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Let $S$ be a scheme and $(G_i,\varphi_{ij})_{i\in I}$ an inductive system of $S$-group schemes. By a sheaf on $S$ I mean an fppf sheaf. Each $G_i$ represents a sheaf on $S$, and the presheaf colimit of the $G_i$ is the functor $T\mapsto\varinjlim G_i(T)$ on locally finitely presented (or locally finite type or whatever you prefer) $S$-schemes $T$. In general, one has to sheafify a colimit. I'm wondering, however, in the case of an inductive system of representable sheaves, if one still needs to sheafify?

While the question above is very general, I'd be happy to have an answer in the situation I am most interested in: $S$ is the spectrum of a Dedekind domain and $G_i=A[p^i]$ for a Neron model $A$ over $S$ (meaning a smooth commutative group scheme whose generic fiber is an abelian variety and such that $A$ has the Neron mapping property for smooth $R$-schemes). I used to believe this was automatic and followed by just "taking limits" since each $G_i$ represents a sheaf, but colimits don't generally commute with infinite direct products, so if I can't always reduce to finite covers, I feel like this argument doesn't work. Perhaps a solution would be to change the site under consideration to one whose objects are all finite type over $R$ (i.e. considering the category of quasi-finite $R$-groups with fppf coverings). I think in such a case the presheaf inductive limit is always already a sheaf.


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