Artin-Winters proof of semi-stable reduction theorem: details

Question

I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail— Let $\mathcal{C}\to S$ be a regular fibered surface over $S = $ Spec $R$, with $R$ a complete discrete valuation ring with maximal ideal $\mathfrak{m}$ and $R/\mathfrak{m} = k$ algebraically closed. Let $X = \mathcal{C}_s$ denote its closed fiber, which we assume connected, and define $$X_n = \mathcal{C}\times_S \text{Spec }R/\mathfrak{m}^{n+1}$$to be the $n^{th}$ infinitesimal neighborhood of $X$ in $\mathcal{C}$. Now from the closed immersions ($i_n:X_{n-1}\hookrightarrow X_n$), we have, for each $n$, an exact sequence: $$0\to K_n\to \mathcal{O}_{X_n}\to {i_n}_*\left(\mathcal{O}_{X_{n-1}}\right)\to 0$$with $K_n^2 = 0$. This induces a map $$1\to 1+K_n\to \mathcal{O}_{X_n}^*\to \mathcal{O}_{X_{n-1}}^*\to 1$$(omitting $i_*$ from the notation since $X_{n}\simeq X_{n-1}$ as topological spaces). Let $J = 1+K_n$. In the proof of proposition of 2.1, they write: "$H^1(X,J)$ is a $k$-vector space, and thus uniquely divisible by $s$" (for $s$ prime to $p=$ char $k$). However, I can't seem to wrap my head around this statement. For one thing, they've defined $J$ as a sheaf of groups on $X_n$, which can only be made into a coherent sheaf via the isomorphism $J = 1+K_n\cong K_n$ for $K_n$ the ideal sheaf cutting out $X_{n-1}$ in $X_{n}$. This seems to give $J$ the structure of a $\mathcal{O}_{X_n}$-module, and thus the structure of a $R/\mathfrak{m}^n$-module. I don't see, however, how we could get a $k$-vector space structure on $J$. Of course, $X\simeq X_n$ topologically, so it makes sense to talk about $H^1(X,J)$ in any case, just as a sheaf of groups. But this won't give it the structure of an $\mathcal{O}_X$-module. Any pointers/corrections are greatly appreciated! Thanks!