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Dima Sustretov
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is a tensor square of a group scheme with $\mathbb{G}_a^n$ fibres reduced?

Let $G$ be a group scheme over $S$ where $S$ is a reduced scheme of finite type over a field $k$ of characteristic 0, and let every fibre $G_s$ over a closed point of $S$ be isomorphic to $\mathbb{G}_a^n$ for some $n$ that varies with $s$.

Is it true then that $G$ is reduced?

Is it true that $G \otimes_S G$ reduced?

(This question is a continuation of this one; the motivation comes from this question)

Dima Sustretov
  • 4.1k
  • 20
  • 35