Lemma 11.15.10EEC7E1B2FB0

Proof. At a point of $X\times \{ t_0\} $, lift a local basis of the fiber to $\mathcal E$. By Nakayama's lemma, the resulting map from a free module of finite rank is surjective near that point. Tensor its kernel sequence with the residue field of $t_0$. Flatness over $T$ makes the first Tor term zero, so the kernel restricts to the kernel of the chosen isomorphism on the fiber, namely zero, and Nakayama's lemma shows that the kernel vanishes near that point. Thus $\mathcal E$ is locally free near every point of the central fiber. The locus where $\mathcal E$ is not locally free is closed. Since the projection $X\times T\to T$ is proper, the image of that locus is closed and misses $t_0$, and we remove this image. $\square$