Let S be a scheme. Suppose we have sheaves in abelian groups $A,B,C$ over the big étale site of $S$. Suppose that $A$ and $C$ are representable by algebraic spaces in groups locally of finite type over $S$. Suppose we have a complex $A\rightarrow B\rightarrow C$ of sheaves in groups such that when we restrict this complex on the small étale site of $S$ it becomes an exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ of sheaves. Can we conclude that also $B$ is representable by an algebraic space? I know that this would be true if the exact sequence lifts to an exact sequence also on the big étale site, but I cannot imagine that this is true in general. We can also assume that $A$ and $C$ are smooth over $S$ if this may help.
1
-
3$\begingroup$ Take $A=C=0$ and $B\ne 0$ whose values on etale $S$-schemes vanish. For example, $S= {\rm{Spec}}(k)$ for a field $k$ and $B = \underline{\rm{Hom}}(\mathbf{G}_a, \mathbf{G}_m)$ the functor of group scheme homomorphisms from $\mathbf{G}_a$ to $\mathbf{G}_m$ (nontrivial on $k$-algebras containing $t \ne 0$ satisfying $t^2=0$ via $x \mapsto 1+tx$). The functor $B$ satisfies Grothendieck's functorial criterion to be locally of finite presentation (which applies to algebraic spaces) yet its tangent space at 0 is infinite-dimensional when ${\rm{char}}(k)>0$, so it is then not an algebraic space. $\endgroup$– nfdc23May 31, 2016 at 13:26
Add a comment
|