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Let $S$ be a scheme, $(\text{Sch}/S)_{\rm Ét}$ a big étale site, and $A$ a representable (either in schemes or algebraic spaces over $S$) abelian sheaf on $(\text{Sch}/S)_{\rm Ét}$.

Suppose there is a short exact sequence of abelian sheaves on $(\text{Sch}/S)_{Ét}$:

$$0\to A_1\to A\to A_2\to 0.$$

  • Are $A_1$ and $A_2$ representable?

  • are the maps $A_1\to A$ and $A\to A_1$ representable, the former by a closed immersion?

If both maps are representable, then $A_1$ is representable if and only if $A_2$ is.

Conversely:

if $A_1$ and $A_2$ are representable, is $A$ too?

In other words, is the full subcategory of the category of abelian sheaves on $(\text{Sch}/S)_{Ét}$ given by representable abelian sheaves and morphisms/representable morphisms of abelian sheaves, a weak Serre subcategory?

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    $\begingroup$ I don't see your "clearly one is representable if and only if..." If by representable you mean representable by a group object in the category of scheme/S then any discrete group is representable, and I can look at thing very far from geometry like the quotient of $GL_n$ by the discrete group of points $GL_n(S)$. $\endgroup$ – Simon Henry Jan 13 '18 at 10:45

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