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I was working through Bhargav's notes on $\delta$-rings and prismatic cohomology, specifically lecture 2, page 2, point 5 where he claims that the ring $\mathbb Z[x]/(px,x^p)$ has a unique $\delta$-structure given by $\delta(x) = 0$.

While it's easy to verify that this is a $\delta$ structure, I don't see why it has to be unique. In fact, isn't $\delta(x) = x$ also a $\delta$-structure? (At the level of the Frobenius, they are the same).

In general, what is the easiest way to check or define a $\delta$-structure?

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    $\begingroup$ The notes don't say $\delta$ is unique, it says there is a unique $\delta$-structure such that $\delta(x) = 0$. $\endgroup$
    – Nicolas Hemelsoet
    Commented Jul 17, 2020 at 7:32
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    $\begingroup$ @NicolasHemelsoet You are certainly right, but it's true that the sentence is slightly ambiguous. I'm not a specialist, but if I did not make a silly mistake the $\delta$-structures on that ring are exactly those given by $\delta(x) =$ a polynomial with zero constant term. $\endgroup$
    – Aurel
    Commented Jul 17, 2020 at 7:55
  • $\begingroup$ @aurel that's what I got too. hmm. That's slightly confusing wording! $\endgroup$
    – Asvin
    Commented Jul 17, 2020 at 13:46

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