Let $\varphi: G \to G^\prime$ be a homomorphism of finite flat group schemes over a locally noetherian base scheme $S$, and let $h_\varphi: h_G \to h_{G^\prime}$ be the induced homomorphism of sheaves of abelian groups on $(\mathfrak{Sch}/S)_{\mathrm{fppf}}$. I wonder what are the equivalent conditions for $h_{\varphi}$ to be injective (surjective).

More specifically, are the following equivalent:

  1. $\varphi$ is a closed immersion;
  2. $h_\varphi$ is injective.

And are the following equivalent:

  1. $\varphi$ is faithfully flat;
  2. $h_\varphi$ is surjective.

Furthermore, is the category of finite flat group schemes over $S$ an abelian category?

  • 3
    $\begingroup$ The first is true by (stacks.math.columbia.edu/tag/04XV) which tells you that closed immersions are the same thing as proper monomorphisms. For your last question, see mathoverflow.net/questions/7688/… $\endgroup$ May 11, 2020 at 13:48
  • 2
    $\begingroup$ Finite flat group schemes cannot be an abelian category because finite groups is not an abelian category. Finite flat commutative group schemes on the other hand has a chance of being abelian, and in the post cited by @user45878 it is explained that this is true when $S$ is the spectrum of a field and false for some other bases (e.g. a mixed characteristic DVR of large ramification index). $\endgroup$ May 11, 2020 at 16:46


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