Let $S$ be a noetherian scheme (possibly nicer assumptions) and $G$ a smooth group scheme of finite type over $S$ with geometrically integral fibers with structure morphism $f: G\rightarrow S$ and $f\_\*\mathcal O\_G =\mathcal O\_S$ universally.


For $n\geq 0$ let $G\_n$ denote the $n$-th infinitesimal neighborhood of the zero section of $S$ in $G$ and by denote $f\_n: G\_n \rightarrow S$ its structure morphism and denote by $g\_n: S \rightarrow G\_n$ the obvious closed immersion.


Let $\mathcal V$ be a coherent sheaf on $G\_n$ with the following property:


$(\*)$ The direct image of $\mathcal V$ along $f\_n$ is locally free of finite type on $S$.


The question is if one can conclude that then the following holds:


$(\*\*)$ The pullback $g\_n^\* \mathcal V$ of $\mathcal V$ along $g\_n$ is locally free of finite type on $S$.


One may feel free to add additional assumptions on $G$ or $S$ or $\mathcal V$ in a discussion about when $(\*\*)$ would hold, but the general case is of course very interesting.


Note: One could of course also ask if $(\*)$ already implies that $\mathcal V$ itself is locally free of finite rank, but I guess this would be false in general.



Locally free sheaves on group schemes and thickenings
No. Let $G=\mathbb A^2$ with the projection map to $S=\mathbb A\_1$. It is a relative group scheme under addition of the $y$ coordinate. Take the module on $G\_2=\textrm{Spec} k[x,y]/y^2$ with two generators, $a$ and $b$, under the relations $ya-xb=0$, $yb=0$. This gives a coherent sheaf. The direct image is the free $k[x]$-module on two generators, defining a locally free sheaf. The pullback is $k[x]\oplus k[x]/x$, which does not define a locally free sheaf.


Edit: Let $E$ be an elliptic curve and $G=E\times S$. Then $G\_n$, for all $n$, is the same as in this example, and $f\_\* \mathcal O\_G=\mathcal O\_S$.


