Return to Revisions

The automorphism group functor is not representable in general. Consider for instance the case of an algebraically closed field $k$ and $\mathbf A^2=\mathrm{Spec}\ k[x,y]$. Assume the automorphism group functor for this scheme is representable by an algebraic space $Aut$. For each $n$ we have an automorphism $(x,y)\mapsto(x+\sum_{1\leq i\leq n}t^iy^i,y)$ over $k[t]/(t^{n+1})$ giving a set of compatible morphisms $\mathrm{Spec}k[t]/(t^{n+1})\to Aut$. That system comes from a morphism $\mathrm{Spec}k[[t]]\to Aut$ (see Emerton's response to this question). However, the corresponding automorphism would have to be $(x,y)\mapsto(x+\sum_{1\leq i}t^iy^i,y)$ which doesn't make sense.