TITLE: The category of finite locally-free commutative group schemes
QUESTION [11 upvotes]: I'm trying to understand the properties of the category $\mathcal{FL}/S$ of finite locally-free commutative group schemes over an arbitrary base-scheme $S$. I know it is not in general an abelian category: Over the integers the morphism $\mathbb{Z}/2\mathbb{Z}\to\mu_2$ given by the ring homomorphism $\mathbb{Z}/(T^2-1)\to\mathbb{Z}\times\mathbb{Z}$ with $T\mapsto (1,-1)$ is a monomorphism and an epimorphism in $\mathcal{FL}/\mathbb{Z}$ but it is clearly not an isomorphism.
What I don't know: Does every morphism in $\mathcal{FL}/S$ have a kernel and/or a cokernel?
One has a notion of short exact sequences in $\mathcal{FL}/S$: If $f:G'\to G$ and $g:G\to G''$ are morphisms in $\mathcal{FL}/S$ we say that the sequence $0\to G'\to G\to G''\to 0$ is exact if its image under the embedding of $\mathcal{FL}/S$ into the category of abelian fppf-sheaves is exact. How can we detect in $\mathcal{FL}/S$ whether a sequence is exact? Specifically: If $f$ is a kernel of $g$ and if $g$ is a cokernel of $f$ (both in $\mathcal{FL}/S$), is the corresponding sequence exact?
I'd appreciate (references for) answers to any of these questions.

REPLY [2 votes]: If the base $S$ is Spec of a Dedekind domain, say $A$, and one restricts to finite flat groups schemes with etale generic fibres (which is no restriction if the fraction field of $A$ has char. 0), then one can form kernels and cokernels in the category of finite flat group schemes by taking scheme-theoretic closures of the corresponding notions on the generic fibre, and this can be useful.  (Of course, one doesnt' get an abelian category: the map
${\mathbb Z}/2 {\mathbb Z} \rightarrow \mu_2$ has trivial kernel and cokernel, but is not an isomorhpism.)  
An illustration of how this method can be used is given in e.g. in my joint paper with Calegari On the
ramification of Hecke algebras at Einstein primes.
(The point of the restriction to a Dedekind base is that in this case, torsion free, which is what you get with scheme-theoretic closure from the generic fibre, coincides with flat.)