rangle \\ &= \langle \xi_1^{2^k-1}, \Sq^r \rangle \\ &= \delta_{r = 2^k-1}. \end{align}

Proposition. $\Sq^r_* \chi \xi_i = \begin{cases} \chi \xi_{i-k}^{2^k}, & \text{if } r = 2^k - 1 \\ 0, & \text{otherwise}. \end{cases}$

Proof. For any $x \in \mathcal{A}^*$, we have \begin{align} \langle \Sq^r_* \chi \xi_i, x \rangle &= \langle \chi \xi_i, \Sq^r x \rangle \\ &= \langle \chi \xi_i, \mu(\Sq^r \otimes x) \rangle \\ &= \langle \xi_i, \chi \mu(\Sq^r \otimes x) \rangle \\ &= \langle \xi_i, \mu(\chi x \otimes \chi \Sq^r) \rangle \\ &= \langle \Delta \xi_i, \chi x \otimes \chi \Sq^r \rangle \\ &= \langle \sum_k \xi_{i-k}^{2^k} \otimes \xi_k, \chi x \otimes \chi \Sq^r \rangle \\ &= \sum_k \langle \xi_{i-k}^{2^k}, \chi x \rangle \cdot \langle \xi_k, \chi \Sq^r \rangle \\ &= \sum_k \langle \chi \xi_{i-k}^{2^k}, x \rangle \cdot \langle \chi \xi_k, \Sq^r \rangle \\ &= \sum_k \langle \chi \xi_{i-k}^{2^k}, x \rangle \cdot \delta_{r=2^k-1} \\ &= \begin{cases} \langle \chi \xi_{i-k}^{2^k}, x \rangle, & \text{if } r = 2^k - 1 \\ 0, & \text{otherwise}. \end{cases} \end{align}

JHF
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