  Let $X$ be a locally Noetherian scheme with a closed subscheme $Z$. Let
$\mathcal{X}$ be the completion of $X$ at $Z$, considered as a formal scheme.
We show that a coherent sheaf on $X$ is equivalently given by a coherent sheaf
on $\mathcal{X}$, a coherent sheaf on the complement of $Z$, and an isomorphism
of pullbacks of these sheaves to a certain adic space $W$. By defining $W$ as
an adic space instead of as a Berkovich space we are able to generalize the
descent result of Ben-Bassat and Temkin from finite type $k$-schemes to locally
Noetherian schemes.
