Let $S$ be affine. A bit of background: Let us think of $S$-group schemes as abelian sheaves over a given site (etale, Zariski, fppf, etc). When we take a cokernel of a morphism $\phi$ this category: $$A \xrightarrow{\phi} B \xrightarrow{\beta} B/\phi(A)$$ it will often be a presheaf, and we must sheafify it wrt the given site in order to obtain a sheaf.

Is there an example of a cokernel where (a) we must work over the fppf site for the sheafification of the cokernel to still give an exact sequence (and the sequence is not exact as sheaves when we sheafify over the etale or zariski site) and (b) doesn't have $\mu_d$ as the kernel.

The only examples satisfying (a) that I, and those around me, can come up with are the following:

  1. $\mu_{d} \to \mu_{n} \xrightarrow{z \mapsto z^d} \mu_{n/d}$; if $d \notin \mathcal{O}_S^*$
  2. $\mu_{d} \to \mathbb{G}_{m} \xrightarrow{z \mapsto z^d} \mathbb{G}_{m}$; if $d \notin \mathcal{O}_S^*$
  3. $\mu_{d}(k) \to SL_d(k) \to PGL_d(k)$; if char $k \mid d$

But these feel essentially the same. Are there any different examples?

  • 2
    $\begingroup$ Can you clarify (a)? $\endgroup$ – Laurent Moret-Bailly Feb 5 at 18:49
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    $\begingroup$ For $S$ a scheme over a characteristic $p$ field $k$, there is a short exact sequence of fppf sheaves as follows (I think this is what you are asking): $$0 \to \alpha_{p,S} \to \mathbb{G}_{a,S} \xrightarrow{(\bullet)^p} \mathbb{G}_{a,S} \to 0.$$ $\endgroup$ – Jason Starr Feb 5 at 18:51
  • $\begingroup$ @LaurentMoret-Bailly edited, hopefully this is clearer $\endgroup$ – Catherine Ray Feb 5 at 19:19

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