Let $k$ be an algebraically closed field of characteristic zero. We consider $f:\mathfrak{X}\to \mathbb{A}^1$ an smooth projective morphism of pure relative dimension $n$, where $\mathfrak{X}$ is an smooth algebraic variety.


Suppose that $\mathcal{F}$ is a torsion-free coherent sheaf on $\mathfrak{X}$ which is flat over $\mathbb{A}^1$. Then, $f\_\*\mathcal{F}\cong \text{H}^0(\mathfrak{X},\mathcal{F})^{\sim}$ is a torsion-free coherent sheaf on the smooth curve $\mathbb{A}^1$, which is therefore locally free. Even better, algebraic vector bundles over the affine line are trivial.


**My question is the following**: Suppose that $\text{H}^0(\mathfrak{X}\_t,\mathcal{F}\_t)$ are all isomorphic and that the map $f\_\*\mathcal{F}\otimes \kappa(t) \to \text{H}^0(\mathfrak{X}\_t,\mathcal{F}\_t)$ is an isomorphism, for every closed point $t\neq 0$. 


What can we say about $h^0(\mathfrak{X}\_0,\mathcal{F}\_0)$ in this particular situation or, equivalently (by Mumford "Abelian varieties" II.5 Corollary 2), can we expect $f\_\*\mathcal{F}\otimes \kappa(0)\to \text{H}^0(\mathfrak{X}\_0,\mathcal{F}\_0)$ to be an isomorphism ? Maybe an injection ("by semi-continuity") ? Under additional hypotheses ?


Thank you in advance for any comment.



Flat family of torsion-free coherent sheaves over $\mathbb{A}^1$
Let $i:Spec(k) \to A^1$ be the embedding if the point zero, $j:X\_0 \to X$ the embedding of its fiber, and $p:X\_0 \to Spec(k)$ the restriction of $f$. Then we have a Cartesian diagram. A flat base change implies that
$$
Li^\*\circ Rf\_\* \cong Rp\_\* \circ Lj^\*,
$$
an equality of derived functors. Since $i$ and $j$ are divisorrial embeddings, one has only $L\_0$ and $L\_1$ nontrivial. Therefore, the spectral sequence for the left hand side gives in degree zero an exact sequence
$$
0 \to i^\*f\_\*(F) \to H^0(Li^\*(Rf\_\*(F)) \to L\_1i^\*(R^1f\_\*(F)) \to 0,
$$
and the spectral sequence for the right hand side gives
$$
0 \to R^1p\_\*(L\_1j^\*(F)) \to H^0(Li^\*(Rp\_\*(F)) \to p\_\*(j^\*(F)) \to R^2p\_\*(L\_1j^\*(F)).
$$


If $F$ is torsion free then $L\_1j^\*(F) = 0$, hence $H^0$ equals $p\_\*(F\_0)$. On the other hand, the first term of the first sequence is $f\_\*(F) \otimes k$. So, the required injectivity follows.


