Let $Y \to X$ be a closed immersion of smooth schemes over, say, the ${\rm Spec}(\mathbb{Z}_p)$. The completion map $$X_{/Y}\to X$$ is an ind-closed immersion (sometimes called pseudo-closed immersion) and in particular étale in the category of noetherian formal schemes. (It is unramified being an ind-closed immersion and flat, thus étale as e.g. to find in http://arxiv.org/abs/math/0605115 ).
Let us denote by $I:= I_Y$ the ideal sheaf defining $Y$ and $\mathcal{D}:= \mathcal{D}_{\mathcal{O}_Y, \gamma}(\mathcal{O}_X)$ the sheaf version divided power envelope of $Y$ in $X$ which comes together with an ideal sheaf $J$ equipped with divided power structures that are compatible with the divided power structures $\gamma$ of $(p)\subset \mathbb{Z}_p$.
Let $D_n:= \underline{{\rm Spec}}(\mathcal{D}/J^{[n+1]})$ be the $n$-th order nilpotent divided power thickening and $D:= {\rm colim}_n (D_n)$ the divided power completion of $Y$ in $X$ (though neither the $D_n$ nor $D$ are subschemes of $X$).
I've been searching literature a lot trying to find good properties of the map $$ D\to X_{/Y}$$ or perhaps of the map $D\to X$. But haven't been able to find any. I hope that people working with these sort of things have a better understanding of this map and only lack the desire to write things up.
Let me also say that I'm happy with the case where $Y$ is a Cartier divisor.
Here is the little I know: Since I assumed that we work over the $p$-adics, the relevant divided power structures to be added to $I$ ($=(x)$, locally) are locally of the form $x^{[n]} "= \frac{x^n}{n!}"$ where $n=pk$ for $k\in \mathbb{N}$. This means that the maps $$ \mathcal{O}_X/I^n \to \mathcal{D}/J^{[n]}$$ are injective with cokernel (locally) killed by a power of $p$ (in fact isomorphisms for $n<p$). I have very little reason to expect any good properties, but also very little intuition so there might be something known to experts.
Before there is the question of what I mean by 'good' property, let me say e.g. flat would be nice, but I'd be happy to know anything.
Thanks.
