The Stacks project

Remark 38.29.7. The result in this section can be generalized. It is probably correct if we only assume $X \to \mathop{\mathrm{Spec}}(A)$ to be separated, of finite presentation, and $K_ n$ pseudo-coherent relative to $A_ n$ supported on a closed subset of $X_ n$ proper over $A_ n$. The outcome will be a $K$ which is pseudo-coherent relative to $A$ supported on a closed subset proper over $A$. If we ever need this, we will formulate a precise statement and prove it here.