I have been reading Hyodo-Kato's paper on log-crystalline cohomology, and there is one statement there that has been troubling me.
To explain this, suppose we have a perfect field $k$ of characteristic $p>0$, with ring of Witt vectors $W$. Let $X/W$ be a semistable scheme, that is a regular, proper scheme over $W$, étale locally smooth over some $W[x_1,\ldots,x_d]/(x_1\ldots x_d-p)$. Let $X_k$ denote the special fibre.
Then we may put a log structure on $X$, which is simply that given by the special fibre, in other words the sheaf of monoids $M$ is the sheaf of functions on $X$ invertible outside of $X_k$, and the map $M\rightarrow\mathcal{O}_X$ is the inclusion. Let $(X,M)$ denote the associated log scheme, and let $(X_k^\times,\overline{M})$ denote the pullback of this log structure to the special fibre $X_k$.
Then the following fact is implicit in their paper: the log scheme $(X_k^\times,\overline{M})$ depends on the scheme $X\otimes_W W/(p^2)$.
My question is essentially: why is this the case? Naively, I would have thought that this should be described étale locally by the chart $\mathbb{N}^d\mapsto \mathcal{O}_{X_s}$ given by $(0,\ldots,0,1,0,\ldots,0)\mapsto x_i$ (with $1$ in the $i$th place), which clearly only depends on the special fibre $X_s^\times$. So why is this not the correct description, and is there a simple description in terms of charts?
