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I am reading through the calculations in Hu-Kriz "Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence" and I've got a small problem in understanding the computations of coefficients of the "first version" of $C_2$-equivariant Steenrod algebra.

Notation:

  • $H$ is a $C_2$-spectrum constructed as follows: we take the naive Eilenberg-MacLane spectrum associated to the constant Mackey functor $\underline{\mathbb{Z}/2}$ and extend it to a complete universe.
  • $A_\star=(H\wedge H)_\star$ is a dual Steenrod algebra associated to $H$ ($RO(C_2)$-graded) and $A_\ast$ is its part graded over the integers.
  • $H^f_\star=(H\wedge EQ_+)_\star$ - the coefficients of homotopy orbits.

Now in Proposition 6.6 the authors compute $A_\star$ by smashing $H\wedge H$ with the isotropy separation sequence. This yields the exact sequence $$ \ldots\to (H\wedge H\wedge EC_{2+})_\star\to A_\star\to (H\wedge H\wedge\widetilde{EC_2})\to\ldots. $$ The authors claim that $(H\wedge H\wedge EC_{2+})_\star\cong A_\ast\otimes H^f_\star$. I cannot see why this is true; I suppose it might come from applying homotopy orbits spectral sequence, but it seems quite complicated to me.

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    $\begingroup$ It is just an application of the Kunneth theorem for mod 2 homology: $(H \wedge X)_*$ is the mod 2 homology of $X$. $\endgroup$
    – Nicholas Kuhn
    Commented Apr 20, 2021 at 13:14
  • $\begingroup$ Ah, ok! As always, the answer is simpler than I thought. Thank you! $\endgroup$
    – Igor Sikora
    Commented Apr 20, 2021 at 13:33

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