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Suppose we know the stable Picard groups (=Picard group of the stable module category) of two cocommunicative Hopf $k$-algebras $G$ and $H$. Is it possible to deduce the stable Picard group of $G\otimes_kH$ from them?

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    $\begingroup$ I don't think so. For example for $A=B=\mathbb{F}_2[C_2]$, the respective Picard groups are trivial, but for $A\otimes B$ the Picard group is given by $\mathbb{Z}$. $\endgroup$
    – Achim Krause
    Commented May 9, 2023 at 8:26
  • $\begingroup$ @AchimKrause Great Couterexample! Thanks! $\endgroup$
    – Syu Gau
    Commented May 10, 2023 at 0:02

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